Daily Papers

Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were developed over the centuries by mathematicians, who emphasized the attainability of arbitrary precision. Computers, however, operate on few limited precision types, such as the popular float32. In this study, we show that when aiming for limited precision, existing approximation methods can be outperformed by programs automatically discovered from scratch by a simple evolutionary algorithm. In particular, over real numbers, our method can approximate the exponential function reaching orders of magnitude more precision for a given number of operations when compared to previous approaches. More practically, over float32 numbers and constrained to less than 1 ULP of error, the same method attains a speedup over baselines by generating code that triggers better XLA/LLVM compilation paths. In other words, in both cases, evolution searched a vast space of possible programs, without knowledge of mathematics, to discover previously unknown optimized approximations to high precision, for the first time. We also give evidence that these results extend beyond the exponential. The ubiquity of transcendental functions suggests that our method has the potential to reduce the cost of scientific computing applications.

In this paper, we present techniques used to implement our portable vectorized library of C standard mathematical functions written entirely in C language. In order to make the library portable while maintaining good performance, intrinsic functions of vector extensions are abstracted by inline functions or preprocessor macros. We implemented the functions so that they can use sub-features of vector extensions such as fused multiply-add, mask registers and extraction of mantissa. In order to make computation with SIMD instructions efficient, the library only uses a small number of conditional branches, and all the computation paths are vectorized. We devised a variation of the Payne-Hanek argument reduction for trigonometric functions and a floating point remainder, both of which are suitable for vector computation. We compare the performance of our library to Intel SVML.

A single two-input gate suffices for all of Boolean logic in digital hardware. No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations. Here I show that a single binary operator, eml(x,y)=exp(x)-ln(y), together with the constant 1, generates the standard repertoire of a scientific calculator. This includes constants such as e, pi, and i; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions. For example, exp(x)=eml(x,1), ln(x)=eml(1,eml(eml(1,x),1)), and likewise for all other operations. That such an operator exists was not anticipated; I found it by systematic exhaustive search and established constructively that it suffices for the concrete scientific-calculator basis. In EML (Exp-Minus-Log) form, every such expression becomes a binary tree of identical nodes, yielding a grammar as simple as S -> 1 | eml(S,S). This uniform structure also enables gradient-based symbolic regression: using EML trees as trainable circuits with standard optimizers (Adam), I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4. The same architecture can fit arbitrary data, but when the generating law is elementary, it may recover the exact formula.

In this paper, we introduce BSRBF-KAN, a Kolmogorov Arnold Network (KAN) that combines B-splines and radial basis functions (RBFs) to fit input vectors during data training. We perform experiments with BSRBF-KAN, multi-layer perception (MLP), and other popular KANs, including EfficientKAN, FastKAN, FasterKAN, and GottliebKAN over the MNIST and Fashion-MNIST datasets. BSRBF-KAN shows stability in 5 training runs with a competitive average accuracy of 97.55% on MNIST and 89.33% on Fashion-MNIST and obtains convergence better than other networks. We expect BSRBF-KAN to open many combinations of mathematical functions to design KANs. Our repo is publicly available at: https://github.com/hoangthangta/BSRBF_KAN.

Ongoing efforts that span over decades show a rise of AI methods for accelerating scientific discovery, yet accelerating discovery in mathematics remains a persistent challenge for AI. Specifically, AI methods were not effective in creation of formulas for mathematical constants because each such formula must be correct for infinite digits of precision, with "near-true" formulas providing no insight toward the correct ones. Consequently, formula discovery lacks a clear distance metric needed to guide automated discovery in this realm. In this work, we propose a systematic methodology for categorization, characterization, and pattern identification of such formulas. The key to our methodology is introducing metrics based on the convergence dynamics of the formulas, rather than on the numerical value of the formula. These metrics enable the first automated clustering of mathematical formulas. We demonstrate this methodology on Polynomial Continued Fraction formulas, which are ubiquitous in their intrinsic connections to mathematical constants, and generalize many mathematical functions and structures. We test our methodology on a set of 1,768,900 such formulas, identifying many known formulas for mathematical constants, and discover previously unknown formulas for pi, ln(2), Gauss', and Lemniscate's constants. The uncovered patterns enable a direct generalization of individual formulas to infinite families, unveiling rich mathematical structures. This success paves the way towards a generative model that creates formulas fulfilling specified mathematical properties, accelerating the rate of discovery of useful formulas.

Unsigned distance fields (UDFs) provide a versatile framework for representing a diverse array of 3D shapes, encompassing both watertight and non-watertight geometries. Traditional UDF learning methods typically require extensive training on large 3D shape datasets, which is costly and necessitates re-training for new datasets. This paper presents a novel neural framework, LoSF-UDF, for reconstructing surfaces from 3D point clouds by leveraging local shape functions to learn UDFs. We observe that 3D shapes manifest simple patterns in localized regions, prompting us to develop a training dataset of point cloud patches characterized by mathematical functions that represent a continuum from smooth surfaces to sharp edges and corners. Our approach learns features within a specific radius around each query point and utilizes an attention mechanism to focus on the crucial features for UDF estimation. Despite being highly lightweight, with only 653 KB of trainable parameters and a modest-sized training dataset with 0.5 GB storage, our method enables efficient and robust surface reconstruction from point clouds without requiring for shape-specific training. Furthermore, our method exhibits enhanced resilience to noise and outliers in point clouds compared to existing methods. We conduct comprehensive experiments and comparisons across various datasets, including synthetic and real-scanned point clouds, to validate our method's efficacy. Notably, our lightweight framework offers rapid and reliable initialization for other unsupervised iterative approaches, improving both the efficiency and accuracy of their reconstructions. Our project and code are available at https://jbhu67.github.io/LoSF-UDF.github.io.

What mathematical functions do neural network components learn? Symbolic distillation addresses this question by expressing neural network components with interpretable, closed-form mathematical expressions that expose the functional structure learned during training. We develop symbolic distillation as a systematic, architecture-agnostic methodology, and release our approach as the open-source SymTorch package - a PySR-powered library built natively for the PyTorch ecosystem. Applying this methodology across diverse architectures, we find that SymTorch is successful in the automated discovery of physical laws. Specifically, our approach (1) recovers pairwise interaction forces from graph neural networks trained on empirical n-body observations, (2) distills the exact closed-form PDE/ODE solutions of multiple physical systems, including the value of constants, from physics-informed neural networks trained on sparse data, and (3) uncovers the chaotic dynamics of the Lorenz system from high-dimensional data, ultimately outperforming the base neural network on downstream prediction tasks. We further demonstrate the utility of our framework for model interpretability by providing an optimized implementation of SLIME - a symbolic extension to the LIME explainability method. SLIME consistently outperforms LIME across predictive metrics across eight popular classification and regression benchmarks, while still providing an interpretable local symbolic model. Lastly, we investigate replacing transformer MLP layers with symbolic surrogates: replacing 1-7 layers with symbolic approximations yields 2-19\% throughput improvements and up to 18.7\% VRAM reduction, with the resulting hybrid models lying on the Pareto front of throughput versus perplexity among open-source LLMs of comparable scale.

Normalization layers and activation functions are fundamental components in deep networks and typically co-locate with each other. Here we propose to design them using an automated approach. Instead of designing them separately, we unify them into a single tensor-to-tensor computation graph, and evolve its structure starting from basic mathematical functions. Examples of such mathematical functions are addition, multiplication and statistical moments. The use of low-level mathematical functions, in contrast to the use of high-level modules in mainstream NAS, leads to a highly sparse and large search space which can be challenging for search methods. To address the challenge, we develop efficient rejection protocols to quickly filter out candidate layers that do not work well. We also use multi-objective evolution to optimize each layer's performance across many architectures to prevent overfitting. Our method leads to the discovery of EvoNorms, a set of new normalization-activation layers with novel, and sometimes surprising structures that go beyond existing design patterns. For example, some EvoNorms do not assume that normalization and activation functions must be applied sequentially, nor need to center the feature maps, nor require explicit activation functions. Our experiments show that EvoNorms work well on image classification models including ResNets, MobileNets and EfficientNets but also transfer well to Mask R-CNN with FPN/SpineNet for instance segmentation and to BigGAN for image synthesis, outperforming BatchNorm and GroupNorm based layers in many cases.

The field of neuroscience and the development of artificial neural networks (ANNs) have mutually influenced each other, drawing from and contributing to many concepts initially developed in statistical mechanics. Notably, Hopfield networks and Boltzmann machines are versions of the Ising model, a model extensively studied in statistical mechanics for over a century. In the first part of this chapter, we provide an overview of the principles, models, and applications of ANNs, highlighting their connections to statistical mechanics and statistical learning theory. Artificial neural networks can be seen as high-dimensional mathematical functions, and understanding the geometric properties of their loss landscapes (i.e., the high-dimensional space on which one wishes to find extrema or saddles) can provide valuable insights into their optimization behavior, generalization abilities, and overall performance. Visualizing these functions can help us design better optimization methods and improve their generalization abilities. Thus, the second part of this chapter focuses on quantifying geometric properties and visualizing loss functions associated with deep ANNs.

Transformer-like models for vision tasks have recently proven effective for a wide range of downstream applications such as segmentation and detection. Previous works have shown that segmentation properties emerge in vision transformers (ViTs) trained using self-supervised methods such as DINO, but not in those trained on supervised classification tasks. In this study, we probe whether segmentation emerges in transformer-based models solely as a result of intricate self-supervised learning mechanisms, or if the same emergence can be achieved under much broader conditions through proper design of the model architecture. Through extensive experimental results, we demonstrate that when employing a white-box transformer-like architecture known as CRATE, whose design explicitly models and pursues low-dimensional structures in the data distribution, segmentation properties, at both the whole and parts levels, already emerge with a minimalistic supervised training recipe. Layer-wise finer-grained analysis reveals that the emergent properties strongly corroborate the designed mathematical functions of the white-box network. Our results suggest a path to design white-box foundation models that are simultaneously highly performant and mathematically fully interpretable. Code is at https://github.com/Ma-Lab-Berkeley/CRATE.

We study the potential of using large language models (LLMs) as an interactive optimizer for solving maximization problems in a text space using natural language and numerical feedback. Inspired by the classical optimization literature, we classify the natural language feedback into directional and non-directional, where the former is a generalization of the first-order feedback to the natural language space. We find that LLMs are especially capable of optimization when they are provided with {directional feedback}. Based on this insight, we design a new LLM-based optimizer that synthesizes directional feedback from the historical optimization trace to achieve reliable improvement over iterations. Empirically, we show our LLM-based optimizer is more stable and efficient in solving optimization problems, from maximizing mathematical functions to optimizing prompts for writing poems, compared with existing techniques.

In this work, we explore the mechanism of in-context learning (ICL) on out-of-distribution (OOD) tasks that were not encountered during training. To achieve this, we conduct synthetic experiments where the objective is to learn OOD mathematical functions through ICL using a GPT-2 model. We reveal that Transformers may struggle to learn OOD task functions through ICL. Specifically, ICL performance resembles implementing a function within the pretraining hypothesis space and optimizing it with gradient descent based on the in-context examples. Additionally, we investigate ICL's well-documented ability to learn unseen abstract labels in context. We demonstrate that such ability only manifests in the scenarios without distributional shifts and, therefore, may not serve as evidence of new-task-learning ability. Furthermore, we assess ICL's performance on OOD tasks when the model is pretrained on multiple tasks. Both empirical and theoretical analyses demonstrate the existence of the low-test-error preference of ICL, where it tends to implement the pretraining function that yields low test error in the testing context. We validate this through numerical experiments. This new theoretical result, combined with our empirical findings, elucidates the mechanism of ICL in addressing OOD tasks.

The analysis of non-stationary time-series data requires insight into its local and global patterns with physical interpretability. However, traditional smoothing algorithms, such as B-splines, Savitzky-Golay filtering, and Empirical Mode Decomposition (EMD), lack the ability to perform parametric optimization with guaranteed continuity. In this paper, we propose Functional Continuous Decomposition (FCD), a JAX-accelerated framework that performs parametric, continuous optimization on a wide range of mathematical functions. By using Levenberg-Marquardt optimization to achieve up to C^1 continuous fitting, FCD transforms raw time-series data into M modes that capture different temporal patterns from short-term to long-term trends. Applications of FCD include physics, medicine, financial analysis, and machine learning, where it is commonly used for the analysis of signal temporal patterns, optimized parameters, derivatives, and integrals of decomposition. Furthermore, FCD can be applied for physical analysis and feature extraction with an average SRMSE of 0.735 per segment and a speed of 0.47s on full decomposition of 1,000 points. Finally, we demonstrate that a Convolutional Neural Network (CNN) enhanced with FCD features, such as optimized function values, parameters, and derivatives, achieved 16.8% faster convergence and 2.5% higher accuracy over a standard CNN.

In this study, we introduce a new single point metaheuristic optimization approach suitable for both continuous and discrete domains. The proposed algorithm, entitled Blindfolded Spiderman Optimization, follows a piecewise linear search trajectory where each line segment considers a move to an improved solution point. The trajectory resembles spiderman jumping from one building to the highest neighbor building in a blindfolded manner. Blindfolded Spiderman Optimization builds on top of the Buggy Pinball Optimization algorithm. Blindfolded Spiderman Optimization is tested on 16 mathematical optimization functions and one discrete problem of Unbounded Knapsack. We perform a thorough evaluation of Blindfolded Spiderman Optimization against established and state-of-the-art metaheuristic optimization methods, including Whale Optimization, Grey Wolf Optimization, Particle Swarm Optimization, Simulated Annealing, Threshold Accepting, and Buggy Pinball Optimization considering various optimization domains and dimensions. We show that Blindfolded Spiderman Optimization achieves great performance on both continuous and discrete spaces, and superior performance compared to all single-point metaheuristic approaches considered.

This paper develops a new mathematical-statistical approach to analyze a class of Flajolet-Martin algorithms (FMa), and provides analytical confidence intervals for the number F0 of distinct elements in a stream, based on Chernoff bounds. The class of FMa has reached a significant popularity in bigdata stream learning, and the attention of the literature has mainly been based on algorithmic aspects, basically complexity optimality, while the statistical analysis of these class of algorithms has been often faced heuristically. The analysis provided here shows deep connections with mathematical special functions and with extreme value theory. The latter connection may help in explaining heuristic considerations, while the first opens many numerical issues, faced at the end of the present paper. Finally, the algorithms are tested on an anonymized real data stream and MonteCarlo simulations are provided to support our analytical choice in this context.

Human language is one of the most expressive tools for conveying intent, yet most artificial or biological systems lack mechanisms to interpret or respond meaningfully to it. Bridging this gap could enable more natural forms of control over complex, decentralized systems. In AI and artificial life, recent work explores how language can specify high-level goals, but most systems still depend on engineered rewards, task-specific supervision, or rigid command sets, limiting generalization to novel instructions. Similar constraints apply in synthetic biology and bioengineering, where the locus of control is often genomic rather than environmental perturbation. A key open question is whether artificial or biological collectives can be guided by free-form natural language alone, without task-specific tuning or carefully designed evaluation metrics. We provide one possible answer here by showing, for the first time, that simple agents' collective behavior can be guided by free-form language prompts: one AI model transforms an imperative prompt into an intervention that is applied to simulated cells; a second AI model scores how well the prompt describes the resulting cellular dynamics; and the former AI model is evolved to improve the scores generated by the latter. Unlike previous work, our method does not require engineered fitness functions or domain-specific prompt design. We show that the evolved system generalizes to unseen prompts without retraining. By treating natural language as a control layer, the system suggests a future in which spoken or written prompts could direct computational, robotic, or biological systems to desired behaviors. This work provides a concrete step toward this vision of AI-biology partnerships, in which language replaces mathematical objective functions, fixed rules, and domain-specific programming.

Numerical approximations of partial differential equations (PDEs) are routinely employed to formulate the solution of physics, engineering and mathematical problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, and more. While this has led to solving many complex phenomena, there are some limitations. Conventional approaches such as Finite Element Methods (FEMs) and Finite Differential Methods (FDMs) require considerable time and are computationally expensive. In contrast, data driven machine learning-based methods such as neural networks provide a faster, fairly accurate alternative, and have certain advantages such as discretization invariance and resolution invariance. This article aims to provide a comprehensive insight into how data-driven approaches can complement conventional techniques to solve engineering and physics problems, while also noting some of the major pitfalls of machine learning-based approaches. Furthermore, we highlight, a novel and fast machine learning-based approach (~1000x) to learning the solution operator of a PDE operator learning. We will note how these new computational approaches can bring immense advantages in tackling many problems in fundamental and applied physics.

Conventional Applicant Tracking Systems (ATS) tend to be inflexible keyword-matchers, and deny gifted candidates a role due to a few minor semantic mismatches. This article describes a new two-step process to design a more refined resume evaluation model based on a small language model (<600M parameters) that is finetuned using GRPO on a custom reward function. To begin with, Supervised Fine-Tuning (SFT) was used to build a solid baseline model. Second, this SFT model was also optimized with the help of Reinforcement Learning (RL) through GRPO under the guidance of a new, multi-component reward function that can holistically assess candidates beyond simple keyword matching. We indicate that the RL application presents a critical problem of reward hacking due to the initial experiments of aggressive penalties, which produces faulty, excessively negative model behaviors. We have overcome this challenge by refining the reward function repeatedly and training hyperparameters into a stable "gentle polishing process" of the reward function. Our resulting GRPO-polished model demonstrates significant real-world efficacy, achieving a final accuracy of 91% on unseen test data. The model shows a strong ability to correctly identify qualified candidates (recall of 0.85 for the 'SELECTED' class) while also showing exceptional precision (1.0), confirming its reliability. These results indicate that a properly executed, two-step fine-tuning procedure can indeed effectively refine a small language model to be able to conduct fine-tuned and human-like candidate scoring, overcoming the drawbacks of both traditional ATS and naive RL usage.

Recent advances in large language models (LLMs) highlight the importance of post training techniques for improving reasoning and mathematical ability. Group Relative Policy Optimization (GRPO) has shown promise in this domain by combining group relative advantage estimation, PPO style clipping, and KL regularization. However, its complexity raises the question of whether all components are necessary for fostering reasoning behaviors. We conduct a systematic analysis of GRPO and identify two key findings: (1) incorporating negative feedback is essential training solely on actions above a baseline limits learning; and (2) PPO style constraints, such as policy ratio clipping, are not required to improve mathematical reasoning or performance. Building on these insights, we propose REINFORCE with Group Relative Advantage (RGRA), a simplified variant that retains group relative advantage estimation but removes PPO style clipping and policy ratio terms. Experiments across standard mathematical benchmarks indicate that RGRA has the potential to achieve stronger performance than GRPO. Our results suggest that simpler REINFORCE based approaches can effectively enhance reasoning in LLMs, offering a more transparent and efficient alternative to GRPO.

Modern neural networks rely on generic activation functions (ReLU, GELU, SiLU) that ignore the mathematical structure inherent in scientific data. We propose Neuro-Symbolic Activation Discovery, a framework that uses Genetic Programming to extract interpretable mathematical formulas from data and inject them as custom activation functions. Our key contribution is the discovery of a Geometric Transfer phenomenon: activation functions learned from particle physics data successfully generalize to ecological classification, outperforming standard activations (ReLU, GELU, SiLU) in both accuracy and parameter efficiency. On the Forest Cover dataset, our Hybrid Transfer model achieves 82.4% accuracy with only 5,825 parameters, compared to 83.4% accuracy requiring 31,801 parameters for a conventional heavy network -- a 5.5x parameter reduction with only 1% accuracy loss. We introduce a Parameter Efficiency Score (E_{param} = AUC / log_{10}(Params)) and demonstrate that lightweight hybrid architectures consistently achieve 18-21% higher efficiency than over-parameterized baselines. Crucially, we establish boundary conditions: while Physics to Ecology transfer succeeds (both involve continuous Euclidean measurements), Physics to Text transfer fails (discrete word frequencies require different mathematical structures). Our work opens pathways toward domain-specific activation libraries for efficient scientific machine learning.

Large language models increasingly expose reasoning traces, yet their underlying cognitive structure and steps remain difficult to identify and analyze beyond surface-level statistics. We adopt Schoenfeld's Episode Theory as an inductive, intermediate-scale lens and introduce ThinkARM (Anatomy of Reasoning in Models), a scalable framework that explicitly abstracts reasoning traces into functional reasoning steps such as Analysis, Explore, Implement, Verify, etc. When applied to mathematical problem solving by diverse models, this abstraction reveals reproducible thinking dynamics and structural differences between reasoning and non-reasoning models, which are not apparent from token-level views. We further present two diagnostic case studies showing that exploration functions as a critical branching step associated with correctness, and that efficiency-oriented methods selectively suppress evaluative feedback steps rather than uniformly shortening responses. Together, our results demonstrate that episode-level representations make reasoning steps explicit, enabling systematic analysis of how reasoning is structured, stabilized, and altered in modern language models.

A key frontier for Multimodal Large Language Models (MLLMs) is the ability to perform deep mathematical and spatial reasoning directly from images, moving beyond their established success in semantic description. Mathematical surface plots provide a rigorous testbed for this capability, as they isolate the task of reasoning from the semantic noise common in natural images. To measure progress on this frontier, we introduce MaRVL-QA (Mathematical Reasoning over Visual Landscapes), a new benchmark designed to quantitatively evaluate these core reasoning skills. The benchmark comprises two novel tasks: Topological Counting, identifying and enumerating features like local maxima; and Transformation Recognition, recognizing applied geometric transformations. Generated from a curated library of functions with rigorous ambiguity filtering, our evaluation on MaRVL-QA reveals that even state-of-the-art MLLMs struggle significantly, often resorting to superficial heuristics instead of robust spatial reasoning. MaRVL-QA provides a challenging new tool for the research community to measure progress, expose model limitations, and guide the development of MLLMs with more profound reasoning abilities.

Practical identifiability is a critical concern in data-driven modeling of mathematical systems. In this paper, we propose a novel framework for practical identifiability analysis to evaluate parameter identifiability in mathematical models of biological systems. Starting with a rigorous mathematical definition of practical identifiability, we demonstrate its equivalence to the invertibility of the Fisher Information Matrix. Our framework establishes the relationship between practical identifiability and coordinate identifiability, introducing a novel metric that simplifies and accelerates the evaluation of parameter identifiability compared to the profile likelihood method. Additionally, we introduce new regularization terms to address non-identifiable parameters, enabling uncertainty quantification and improving model reliability. To guide experimental design, we present an optimal data collection algorithm that ensures all model parameters are practically identifiable. Applications to Hill functions, neural networks, and dynamic biological models demonstrate the feasibility and efficiency of the proposed computational framework in uncovering critical biological processes and identifying key observable variables.

Quantified formulas with Uninterpreted Functions (UFs) over non-linear real arithmetic pose fundamental challenges for Satisfiability Modulo Theories (SMT) solving. Traditional quantifier instantiation methods struggle because they lack semantic understanding of UF constraints, forcing them to search through unbounded solution spaces with limited guidance. We present AquaForte, a framework that leverages Large Language Models to provide semantic guidance for UF instantiation by generating instantiated candidates for function definitions that satisfy the constraints, thereby significantly reducing the search space and complexity for solvers. Our approach preprocesses formulas through constraint separation, uses structured prompts to extract mathematical reasoning from LLMs, and integrates the results with traditional SMT algorithms through adaptive instantiation. AquaForte maintains soundness through systematic validation: LLM-guided instantiations yielding SAT solve the original problem, while UNSAT results generate exclusion clauses for iterative refinement. Completeness is preserved by fallback to traditional solvers augmented with learned constraints. Experimental evaluation on SMT-COMP benchmarks demonstrates that AquaForte solves numerous instances where state-of-the-art solvers like Z3 and CVC5 timeout, with particular effectiveness on satisfiable formulas. Our work shows that LLMs can provide valuable mathematical intuition for symbolic reasoning, establishing a new paradigm for SMT constraint solving.

Large Language Models (LLMs) have limited performance when solving arithmetic reasoning tasks and often provide incorrect answers. Unlike natural language understanding, math problems typically have a single correct answer, making the task of generating accurate solutions more challenging for LLMs. To the best of our knowledge, we are not aware of any LLMs that indicate their level of confidence in their responses which fuels a trust deficit in these models impeding their adoption. To address this deficiency, we propose `MathPrompter', a technique that improves performance of LLMs on arithmetic problems along with increased reliance in the predictions. MathPrompter uses the Zero-shot chain-of-thought prompting technique to generate multiple Algebraic expressions or Python functions to solve the same math problem in different ways and thereby raise the confidence level in the output results. This is in contrast to other prompt based CoT methods, where there is no check on the validity of the intermediate steps followed. Our technique improves over state-of-the-art on the MultiArith dataset (78.7%rightarrow92.5%) evaluated using 175B parameter GPT-based LLM.

We present AXIOM, a trust-first neuro-symbolic execution architecture for natural-language mathematical reasoning. In AXIOM, the language model functions strictly as a canonicalizer: it rewrites informal problem text into a narrow schema consumed by a deterministic Computer-Algebra-System (CAS) pipeline, which derives and verifies the answer or abstains as a first-class output. Routing follows a 1:1:1 alignment between problem-shape regex, schema-specific prompt, and closed-form CAS handler, with 3,100+ such routes shipped and zero LOST_CORRECT regressions across 250+ consecutive ship commits. We report empirical results on 4 MATH categories with a cumulative correctness of 94.36% (2,592/2,747) at 100.00% trust on parseable (zero confident-wrong answers across the full 2,747-record benchmark), all four domains above the per-domain 70/90/70 floor with per-domain trust at 100.0%, and median latency of 1 ms on rule-only handlers (88% of records on the lm-eval arithmetic 20,000-record benchmark). The architecture has served ~30,000 production queries through a public deployment. The contribution we emphasize is not a final accuracy figure but the forward dynamic the architecture establishes: every logged abstain in production is a candidate correct after one ship cycle, since new tasks compose without regressing the registry. The operational discipline behind this property -- math-template bucketing, LOST_CORRECT scan as regression oracle, parseable-first onboarding, and abstain as first-class output -- constitutes a transferable framework for trustworthy neuro-symbolic systems beyond mathematics.

Selecting the most suitable activation function is a critical factor in the effectiveness of deep learning models, as it influences their learning capacity, stability, and computational efficiency. In recent years, the Gaussian Error Linear Unit (GELU) activation function has emerged as a dominant method, surpassing traditional functions such as the Rectified Linear Unit (ReLU) in various applications. This study presents a rigorous mathematical investigation of the GELU activation function, exploring its differentiability, boundedness, stationarity, and smoothness properties in detail. Additionally, we conduct an extensive experimental comparison of the GELU function against a broad range of alternative activation functions, utilizing a residual convolutional network trained on the CIFAR-10, CIFAR-100, and STL-10 datasets as the empirical testbed. Our results demonstrate the superior performance of GELU compared to other activation functions, establishing its suitability for a wide range of deep learning applications. This comprehensive study contributes to a more profound understanding of the underlying mathematical properties of GELU and provides valuable insights for practitioners aiming to select activation functions that optimally align with their specific objectives and constraints in deep learning.

Recent progress in large-scale reinforcement learning (RL) has notably enhanced the reasoning capabilities of large language models (LLMs), especially in mathematical domains. However, current multimodal LLMs (MLLMs) for mathematical reasoning often rely on one-to-one image-text pairs and single-solution supervision, overlooking the diversity of valid reasoning perspectives and internal reflections. In this work, we introduce MathV-DP, a novel dataset that captures multiple diverse solution trajectories for each image-question pair, fostering richer reasoning supervision. We further propose Qwen-VL-DP, a model built upon Qwen-VL, fine-tuned with supervised learning and enhanced via group relative policy optimization (GRPO), a rule-based RL approach that integrates correctness discrimination and diversity-aware reward functions. Our method emphasizes learning from varied reasoning perspectives and distinguishing between correct yet distinct solutions. Extensive experiments on the MathVista's minitest and Math-V benchmarks demonstrate that Qwen-VL-DP significantly outperforms prior base MLLMs in both accuracy and generative diversity, highlighting the importance of incorporating diverse perspectives and reflective reasoning in multimodal mathematical reasoning.

Recent advances in Large Language Models (LLMs) and Vision Language Models (VLMs) have shown significant progress in mathematical reasoning, yet they still face a critical bottleneck with problems requiring visual assistance, such as drawing auxiliary lines or plotting functions to solve the problems. Most LLMs and VLMs are constrained to text-only reasoning chains, while multimodal unified models that can generate interleaved text and images lack the necessary precision and controllability for such tasks. To address this, we propose CodePlot-CoT, a code-driven Chain-of-Thought paradigm for "thinking with images" in mathematics. Our approach leverages the VLM to generate text reasoning as well as executable plotting code, which is then rendered into images as "visual thought", to solve mathematical problems. To achieve this, we first construct Math-VR, the first large-scale, bilingual dataset and benchmark for Mathematics problems with Visual Reasoning, comprising 178K samples. Second, to create high-quality training data, we develop a state-of-the-art image-to-code converter specialized for parsing complex mathematical figures into codes. Finally, using these training data, we train the CodePlot-CoT model for solving mathematical problems. Experimental results show that our model achieves up to 21% increase over base model on our new benchmark, fully validating the efficacy of our proposed code-driven reasoning paradigm. Our work opens a new direction for multimodal mathematical reasoning and provides the community with the first large-scale dataset, comprehensive benchmark, and strong approach for such problems. To facilitate future research, we make our datasets, code, and pretrained models publicly available at https://github.com/HKU-MMLab/Math-VR-CodePlot-CoT.

Deep learning-based 3D medical image segmentation methods relies on large-scale labeled datasets, yet acquiring such data is difficult due to privacy constraints and the high cost of expert annotation. Formula-Driven Supervised Learning (FDSL) offers an appealing alternative by generating training data and labels directly from mathematical formulas. However, existing voxel-based approaches are limited in geometric expressiveness and cannot synthesize realistic textures. We introduce Formula-Driven supervised learning with Implicit Functions (FDIF), a framework that enables scalable pre-training without using any real data and medical expert annotations. FDIF introduces an implicit-function representation based on signed distance functions (SDFs), enabling compact modeling of complex geometries while exploiting the surface representation of SDFs to support controllable synthesis of both geometric and intensity textures. Across three medical image segmentation benchmarks (AMOS, ACDC, and KiTS) and three architectures (SwinUNETR, nnUNet ResEnc-L, and nnUNet Primus-M), FDIF consistently improves over a formula-driven method, and achieves performance comparable to self-supervised approaches pre-trained on large-scale real datasets. We further show that FDIF pre-training also benefits 3D classification tasks, highlighting implicit-function-based formula supervision as a promising paradigm for data-free representation learning. Code is available at https://github.com/yamanoko/FDIF.

Deep convolutional neural networks have led to breakthrough results in numerous practical machine learning tasks such as classification of images in the ImageNet data set, control-policy-learning to play Atari games or the board game Go, and image captioning. Many of these applications first perform feature extraction and then feed the results thereof into a trainable classifier. The mathematical analysis of deep convolutional neural networks for feature extraction was initiated by Mallat, 2012. Specifically, Mallat considered so-called scattering networks based on a wavelet transform followed by the modulus non-linearity in each network layer, and proved translation invariance (asymptotically in the wavelet scale parameter) and deformation stability of the corresponding feature extractor. This paper complements Mallat's results by developing a theory that encompasses general convolutional transforms, or in more technical parlance, general semi-discrete frames (including Weyl-Heisenberg filters, curvelets, shearlets, ridgelets, wavelets, and learned filters), general Lipschitz-continuous non-linearities (e.g., rectified linear units, shifted logistic sigmoids, hyperbolic tangents, and modulus functions), and general Lipschitz-continuous pooling operators emulating, e.g., sub-sampling and averaging. In addition, all of these elements can be different in different network layers. For the resulting feature extractor we prove a translation invariance result of vertical nature in the sense of the features becoming progressively more translation-invariant with increasing network depth, and we establish deformation sensitivity bounds that apply to signal classes such as, e.g., band-limited functions, cartoon functions, and Lipschitz functions.

Mathematical problem-solving is a key field in artificial intelligence (AI) and a critical benchmark for evaluating the capabilities of large language models (LLMs). While extensive research has focused on mathematical problem-solving, most existing work and datasets concentrate on computational tasks, leaving gaps in areas like mathematical analysis, which demands rigorous proofs and formal reasoning. We developed the DEMI-MathAnalysis dataset, comprising proof-based problems from mathematical analysis topics such as Sequences and Limits, Infinite Series, and Convex Functions. We also designed a guiding framework to rigorously enhance LLMs' ability to solve these problems. Through fine-tuning LLMs on this dataset and employing our framework, we observed significant improvements in their capability to generate logical, complete, and elegant proofs. This work addresses critical gaps in mathematical reasoning and contributes to advancing trustworthy AI capable of handling formalized mathematical language. The code is publicly accessible at LLMs for Mathematical Analysis.

One of the most common and universal problems in science is to investigate a function. The prediction can be made by an Artificial Neural Network (ANN) or a mathematical model. Both approaches have their advantages and disadvantages. Mathematical models were sought as more trustworthy as their prediction is based on the laws of physics expressed in the form of mathematical equations. However, the majority of existing mathematical models include different empirical parameters, and both approaches inherit inevitable experimental errors. At the same time, the approximation of neural networks can reproduce the solution extremely well if fed with a sufficient amount of data. The difference is that an ANN requires big data to build its accurate approximation whereas a typical mathematical model needs just several data points to estimate an empirical constant. Therefore, the common problem that developer meet is the inaccuracy of mathematical models and artificial neural network. An another common challenge is the computational complexity of the mathematical models, or lack of data for a sufficient precision of the Artificial Neural Networks. In the presented paper those problems are addressed using the integration of a mathematical model with an artificial neural network. In the presented analysis, an ANN predicts just a part of the mathematical model and its weights and biases are adjusted based on the output of the mathematical model. The performance of Integrated Mathematical modeling - Artificial Neural Network (IMANN) is compared to a Dense Neural Network (DNN) with the use of the benchmarking functions. The obtained calculation results indicate that such an approach could lead to an increase of precision as well as limiting the data-set required for learning.

The advancement of reasoning capabilities in Large Language Models (LLMs) requires substantial amounts of high-quality reasoning data, particularly in mathematics. Existing data synthesis methods, such as data augmentation from annotated training sets or direct question generation based on relevant knowledge points and documents, have expanded datasets but face challenges in mastering the inner logic of the problem during generation and ensuring the verifiability of the solutions. To address these issues, we propose RV-Syn, a novel Rational and Verifiable mathematical Synthesis approach. RV-Syn constructs a structured mathematical operation function library based on initial seed problems and generates computational graphs as solutions by combining Python-formatted functions from this library. These graphs are then back-translated into complex problems. Based on the constructed computation graph, we achieve solution-guided logic-aware problem generation. Furthermore, the executability of the computational graph ensures the verifiability of the solving process. Experimental results show that RV-Syn surpasses existing synthesis methods, including those involving human-generated problems, achieving greater efficient data scaling. This approach provides a scalable framework for generating high-quality reasoning datasets.

Formulas involving fundamental mathematical constants had a great impact on various fields of science and mathematics, for example aiding in proofs of irrationality of constants. However, the discovery of such formulas has historically remained scarce, often perceived as an act of mathematical genius by great mathematicians such as Ramanujan, Euler, and Gauss. Recent efforts to automate the discovery of formulas for mathematical constants, such as the Ramanujan Machine project, relied on exhaustive search. Despite several successful discoveries, exhaustive search remains limited by the space of options that can be covered and by the need for vast amounts of computational resources. Here we propose a fundamentally different method to search for conjectures on mathematical constants: through analysis of integer sequences. We introduce the Enumerated Signed-continued-fraction Massey Approve (ESMA) algorithm, which builds on the Berlekamp-Massey algorithm to identify patterns in integer sequences that represent mathematical constants. The ESMA algorithm found various known formulas for e, e^2, tan(1), and ratios of values of Bessel functions. The algorithm further discovered a large number of new conjectures for these constants, some providing simpler representations and some providing faster numerical convergence than the corresponding simple continued fractions. Along with the algorithm, we present mathematical tools for manipulating continued fractions. These connections enable us to characterize what space of constants can be found by ESMA and quantify its algorithmic advantage in certain scenarios. Altogether, this work continues in the development of augmenting mathematical intuition by computer algorithms, to help reveal mathematical structures and accelerate mathematical research.

Diffusion models have emerged as a powerful tool rivaling GANs in generating high-quality samples with improved fidelity, flexibility, and robustness. A key component of these models is to learn the score function through score matching. Despite empirical success on various tasks, it remains unclear whether gradient-based algorithms can learn the score function with a provable accuracy. As a first step toward answering this question, this paper establishes a mathematical framework for analyzing score estimation using neural networks trained by gradient descent. Our analysis covers both the optimization and the generalization aspects of the learning procedure. In particular, we propose a parametric form to formulate the denoising score-matching problem as a regression with noisy labels. Compared to the standard supervised learning setup, the score-matching problem introduces distinct challenges, including unbounded input, vector-valued output, and an additional time variable, preventing existing techniques from being applied directly. In this paper, we show that with proper designs, the evolution of neural networks during training can be accurately modeled by a series of kernel regression tasks. Furthermore, by applying an early-stopping rule for gradient descent and leveraging recent developments in neural tangent kernels, we establish the first generalization error (sample complexity) bounds for learning the score function with neural networks, despite the presence of noise in the observations. Our analysis is grounded in a novel parametric form of the neural network and an innovative connection between score matching and regression analysis, facilitating the application of advanced statistical and optimization techniques.

Representation learning methods have revolutionized machine learning on networks by converting discrete network structures into continuous domains. However, dynamic networks that evolve over time pose new challenges. To address this, dynamic representation learning methods have gained attention, offering benefits like reduced learning time and improved accuracy by utilizing temporal information. T-batching is a valuable technique for training dynamic network models that reduces training time while preserving vital conditions for accurate modeling. However, we have identified a limitation in the training loss function used with t-batching. Through mathematical analysis, we propose two alternative loss functions that overcome these issues, resulting in enhanced training performance. We extensively evaluate the proposed loss functions on synthetic and real-world dynamic networks. The results consistently demonstrate superior performance compared to the original loss function. Notably, in a real-world network characterized by diverse user interaction histories, the proposed loss functions achieved more than 26.9% enhancement in Mean Reciprocal Rank (MRR) and more than 11.8% improvement in Recall@10. These findings underscore the efficacy of the proposed loss functions in dynamic network modeling.

We introduce a novel mathematical framework that unifies neural population dynamics, hippocampal sharp wave-ripple (SpWR) generation, and cognitive consistency constraints inspired by Heider's theory. Our model leverages low-dimensional manifold representations to capture structured neural drift and incorporates a balance energy function to enforce coherent synaptic interactions, effectively simulating the memory consolidation processes observed in biological systems. Simulation results demonstrate that our approach not only reproduces key features of SpWR events but also enhances network interpretability. This work paves the way for scalable neuromorphic architectures that bridge neuroscience and artificial intelligence, offering more robust and adaptive learning mechanisms for future intelligent systems.

State of the art Symbolic Regression (SR) methods currently build specialized models, while the application of Large Language Models (LLMs) remains largely unexplored. In this work, we introduce the first comprehensive framework that utilizes LLMs for the task of SR. We propose In-Context Symbolic Regression (ICSR), an SR method which iteratively refines a functional form with an LLM and determines its coefficients with an external optimizer. ICSR leverages LLMs' strong mathematical prior both to propose an initial set of possible functions given the observations and to refine them based on their errors. Our findings reveal that LLMs are able to successfully find symbolic equations that fit the given data, matching or outperforming the overall performance of the best SR baselines on four popular benchmarks, while yielding simpler equations with better out of distribution generalization.

In the mid-twentieth century, mathematician and polymath John von Neumann created a computational system on an array of cells as a simple model of the human brain, where each cell had one of a finite set of roles or states that he predicted would be modelled by a diffusion process. In this work, we show that such a system, when developed in a modern deep learning setting, enables the construction of an artificial neuron having specialized roles that can be learnt. We refer to this neuron as the Von Neumann neuron, and the resulting neural network from such neurons result in a self-engineered design whose architecture is only dependent on the structure and locations of its inputs and outputs on this cellular array. The mathematical framework for these Von Neumann Networks (VNNs) is also constructed and shows that they are based on the extension of neural operators and the learning of Green's functions with convolutions on a cellular topology having a diffusion signature. We also prove that these VNNs are part of a more general computational system called Cellular Machines that are computationally universal. Initial experiments show that VNN based multi-layered perceptrons outperform their equivalent deep learning variant on basic tasks, while being more parameter efficient and are capable of learning new types of tasks. This includes the ability to solve for and construct an extension of the Von Neumann (hardware) architecture common to all modern computers to cells and suggests new opportunities that could be explored.

A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. Motivated by the underlying mathematical structure, NIO is based on a suitable composition of DeepONets and FNOs to approximate mappings from operators to functions. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods.

We present Universal Conditional Logic (UCL), a mathematical framework for prompt optimization that transforms prompt engineering from heuristic practice into systematic optimization. Through systematic evaluation (N=305, 11 models, 4 iterations), we demonstrate significant token reduction (29.8%, t(10)=6.36, p < 0.001, Cohen's d = 2.01) with corresponding cost savings. UCL's structural overhead function O_s(A) explains version-specific performance differences through the Over-Specification Paradox: beyond threshold S* = 0.509, additional specification degrades performance quadratically. Core mechanisms -- indicator functions (I_i in {0,1}), structural overhead (O_s = gamma * sum(ln C_k)), early binding -- are validated. Notably, optimal UCL configuration varies by model architecture -- certain models (e.g., Llama 4 Scout) require version-specific adaptations (V4.1). This work establishes UCL as a calibratable framework for efficient LLM interaction, with model-family-specific optimization as a key research direction.

Permutation Entropy and statistiCal Complexity Analysis for astRophYsics (PECCARY) is a computationally inexpensive, statistical method by which any time-series can be characterized as predominantly regular, complex, or stochastic. Elements of the PECCARY method have been used in a variety of physical, biological, economic, and mathematical scenarios, but have not yet gained traction in the astrophysical community. This study introduces the PECCARY technique with the specific aims to motivate its use in and optimize it for the analysis of astrophysical orbital systems. PECCARY works by decomposing a time-dependent measure, such as the x-coordinate or orbital angular momentum time-series, into ordinal patterns. Due to its unique approach and statistical nature, PECCARY is well-suited for detecting preferred and forbidden patterns (a signature of chaos), even when the chaotic behavior is short-lived or when working with a relatively short duration time-series or small sets of time-series data. A variety of examples are used to demonstrate the capabilities of PECCARY. These include mathematical examples (sine waves, varieties of noise, sums of sine waves, well-known chaotic functions), a double pendulum system, and astrophysical tracer particle simulations with potentials of varying intricacies. Since the adopted timescale used to diagnose a given time-series can affect the outcome, a method is presented to identify an ideal sampling scheme, constrained by the overall duration and the natural timescale of the system. The accompanying PECCARY Python package and its usage are discussed.

We propose Mish, a novel self-regularized non-monotonic activation function which can be mathematically defined as: f(x)=xtanh(softplus(x)). As activation functions play a crucial role in the performance and training dynamics in neural networks, we validated experimentally on several well-known benchmarks against the best combinations of architectures and activation functions. We also observe that data augmentation techniques have a favorable effect on benchmarks like ImageNet-1k and MS-COCO across multiple architectures. For example, Mish outperformed Leaky ReLU on YOLOv4 with a CSP-DarkNet-53 backbone on average precision (AP_{50}^{val}) by 2.1% in MS-COCO object detection and ReLU on ResNet-50 on ImageNet-1k in Top-1 accuracy by approx1% while keeping all other network parameters and hyperparameters constant. Furthermore, we explore the mathematical formulation of Mish in relation with the Swish family of functions and propose an intuitive understanding on how the first derivative behavior may be acting as a regularizer helping the optimization of deep neural networks. Code is publicly available at https://github.com/digantamisra98/Mish.

A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent homology (PH) provides a delicate balance between data simplification and intrinsic structure characterization, and has been applied to various areas successfully. However, the combination of PH and machine learning has been hindered greatly by three challenges, namely topological representation of data, PH-based distance measurements or metrics, and PH-based feature representation. With the development of topological data analysis, progresses have been made on all these three problems, but widely scattered in different literatures. In this paper, we provide a systematical review of PH and PH-based supervised and unsupervised models from a computational perspective. Our emphasizes are the recent development of mathematical models and tools, including PH softwares and PH-based functions, feature representations, kernels, and similarity models. Essentially, this paper can work as a roadmap for the practical application of PH-based machine learning tools. Further, we consider different topological feature representations in different machine learning models, and investigate their impacts on the protein secondary structure classification.

We propose a novel persona-driven data synthesis methodology that leverages various perspectives within a large language model (LLM) to create diverse synthetic data. To fully exploit this methodology at scale, we introduce Persona Hub -- a collection of 1 billion diverse personas automatically curated from web data. These 1 billion personas (~13% of the world's total population), acting as distributed carriers of world knowledge, can tap into almost every perspective encapsulated within the LLM, thereby facilitating the creation of diverse synthetic data at scale for various scenarios. By showcasing Persona Hub's use cases in synthesizing high-quality mathematical and logical reasoning problems, instructions (i.e., user prompts), knowledge-rich texts, game NPCs and tools (functions) at scale, we demonstrate persona-driven data synthesis is versatile, scalable, flexible, and easy to use, potentially driving a paradigm shift in synthetic data creation and applications in practice, which may have a profound impact on LLM research and development.

Understanding what individual neurons encode is a core question in neuroscience. In primary visual cortex (V1), mathematical models (e.g., Gabor functions) capture neural selectivity, but no comparable framework exists for higher areas. We show that natural language can fill this role: across macaque V1 and V4, the selectivity of most neurons is captured by concise, verifiable semantic descriptions. Using digital twins of V1 and V4, we develop a closed-loop framework that translates each neuron's high- and low-activating images into dense captions, generates a semantic hypothesis and synthesized images, and verifies the hypothesis in silico. Descriptions range from oriented edges and spatial frequency in V1 to conjunctions of form, color, and texture in V4. In V4, images generated from activating and suppressing hypotheses drove 96.1% of neurons above the 95th and 97.6% below the 5th percentile of natural-image responses, respectively (vs. ~10% for random images); V1 activation results matched V4, while V1 suppression was less describable in language. Representational similarity analysis reveals partial alignment between neural activity, vision embeddings, and language embeddings, with vision most aligned to neural activity; alignment lost in the text bottleneck is recovered when hypotheses are rendered back into images, showing that linguistic compression is lossy yet semantically faithful. Together, these results show that combining generative models with neural digital twins enables interpretable, testable descriptions of neural function at scale, toward agentic scientific discovery.

As part of a series on first-principles modeling of cognitive functions, this paper attempts to provide a mathematical formulation of thinking and perception. It formally derives slow thinking or more generally, active perception, and encompasses the design, training and inference of slow thinking large language models. Our starting point is the lifting and projection of probability distributions on the observable and latent spaces, with the objective of representing complex data distributions by simple function families such as neural networks. A theory called "active lifting" is proposed, based on the sampling of latent sequences and an intrinsic drive to reduce uncertainty with maximum rate. It derives a large design space, containing the slow thinking models in a subspace that we call the static theory. These models are positioned on the representation hierarchy and sampler hierarchy induced by the static theory, and can be upgraded by climbing the two hierarchies. Active lifting further derives an inference process with an internal time axis, and a training objective that resembles minimum-length coding as well as the invention of languages. Thus, it characterizes the agency of perception, including the emergence of the slow thinking formats. Technical by-products of this theory include a three-stage pathway for improving slow thinking models, a unified approach to constructing encoders and generative models for all data modalities, a priori formation of human-like visual representations, and a possible solution to policy collapse.

Recent findings suggest that consecutive layers of neural networks with the ReLU activation function fold the input space during the learning process. While many works hint at this phenomenon, an approach to quantify the folding was only recently proposed by means of a space folding measure based on Hamming distance in the ReLU activation space. We generalize this measure to a wider class of activation functions through introduction of equivalence classes of input data, analyse its mathematical and computational properties and come up with an efficient sampling strategy for its implementation. Moreover, it has been observed that space folding values increase with network depth when the generalization error is low, but decrease when the error increases. This underpins that learned symmetries in the data manifold (e.g., invariance under reflection) become visible in terms of space folds, contributing to the network's generalization capacity. Inspired by these findings, we outline a novel regularization scheme that encourages the network to seek solutions characterized by higher folding values.

Latent variable collaborative filtering methods have been a standard approach to modelling user-click interactions due to their simplicity and effectiveness. However, there is limited work on analyzing the mathematical properties of these methods in particular on preventing the overfitting towards the identity, and such methods typically utilize loss functions that overlook the geometry between items. In this work, we introduce a notion of generalization gap in collaborative filtering and analyze this with respect to latent collaborative filtering models. We present a geometric upper bound that gives rise to loss functions, and a way to meaningfully utilize the geometry of item-metadata to improve recommendations. We show how these losses can be minimized and gives the recipe to a new latent collaborative filtering algorithm, which we refer to as GeoCF, due to the geometric nature of our results. We then show experimentally that our proposed GeoCF algorithm can outperform other all existing methods on the Movielens20M and Netflix datasets, as well as two large-scale internal datasets. In summary, our work proposes a theoretically sound method which paves a way to better understand generalization of collaborative filtering at large.

Symbolic regression (SR) is an area of interpretable machine learning that aims to identify mathematical expressions, often composed of simple functions, that best fit in a given set of covariates X and response y. In recent years, deep symbolic regression (DSR) has emerged as a popular method in the field by leveraging deep reinforcement learning to solve the complicated combinatorial search problem. In this work, we propose an alternative framework (GFN-SR) to approach SR with deep learning. We model the construction of an expression tree as traversing through a directed acyclic graph (DAG) so that GFlowNet can learn a stochastic policy to generate such trees sequentially. Enhanced with an adaptive reward baseline, our method is capable of generating a diverse set of best-fitting expressions. Notably, we observe that GFN-SR outperforms other SR algorithms in noisy data regimes, owing to its ability to learn a distribution of rewards over a space of candidate solutions.

Transformers have found extensive applications across various domains due to the powerful fitting capabilities. This success can be partially attributed to their inherent nonlinearity. Thus, in addition to the ReLU function employed in the original transformer architecture, researchers have explored alternative modules such as GeLU and SwishGLU to enhance nonlinearity and thereby augment representational capacity. In this paper, we propose a novel category of polynomial composition activations (PolyCom), designed to optimize the dynamics of transformers. Theoretically, we provide a comprehensive mathematical analysis of PolyCom, highlighting its enhanced expressivity and efficacy relative to other activation functions. Notably, we demonstrate that networks incorporating PolyCom achieve the optimal approximation rate, indicating that PolyCom networks require minimal parameters to approximate general smooth functions in Sobolev spaces. We conduct empirical experiments on the pre-training configurations of large language models (LLMs), including both dense and sparse architectures. By substituting conventional activation functions with PolyCom, we enable LLMs to capture higher-order interactions within the data, thus improving performance metrics in terms of accuracy and convergence rates. Extensive experimental results demonstrate the effectiveness of our method, showing substantial improvements over other activation functions. Code is available at https://github.com/BryceZhuo/PolyCom.

Post-training processes are essential phases in grounding pre-trained language models to real-world tasks, with learning from demonstrations or preference signals playing a crucial role in this adaptation. We present a unified theoretical framework bridging Supervised Fine-Tuning (SFT) and preference learning in Large Language Model (LLM) post-training. Through rigorous mathematical derivation, we demonstrate that both SFT and preference learning methods like Direct Preference Optimization (DPO) operate within the same optimal policy-reward subspace, with SFT representing a special case of implicit reward learning. Our analysis reveals a critical limitation in conventional SFT: the KL divergence term in distribution matching becomes constant with respect to the policy during optimization, failing to constrain model updates. To address this, we propose a simple yet effective learning rate reduction approach that yields significant performance improvements (up to 25\% relative gain and 6\% absolute win rate increase in instruction following tasks. Additionally, we derive alternative SFT objectives from various f-divergence functions that preserve the KL term during optimization, further enhancing post-DPO model performance. Finally, we extend the theoretical relationship between LLM logits and Q-functions from preference learning to the SFT context, providing mathematical derivations and experimental validation.

Large language models (LLMs) are known for their exceptional performance in natural language processing, making them highly effective in many human life-related or even job-related tasks. The attention mechanism in the Transformer architecture is a critical component of LLMs, as it allows the model to selectively focus on specific input parts. The softmax unit, which is a key part of the attention mechanism, normalizes the attention scores. Hence, the performance of LLMs in various NLP tasks depends significantly on the crucial role played by the attention mechanism with the softmax unit. In-context learning, as one of the celebrated abilities of recent LLMs, is an important concept in querying LLMs such as ChatGPT. Without further parameter updates, Transformers can learn to predict based on few in-context examples. However, the reason why Transformers becomes in-context learners is not well understood. Recently, several works [ASA+22,GTLV22,ONR+22] have studied the in-context learning from a mathematical perspective based on a linear regression formulation min_x| Ax - b |_2, which show Transformers' capability of learning linear functions in context. In this work, we study the in-context learning based on a softmax regression formulation min_{x} | langle exp(Ax), {bf 1}_n rangle^{-1} exp(Ax) - b |_2 of Transformer's attention mechanism. We show the upper bounds of the data transformations induced by a single self-attention layer and by gradient-descent on a ell_2 regression loss for softmax prediction function, which imply that when training self-attention-only Transformers for fundamental regression tasks, the models learned by gradient-descent and Transformers show great similarity.

This project focuses on a method to extract a frequency comb in mechanical means, for general interest and numerous practical applications in MEMS. The method of execution is the implementation of a beam that is exhibiting non-linear dynamics that is perturbed and analyzed for its transverse vibrations. The perturbation is an external harmonic driver with a chosen small amplitude and frequency (which is slightly detuned from the beam eigenfrequency), that when engaged with the unperturbed beam oscillations, causes it reach a state of "injection pulling" - an effect that occurs when one harmonic oscillator is coupled with a second one and causes it to oscillate in a frequency near its own. This causes the beam to reach SNIC bifurcation, rendering a frequency comb as desired. Theoretical analysis showed that the problem can be modelled using a non-linear equation of the beam, that translates to a form of the non-linear Duffing equation. While a solution to the dynamics function of the beam is hard to obtain in practice due to mathematical difficulties, a slow evolution model is suggested that is composed of functions of a amplitude and phase. Using several additional mathematical assumptions, the amplitude is seen to be related to the phase, while the phase equation solution is seen to be of the form of Adler's equation. These assumptions ultimately reduce the entire behaviour of the beam to a relatively simple solution to the Adler equation, which has a known analytical solution. Computerized numerical simulations are run on it to check the results and compare them to the theory and desired outcome. The results agreed with the theory and produce the expected frequency comb, showing the assumptions to be valid in extracting the comb.

We introduce ShinkaEvolve: a new open-source framework leveraging large language models (LLMs) to advance scientific discovery with state-of-the-art performance and unprecedented efficiency. Recent advances in scaling inference time compute of LLMs have enabled significant progress in generalized scientific discovery. These approaches rely on evolutionary agentic harnesses that leverage LLMs as mutation operators to generate candidate solutions. However, current code evolution methods suffer from critical limitations: they are sample inefficient, requiring thousands of samples to identify effective solutions, and remain closed-source, hindering broad adoption and extension. ShinkaEvolve addresses these limitations, introducing three key innovations: a parent sampling technique balancing exploration and exploitation, code novelty rejection-sampling for efficient search space exploration, and a bandit-based LLM ensemble selection strategy. We evaluate ShinkaEvolve across diverse tasks, demonstrating consistent improvements in sample efficiency and solution quality. ShinkaEvolve discovers a new state-of-the-art circle packing solution using only 150 samples, designs high-performing agentic harnesses for AIME mathematical reasoning tasks, identifies improvements to ALE-Bench competitive programming solutions, and discovers novel mixture-of-expert load balancing loss functions that illuminate the space of optimization strategies. Our results demonstrate that ShinkaEvolve achieves broad applicability with exceptional sample efficiency. By providing open-source accessibility and cost-efficiency, this work democratizes open-ended discovery across diverse computational problems.

Automatic analysis of the enormous sets of images is a critical task in life sciences. This faces many challenges such as: algorithms are highly parameterized, significant human input is intertwined, and lacking a standard meta-visualization approach. This paper proposes an alternative iterative approach for optimizing input parameters, saving time by minimizing the user involvement, and allowing for understanding the workflow of algorithms and discovering new ones. The main focus is on developing an interactive visualization technique that enables users to analyze the relationships between sampled input parameters and corresponding output. This technique is implemented as a prototype called Veni Vidi Vici, or "I came, I saw, I conquered." This strategy is inspired by the mathematical formulas of numbering computable functions and is developed atop ImageJ, a scientific image processing program. A case study is presented to investigate the proposed framework. Finally, the paper explores some potential future issues in the application of the proposed approach in parameter space analysis in visualization.

We propose a data augmentation method for offline reinforcement learning, motivated by active positioning problems. Particularly, our approach enables the training of off-policy models from a limited number of suboptimal trajectories. We introduce a trajectory-based augmentation technique that exploits task structure and the geometric relationship between rewards, value functions, and mathematical properties of logging policies. During data collection, our augmentation supports suboptimal logging policies, leading to higher data quality and improved offline reinforcement learning performance. We provide theoretical justification for these strategies and validate them empirically across positioning tasks of varying dimensionality and under partial observability.

Utility functions or their equivalents (value functions, objective functions, loss functions, reward functions, preference orderings) are a central tool in most current machine learning systems. These mechanisms for defining goals and guiding optimization run into practical and conceptual difficulty when there are independent, multi-dimensional objectives that need to be pursued simultaneously and cannot be reduced to each other. Ethicists have proved several impossibility theorems that stem from this origin; those results appear to show that there is no way of formally specifying what it means for an outcome to be good for a population without violating strong human ethical intuitions (in such cases, the objective function is a social welfare function). We argue that this is a practical problem for any machine learning system (such as medical decision support systems or autonomous weapons) or rigidly rule-based bureaucracy that will make high stakes decisions about human lives: such systems should not use objective functions in the strict mathematical sense. We explore the alternative of using uncertain objectives, represented for instance as partially ordered preferences, or as probability distributions over total orders. We show that previously known impossibility theorems can be transformed into uncertainty theorems in both of those settings, and prove lower bounds on how much uncertainty is implied by the impossibility results. We close by proposing two conjectures about the relationship between uncertainty in objectives and severe unintended consequences from AI systems.

Deep Neural Networks (DNNs) can be represented as graphs whose links and vertices iteratively process data and solve tasks sub-optimally. Complex Network Theory (CNT), merging statistical physics with graph theory, provides a method for interpreting neural networks by analysing their weights and neuron structures. However, classic works adapt CNT metrics that only permit a topological analysis as they do not account for the effect of the input data. In addition, CNT metrics have been applied to a limited range of architectures, mainly including Fully Connected neural networks. In this work, we extend the existing CNT metrics with measures that sample from the DNNs' training distribution, shifting from a purely topological analysis to one that connects with the interpretability of deep learning. For the novel metrics, in addition to the existing ones, we provide a mathematical formalisation for Fully Connected, AutoEncoder, Convolutional and Recurrent neural networks, of which we vary the activation functions and the number of hidden layers. We show that these metrics differentiate DNNs based on the architecture, the number of hidden layers, and the activation function. Our contribution provides a method rooted in physics for interpreting DNNs that offers insights beyond the traditional input-output relationship and the CNT topological analysis.

The use of large language models (LLMs) as automated evaluation tools to assess the quality of generated natural language, known as LLMs-as-Judges, has demonstrated promising capabilities and is rapidly gaining widespread attention. However, when applied to pairwise comparisons of candidate responses, LLM-based evaluators often exhibit selection bias. Specifically, their judgments may become inconsistent when the option positions or ID tokens are swapped, compromising the effectiveness and fairness of the evaluation result. To address this challenge, we introduce CalibraEval, a novel label-free method for mitigating selection bias during inference. Specifically, CalibraEval reformulates debiasing as an optimization task aimed at adjusting observed prediction distributions to align with unbiased prediction distributions. To solve this optimization problem, we propose a non-parametric order-preserving algorithm (NOA). This algorithm leverages the partial order relationships between model prediction distributions, thereby eliminating the need for explicit labels and precise mathematical function modeling.Empirical evaluations of LLMs in multiple representative benchmarks demonstrate that CalibraEval effectively mitigates selection bias and improves performance compared to existing debiasing methods. This work marks a step toward building more robust and unbiased automated evaluation frameworks, paving the way for improved reliability in AI-driven assessments

We introduce a new functional space U designed to contain all classical arithmetic functions (Mobius, von Mangoldt, Euler phi, divisor functions, Dirichlet characters, etc.). The norm of U combines a Hilbert-type component, based on square summability of Dirichlet coefficients for every s > 1, with a Banach component controlling logarithmic averages of partial sums. We prove that U is a complete Banach space which embeds continuously all standard Hilbert spaces of Dirichlet series and allows natural actions of Dirichlet convolution and shift operators. This framework provides a unified analytic setting for classical and modern problems in multiplicative number theory.

Solutions of nonlinear functional equations are generally not expressed as a finite number of combinations and compositions of elementary and known special functions. One of the approaches to study them is, firstly, to find formal solutions (that is, series whose terms are described and ordered in some way but which do not converge apriori) and, secondly, to study the convergence or summability of these formal solutions (the existence and uniqueness of actual solutions with the given asymptotic expansion in a certain domain). In this paper we deal only with the convergence of formal functional series having the form of an infinite sum of power functions with (complex, in general) power exponents and satisfying analytical functional equations of the following three types: a differential, q-difference or Mahler equation.

A comparative analysis of the special shapes (patterns, profiles) of the eclipses applied for the phenomenological modeling of the light curves of eclipsing binary stars is conducted. Families of functions are considered, generalizing local approximations (Andronov, 2010, 2012) and the functions theoretically unlimited in a width, based on a Gaussian (Mikulasek, 2015). For an analysis, the light curve of the star V0882 Car = 2MASS J11080308 - 6145589 of the classic Algol - subtype (\beta Persei) is used. By analyzing dozens of modified functions with additional parameters, it was chosen the 14 best ones according to the criterion of the least sum of squares of deviations. The best are the functions with an additional parameter, describing profiles, which are limited in phase.

This work characterizes large language models' chain-of-thought generation as a structured trajectory through representation space. We show that mathematical reasoning traverses functionally ordered, step-specific subspaces that become increasingly separable with layer depth. This structure already exists in base models, while reasoning training primarily accelerates convergence toward termination-related subspaces rather than introducing new representational organization. While early reasoning steps follow similar trajectories, correct and incorrect solutions diverge systematically at late stages. This late-stage divergence enables mid-reasoning prediction of final-answer correctness with ROC-AUC up to 0.87. Furthermore, we introduce trajectory-based steering, an inference-time intervention framework that enables reasoning correction and length control based on derived ideal trajectories. Together, these results establish reasoning trajectories as a geometric lens for interpreting, predicting, and controlling LLM reasoning behavior.

This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and return a matrix inverse or factorization, derivatives of ODE solutions, and even stochastic derivatives of random functions. It emphasizes practical computational applications, such as large-scale optimization and machine learning, where derivatives must be re-imagined in order to be propagated through complicated calculations. The class also discusses efficiency concerns leading to "adjoint" or "reverse-mode" differentiation (a.k.a. "backpropagation"), and gives a gentle introduction to modern automatic differentiation (AD) techniques.

Reinforcement Learning with Verifiable Rewards (RLVR) has catalyzed significant advances in the reasoning capabilities of Large Language Models (LLMs). However, effectively managing the exploration and exploitation trade-off remains a critical challenge. In this paper, we fully analyze the exploration and exploitation dilemma of extremely hard and easy samples during the training and propose a new fine-grained trade-off mechanism. Concretely, we introduce a perplexity space disentangling strategy that divides the sample space into distinct exploration (high perplexity) and exploitation (low perplexity) subspaces, thereby mining fine-grained samples requiring exploration-exploitation trade-off. Subsequently, we propose a bidirectional reward allocation mechanism with a minimum impact on verification rewards to implement perplexity-guided exploration and exploitation, enabling more stable policy optimization. Finally, we have evaluated our method on two mainstream tasks: mathematical reasoning and function calling, and experimental results demonstrate the superiority of the proposed method, confirming its effectiveness in enhancing LLM performance by fine-grained exploration-exploitation trade-off.

In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and algorithms become more powerful, an intriguing possibility arises - the interplay between human intuition and computer algorithms can lead to discoveries of novel mathematical concepts that would otherwise remain elusive. To realize this perspective, we have developed a massively parallel computer algorithm that discovers an unprecedented number of continued fraction formulas for fundamental mathematical constants. The sheer number of formulas discovered by the algorithm unveils a novel mathematical structure that we call the conservative matrix field. Such matrix fields (1) unify thousands of existing formulas, (2) generate infinitely many new formulas, and most importantly, (3) lead to unexpected relations between different mathematical constants, including multiple integer values of the Riemann zeta function. Conservative matrix fields also enable new mathematical proofs of irrationality. In particular, we can use them to generalize the celebrated proof by Ap\'ery for the irrationality of zeta(3). Utilizing thousands of personal computers worldwide, our computer-supported research strategy demonstrates the power of experimental mathematics, highlighting the prospects of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science.

We discuss methods for modeling eclipsing binary stars using the "physical", "simplified" and "phenomenological" models. There are few realizations of the "physical" Wilson-Devinney (1971) code and its improvements, e.g. Binary Maker, Phoebe. A parameter search using the Monte-Carlo method was realized by Zola et al. (2010), which is efficient in expense of too many evaluations of the test function. We compare existing algorithms of minimization of multi-parametric functions and propose to use a "combined" algorithm, depending on if the Hessian matrix is positively determined. To study methods, a simply fast-computed function resembling the "complete" test function for the physical model. Also we adopt a simplified model of an eclipsing binary at a circular orbit assuming spherical components with an uniform brightness distribution. This model resembles more advanced models in a sense of correlated parameter estimates due to a similar topology of the test function. Such a model may be applied to detached Algol-type systems, where the tidal distortion of components is negligible.

This article establishes a complete approximate axiomatization for the real-closed field R expanded with all differentially-defined functions, including special functions such as sin(x), cos(x), e^x, dots. Every true sentence is provable up to some numerical approximation, and the truth of such approximations converge under mild conditions. Such an axiomatization is a fragment of the axiomatization for differential dynamic logic, and is therefore a finite extension of the axiomatization of real-closed fields. Furthermore, the numerical approximations approximate formulas containing special function symbols by FOL_{R} formulas, improving upon earlier decidability results only concerning closed sentences.

Mathematical formulas are a fundamental and widely used component in various scientific fields, serving as a universal language for expressing complex concepts and relationships. While state-of-the-art transformer models excel in processing and understanding natural language, they encounter challenges with mathematical notation, which involves a complex structure and diverse representations. This study focuses on the development of specialized training datasets to enhance the encoding of mathematical content. We introduce Math Mutator (MAMUT), a framework capable of generating equivalent and falsified versions of a given mathematical formula in LaTeX notation, effectively capturing the mathematical variety in notation of the same concept. Based on MAMUT, we have generated four large mathematical datasets containing diverse notation, which can be used to train language models with enhanced mathematical embeddings.

There are now several comprehensive web applications, stand-alone computer programs and computer algebra functions that, given a floating point number such as 6.518670730718491, can return concise nonfloat constants such as 3 arctan 2 + ln 9 + 1, that closely approximate the float. Examples include AskConstants, Inverse Symbolic Calculator, the Maple identify function, MESearch, OEIS, RIES, and WolframAlpha. Usefully often such a result is the exact limit as the float is computed with increasing precision. Therefore these program results are candidates for proving an exact result that you could not otherwise compute or conjecture without the program. Moreover, candidates that are not the exact limit can be provable bounds, or convey qualitative insight, or suggest series that they truncate, or provide sufficiently close efficient approximations for subsequent computation. This article describes some of these programs, how they work, and how best to use each of them. Almost everyone who uses or should use mathematical software can benefit from acquaintance with several such programs, because these programs differ in the sets of constants that they can return.

Two kinds of novel generalizations of Nesbitt's inequality are explored in various cases regarding dimensions and parameters in this article. Some other cases are also discussed elaborately by using the semiconcave-semiconvex theorem. The general inequalities are then employed to deduce some alternate inequalities and mathematical competition questions. At last, a relation about Hurwitz-Lerch zeta functions is obtained.

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces tX with scale parameter t, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the moment which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.

Code generation is a core application of large language models (LLMs), yet LLMs still frequently fail on complex programming tasks. Given its success in mathematical reasoning, test-time scaling approaches such as Process Reward Model (PRM)-based Best-of-N selection offer a promising way to improve performance. However, existing PRMs remain ineffective for code generation due to the lack of meaningful step decomposition in code and the noise of Monte Carlo-estimated partial-solution correctness scores (rewards). To address these challenges, we propose FunPRM. FunPRM prompts LLMs to encourage modular code generation organized into functions, with functions treated as PRM reasoning steps. Furthermore, FunPRM introduces a novel meta-learning-based reward correction mechanism that leverages clean final-solution rewards obtained via a unit-test-based evaluation system to purify noisy partial-solution rewards. Experiments on LiveCodeBench and BigCodeBench demonstrate that FunPRM consistently outperforms existing test-time scaling methods across five base LLMs, notably achieving state-of-the-art performance on LiveCodeBench when combined with O4-mini. Furthermore, FunPRM produces code that is more readable and reusable for developers.

We introduce Functional Diffusion Processes (FDPs), which generalize score-based diffusion models to infinite-dimensional function spaces. FDPs require a new mathematical framework to describe the forward and backward dynamics, and several extensions to derive practical training objectives. These include infinite-dimensional versions of Girsanov theorem, in order to be able to compute an ELBO, and of the sampling theorem, in order to guarantee that functional evaluations in a countable set of points are equivalent to infinite-dimensional functions. We use FDPs to build a new breed of generative models in function spaces, which do not require specialized network architectures, and that can work with any kind of continuous data. Our results on real data show that FDPs achieve high-quality image generation, using a simple MLP architecture with orders of magnitude fewer parameters than existing diffusion models.

Current mathematical reasoning benchmarks for large language models (LLMs) are approaching saturation, with some achieving > 90% accuracy, and are increasingly compromised by training-set contamination. We introduce Putnam-AXIOM, a benchmark of 522 university-level competition problems drawn from the prestigious William Lowell Putnam Mathematical Competition, and Putnam-AXIOM Variation, an unseen companion set of 100 functional variants generated by programmatically perturbing variables and constants. The variation protocol produces an unlimited stream of equally difficult, unseen instances -- yielding a contamination-resilient test bed. On the Original set, OpenAI's o1-preview -- the strongest evaluated model -- scores 41.9%, but its accuracy drops by 19.6% (46.8% relative decrease) on the paired Variations. The remaining eighteen models show the same downward trend, ten of them with non-overlapping 95% confidence intervals. These gaps suggest memorization and highlight the necessity of dynamic benchmarks. We complement "boxed" accuracy with Teacher-Forced Accuracy (TFA), a lightweight metric that directly scores reasoning traces and automates natural language proof evaluations. Putnam-AXIOM therefore provides a rigorous, contamination-resilient evaluation framework for assessing advanced mathematical reasoning of LLMs. Data and evaluation code are publicly available at https://github.com/brando90/putnam-axiom.

Mathematical reasoning is a primary indicator of large language models (LLMs) intelligence. However, existing LLMs exhibit failures of robustness and generalization. This paper attributes these deficiencies to spurious reasoning, i.e., producing answers from superficial features. To address this challenge, we propose the AdaR framework to enable adaptive reasoning, wherein models rely on problem-solving logic to produce answers. AdaR synthesizes logically equivalent queries by varying variable values, and trains models with RLVR on these data to penalize spurious logic while encouraging adaptive logic. To improve data quality, we extract the problem-solving logic from the original query and generate the corresponding answer by code execution, then apply a sanity check. Experimental results demonstrate that AdaR improves robustness and generalization, achieving substantial improvement in mathematical reasoning while maintaining high data efficiency. Analysis indicates that data synthesis and RLVR function in a coordinated manner to enable adaptive reasoning in LLMs. Subsequent analyses derive key design insights into the effect of critical factors and the applicability to instruct LLMs. Our project is available at https://github.com/LaiZhejian/AdaR

We propose a general framework for conditional sampling in PDE-based inverse problems, targeting the recovery of whole solutions from extremely sparse or noisy measurements. This is accomplished by a function-space diffusion model and plug-and-play guidance for conditioning. Our method first trains an unconditional discretization-agnostic denoising model using neural operator architectures. At inference, we refine the samples to satisfy sparse observation data via a gradient-based guidance mechanism. Through rigorous mathematical analysis, we extend Tweedie's formula to infinite-dimensional Hilbert spaces, providing the theoretical foundation for our posterior sampling approach. Our method (FunDPS) accurately captures posterior distributions in function spaces under minimal supervision and severe data scarcity. Across five PDE tasks with only 3% observation, our method achieves an average 32% accuracy improvement over state-of-the-art fixed-resolution diffusion baselines while reducing sampling steps by 4x. Furthermore, multi-resolution fine-tuning ensures strong cross-resolution generalizability. To the best of our knowledge, this is the first diffusion-based framework to operate independently of discretization, offering a practical and flexible solution for forward and inverse problems in the context of PDEs. Code is available at https://github.com/neuraloperator/FunDPS

We present a unified representation of the most popular neural network activation functions. Adopting Mittag-Leffler functions of fractional calculus, we propose a flexible and compact functional form that is able to interpolate between various activation functions and mitigate common problems in training neural networks such as vanishing and exploding gradients. The presented gated representation extends the scope of fixed-shape activation functions to their adaptive counterparts whose shape can be learnt from the training data. The derivatives of the proposed functional form can also be expressed in terms of Mittag-Leffler functions making it a suitable candidate for gradient-based backpropagation algorithms. By training multiple neural networks of different complexities on various datasets with different sizes, we demonstrate that adopting a unified gated representation of activation functions offers a promising and affordable alternative to individual built-in implementations of activation functions in conventional machine learning frameworks.

Implicit Chain-of-Thought (CoT) reduces the inference cost of large language models by internalizing the explicit rationales. However, existing approaches typically lack alignment with explicit rationales and adaptivity to example complexity. In this work, we propose CIRF (\underline{Chain-of-thoughts Into Reusable Functional units}), an implicit CoT framework that performs reasoning as a dynamic sequence of discrete functional tokens. CIRF assigns a functional token to each semantically coherent reasoning unit in explicit CoT traces. The model is then fine-tuned to autoregressively generate functional tokens and their optional results, followed by the final answer. This design aligns latent reasoning with a sequence of functional units, facilitating parallel training, explicit rationale alignment, and adaptive reasoning. Extensive experiments on mathematical, symbolic, and commonsense reasoning benchmarks show that CIRF provides a favorable accuracy-latency trade-off compared with state-of-the-art implicit CoT methods. Further analyses demonstrate that CIRF constructs distinct, interpretable functional tokens, leading to consistent performance improvements.

We introduce Exhaustive Symbolic Integration (ESI), a method that enumerates all symbolic functions up to a given complexity k within a specified operator basis and determines which admit closed-form antiderivatives within the same class. This allows us to compute the "integrability fraction" ρ(k) (the fraction of functions whose derivatives lie within the same class), which we do for five operator bases including combinations of rational functions, powers, exponentials, logarithms and trigonometric functions. We find that ρ(k) declines at high complexity and that the operator basis has a dramatic effect -- in particular, adding the logarithm boosts ρ(k) by a factor of sim3 and produces or exacerbates a clear peak at k=6. We also deploy ESI as a novel integration algorithm, identifying three integrals that resist SymPy, Mathematica, RUBI, FriCAS, Maxima and Giac under all tested strategies. When an antiderivative can be found by multiple methods, ESI often returns the simplest form. These results reveal that the landscape of symbolic integrability is shaped primarily by the choice of operators, and that exhaustive enumeration can systematically discover integrable forms -- including novel ones -- that elude computer albegra systems.

Recently, deep learning-based algorithms are widely adopted due to the advantage of being able to establish anomaly detection models without or with minimal domain knowledge of the task. Instead, to train the artificial neural network more stable, it should be better to define the appropriate neural network structure or the loss function. For the training anomaly detection model, the mean squared error (MSE) function is adopted widely. On the other hand, the novel loss function, logarithmic mean squared error (LMSE), is proposed in this paper to train the neural network more stable. This study covers a variety of comparisons from mathematical comparisons, visualization in the differential domain for backpropagation, loss convergence in the training process, and anomaly detection performance. In an overall view, LMSE is superior to the existing MSE function in terms of strongness of loss convergence, anomaly detection performance. The LMSE function is expected to be applicable for training not only the anomaly detection model but also the general generative neural network.

Real-world phenomena do not generate arbitrary variability: their signals concentrate on compact, low-variability subsets of functional space, enabling rapid generalization from few examples. A small child can recognize a dog after extremely limited exposure because the perceptual manifold of "dog" is compact, structured, and low-dimensional. We formalize this principle through a deterministic functional-topological framework in which the set of valid realizations produced by a physical process forms a compact subset of a Banach space, endowed with stable invariants, a finite Hausdorff radius, and an induced continuous perceptual functional. This geometry provides explicit limits on knowledge, conditions for identifiability, and guarantees for generalization from sparse evidence -- properties fundamental to both natural and artificial intelligence. Across electromechanical, electrochemical, and physiological domains, we show that real-world processes consistently generate compact perceptual manifolds with the same geometric characteristics. Their boundaries can be discovered in a fully self-supervised manner as the empirical radius saturates with increasing sampling, even when the governing equations are unknown. These results demonstrate that deterministic functional topology offers a unified mathematical foundation for perception, representation, and world-model construction. It provides a geometric explanation for why biological learners and self-supervised AI systems can generalize from few observations, and establishes compact perceptual manifolds as a fundamental building block for future AI architectures. Finally, this work unifies biological perception and modern self-supervised models under a single geometric principle: both derive their generalization ability from the compactness and invariants of real-world perceptual manifolds.

In this work, we propose Reinforced Functional Token Tuning (RFTT), a novel reinforced fine-tuning framework that empowers Large Language Models (LLMs) with self-play learn-to-reason capabilities. Unlike prior prompt-driven reasoning efforts, RFTT embeds a rich set of learnable functional tokens (e.g., <analyze>, <verify>, <refine>) directly into the model vocabulary, enabling chain-of-thought construction with diverse human-like reasoning behaviors. Specifically, RFTT comprises two phases: (1) supervised fine-tuning performs prompt-driven tree search to obtain self-generated training data annotated with functional tokens, which warms up the model to learn these tokens for reasoning; and (2) online reinforcement learning further allows the model to explore different reasoning pathways through functional token sampling without relying on prompts, thereby facilitating effective self-improvement for functional reasoning. Extensive experiments demonstrate the superiority of the proposed RFTT on mathematical benchmarks, significantly boosting Qwen-2.5-7B-Instruct (70.6% to 79.8%) and LLaMA-3.1-8B-Instruct (32.2% to 60.2%) on the MATH dataset. Moreover, the performance of RFTT consistently improves with more search rollouts at inference time. Our code is available at https://github.com/sastpg/RFTT.

While it is still unclear if agents with Artificial General Intelligence (AGI) could ever be built, we can already use mathematical models to investigate potential safety systems for these agents. We present an AGI safety layer that creates a special dedicated input terminal to support the iterative improvement of an AGI agent's utility function. The humans who switched on the agent can use this terminal to close any loopholes that are discovered in the utility function's encoding of agent goals and constraints, to direct the agent towards new goals, or to force the agent to switch itself off. An AGI agent may develop the emergent incentive to manipulate the above utility function improvement process, for example by deceiving, restraining, or even attacking the humans involved. The safety layer will partially, and sometimes fully, suppress this dangerous incentive. The first part of this paper generalizes earlier work on AGI emergency stop buttons. We aim to make the mathematical methods used to construct the layer more accessible, by applying them to an MDP model. We discuss two provable properties of the safety layer, and show ongoing work in mapping it to a Causal Influence Diagram (CID). In the second part, we develop full mathematical proofs, and show that the safety layer creates a type of bureaucratic blindness. We then present the design of a learning agent, a design that wraps the safety layer around either a known machine learning system, or a potential future AGI-level learning system. The resulting agent will satisfy the provable safety properties from the moment it is first switched on. Finally, we show how this agent can be mapped from its model to a real-life implementation. We review the methodological issues involved in this step, and discuss how these are typically resolved.

In deep metric learning, the Triplet Loss has emerged as a popular method to learn many computer vision and natural language processing tasks such as facial recognition, object detection, and visual-semantic embeddings. One issue that plagues the Triplet Loss is network collapse, an undesirable phenomenon where the network projects the embeddings of all data onto a single point. Researchers predominately solve this problem by using triplet mining strategies. While hard negative mining is the most effective of these strategies, existing formulations lack strong theoretical justification for their empirical success. In this paper, we utilize the mathematical theory of isometric approximation to show an equivalence between the Triplet Loss sampled by hard negative mining and an optimization problem that minimizes a Hausdorff-like distance between the neural network and its ideal counterpart function. This provides the theoretical justifications for hard negative mining's empirical efficacy. In addition, our novel application of the isometric approximation theorem provides the groundwork for future forms of hard negative mining that avoid network collapse. Our theory can also be extended to analyze other Euclidean space-based metric learning methods like Ladder Loss or Contrastive Learning.

Theano is a Python library that allows to define, optimize, and evaluate mathematical expressions involving multi-dimensional arrays efficiently. Since its introduction, it has been one of the most used CPU and GPU mathematical compilers - especially in the machine learning community - and has shown steady performance improvements. Theano is being actively and continuously developed since 2008, multiple frameworks have been built on top of it and it has been used to produce many state-of-the-art machine learning models. The present article is structured as follows. Section I provides an overview of the Theano software and its community. Section II presents the principal features of Theano and how to use them, and compares them with other similar projects. Section III focuses on recently-introduced functionalities and improvements. Section IV compares the performance of Theano against Torch7 and TensorFlow on several machine learning models. Section V discusses current limitations of Theano and potential ways of improving it.

Recent progress in large language models (LLMs) highlights the power of scaling test-time compute to achieve strong performance on complex tasks, such as mathematical reasoning and code generation. This raises a critical question: how should model training be modified to optimize performance under a subsequent test-time compute strategy and budget? To explore this, we focus on pass@N, a simple test-time strategy that searches for a correct answer in N independent samples. We show, surprisingly, that training with cross-entropy (CE) loss can be {it misaligned} with pass@N in that pass@N accuracy {it decreases} with longer training. We explain the origins of this misalignment in terms of model overconfidence induced by CE, and experimentally verify our prediction of overconfidence as an impediment to scaling test-time compute via pass@N. Furthermore we suggest a principled, modified training loss that is better aligned to pass@N by limiting model confidence and rescuing pass@N test performance. Our algorithm demonstrates improved mathematical reasoning on MATH and MiniF2F benchmarks under several scenarios: (1) providing answers to math questions; and (2) proving theorems by searching over proof trees of varying shapes. Overall our work underscores the importance of co-designing two traditionally separate phases of LLM development: training-time protocols and test-time search and reasoning strategies.

Large language models (LLMs) have become increasingly capable of following instructions and complex reasoning, making prompting a flexible interface for adapting models without parameter updates. Yet prompt design remains labor-intensive and highly sensitive to formatting, phrasing, and instruction order, motivating automated prompt optimization methods that reduce manual effort while preserving inference-time flexibility. However, existing methods often search over prompt candidates or use fixed critique-refine pipelines driven by individual examples or small batches, limiting their ability to capture systematic error patterns and make targeted edits grounded in failure history. We propose Reflective Prompt Tuning (RPT), a framework that uses LLM function calling to simulate the iterative workflow of human prompt engineers. An LLM optimizer calls a diagnostic function that evaluates the target model over an entire optimization set, summarizes recurring failure modes, and returns a structured diagnostic report. The optimizer uses this report, together with an accumulated memory of prior reports, to revise the prompt for the next iteration. RPT further supports confidence-aware optimization by using calibration signals in diagnostic feedback and final prompt selection. Across three reasoning tasks, RPT improves over initial prompts by up to 12.9 points, remains competitive with state of the art, and improves confidence calibration. Our analyses show that RPT is especially effective on multi-hop and mathematical reasoning, producing targeted prompt revisions that align with diagnosed failure patterns and lead to gains in task performance and calibration.

Although Large Language Models (LLMs) and Large Multimodal Models (LMMs) exhibit impressive skills in various domains, their ability for mathematical reasoning within visual contexts has not been formally examined. Equipping LLMs and LMMs with this capability is vital for general-purpose AI assistants and showcases promising potential in education, data analysis, and scientific discovery. To bridge this gap, we present MathVista, a benchmark designed to amalgamate challenges from diverse mathematical and visual tasks. We first taxonomize the key task types, reasoning skills, and visual contexts from the literature to guide our selection from 28 existing math-focused and visual question answering datasets. Then, we construct three new datasets, IQTest, FunctionQA, and PaperQA, to accommodate for missing types of visual contexts. The problems featured often require deep visual understanding beyond OCR or image captioning, and compositional reasoning with rich domain-specific tools, thus posing a notable challenge to existing models. We conduct a comprehensive evaluation of 11 prominent open-source and proprietary foundation models (LLMs, LLMs augmented with tools, and LMMs), and early experiments with GPT-4V. The best-performing model, Multimodal Bard, achieves only 58% of human performance (34.8% vs 60.3%), indicating ample room for further improvement. Given this significant gap, MathVista fuels future research in the development of general-purpose AI agents capable of tackling mathematically intensive and visually rich real-world tasks. Preliminary tests show that MathVista also presents challenges to GPT-4V, underscoring the benchmark's importance. The project is available at https://mathvista.github.io/.

Best-of-N decoding methods instruct large language models (LLMs) to generate multiple solutions, score each using a scoring function, and select the highest scored as the final answer to mathematical reasoning problems. However, this repeated independent process often leads to the same mistakes, making the selected solution still incorrect. We propose a novel prompting method named Stepwise Correction (StepCo) that helps LLMs identify and revise incorrect steps in their generated reasoning paths. It iterates verification and revision phases that employ a process-supervised verifier. The verify-then-revise process not only improves answer correctness but also reduces token consumption with fewer paths needed to generate. With StepCo, a series of LLMs demonstrate exceptional performance. Notably, using GPT-4o as the backend LLM, StepCo achieves an average accuracy of 94.1 across eight datasets, significantly outperforming the state-of-the-art Best-of-N method by +2.4, while reducing token consumption by 77.8%.

The simulation of human neurons and neurotransmission mechanisms has been realized in deep neural networks based on the theoretical implementations of activation functions. However, recent studies have reported that the threshold potential of neurons exhibits different values according to the locations and types of individual neurons, and that the activation functions have limitations in terms of representing this variability. Therefore, this study proposes a simple yet effective activation function that facilitates different thresholds and adaptive activations according to the positions of units and the contexts of inputs. Furthermore, the proposed activation function mathematically exhibits a more generalized form of Swish activation function, and thus we denoted it as Adaptive SwisH (ASH). ASH highlights informative features that exhibit large values in the top percentiles in an input, whereas it rectifies low values. Most importantly, ASH exhibits trainable, adaptive, and context-aware properties compared to other activation functions. Furthermore, ASH represents general formula of the previously studied activation function and provides a reasonable mathematical background for the superior performance. To validate the effectiveness and robustness of ASH, we implemented ASH into many deep learning models for various tasks, including classification, detection, segmentation, and image generation. Experimental analysis demonstrates that our activation function can provide the benefits of more accurate prediction and earlier convergence in many deep learning applications.

Activation Functions introduce non-linearity in the deep neural networks. This nonlinearity helps the neural networks learn faster and efficiently from the dataset. In deep learning, many activation functions are developed and used based on the type of problem statement. ReLU's variants, SWISH, and MISH are goto activation functions. MISH function is considered having similar or even better performance than SWISH, and much better than ReLU. In this paper, we propose an activation function named APTx which behaves similar to MISH, but requires lesser mathematical operations to compute. The lesser computational requirements of APTx does speed up the model training, and thus also reduces the hardware requirement for the deep learning model. Source code: https://github.com/mr-ravin/aptx_activation

We develop a unified theoretical framework for sparse knowledge distillation based on probability-domain softening operators. While the equivalence p^{1/T} propto softmax(z/T) is well known, our contribution is an operator-level analytical framework built on this foundation rather than the equivalence itself. The framework comprises four core components: (i) operator-agnostic bias--variance decompositions that characterize when sparse students outperform dense teachers, (ii) a homotopy path formalization of multi-stage pruning in function space explaining why iterative compression succeeds where one-shot pruning fails, (iii) convergence guarantees establishing O(1/n) rates for n-stage distillation with explicit parameter dependence, and (iv) equivalence class characterizations identifying distinct probability-domain operators that yield identical student models under capacity constraints. We introduce an axiomatic definition of probability-domain softening operators based on ranking preservation, continuity, entropy monotonicity, identity, and boundary behavior, and show that multiple non-equivalent operator families satisfy these axioms. All learning-theoretic guarantees are shown to hold uniformly across this operator class, independent of implementation details. These results provide theoretical grounding for black-box teacher distillation, partial-access settings such as top-k truncation and text-only outputs, and privacy-preserving model compression.

The rapid development of large language models (LLMs) has spurred extensive research into their domain-specific capabilities, particularly mathematical reasoning. However, most open-source LLMs focus solely on mathematical reasoning, neglecting the integration with visual injection, despite the fact that many mathematical tasks rely on visual inputs such as geometric diagrams, charts, and function plots. To fill this gap, we introduce MultiMath-7B, a multimodal large language model that bridges the gap between math and vision. MultiMath-7B is trained through a four-stage process, focusing on vision-language alignment, visual and math instruction-tuning, and process-supervised reinforcement learning. We also construct a novel, diverse and comprehensive multimodal mathematical dataset, MultiMath-300K, which spans K-12 levels with image captions and step-wise solutions. MultiMath-7B achieves state-of-the-art (SOTA) performance among open-source models on existing multimodal mathematical benchmarks and also excels on text-only mathematical benchmarks. Our model and dataset are available at {blue{https://github.com/pengshuai-rin/MultiMath}}.

We have recently witnessed a number of impressive results on hard mathematical reasoning problems with language models. At the same time, the robustness of these models has also been called into question; recent works have shown that models can rely on shallow patterns in the problem description when generating a solution. Building on the idea of behavioral testing, we propose a novel framework, which pins down the causal effect of various factors in the input, e.g., the surface form of the problem text, the operands, and math operators on the output solution. By grounding the behavioral analysis in a causal graph describing an intuitive reasoning process, we study the behavior of language models in terms of robustness and sensitivity to direct interventions in the input space. We apply our framework on a test bed of math word problems. Our analysis shows that robustness does not appear to continuously improve as a function of size, but the GPT-3 Davinci models (175B) achieve a dramatic improvement in both robustness and sensitivity compared to all other GPT variants.

We generalize quasi-arithmetic means beyond scalars by considering the gradient map of a Legendre type real-valued function. The gradient map of a Legendre type function is proven strictly comonotone with a global inverse. It thus yields a generalization of strictly mononotone and differentiable functions generating scalar quasi-arithmetic means. Furthermore, the Legendre transformation gives rise to pairs of dual quasi-arithmetic averages via the convex duality. We study the invariance and equivariance properties under affine transformations of quasi-arithmetic averages via the lens of dually flat spaces of information geometry. We show how these quasi-arithmetic averages are used to express points on dual geodesics and sided barycenters in the dual affine coordinate systems. We then consider quasi-arithmetic mixtures and describe several parametric and non-parametric statistical models which are closed under the quasi-arithmetic mixture operation.

We present a novel approach to symbolic regression using vision-capable large language models (LLMs) and the ideas behind Google DeepMind's Funsearch. The LLM is given a plot of a univariate function and tasked with proposing an ansatz for that function. The free parameters of the ansatz are fitted using standard numerical optimisers, and a collection of such ansätze make up the population of a genetic algorithm. Unlike other symbolic regression techniques, our method does not require the specification of a set of functions to be used in regression, but with appropriate prompt engineering, we can arbitrarily condition the generative step. By using Kolmogorov Arnold Networks (KANs), we demonstrate that ``univariate is all you need'' for symbolic regression, and extend this method to multivariate functions by learning the univariate function on each edge of a trained KAN. The combined expression is then simplified by further processing with a language model.

The rapid advancements in Vision-Language Models (VLMs) have shown great potential in tackling mathematical reasoning tasks that involve visual context. Unlike humans who can reliably apply solution steps to similar problems with minor modifications, we found that SOTA VLMs like GPT-4o can consistently fail in these scenarios, revealing limitations in their mathematical reasoning capabilities. In this paper, we investigate the mathematical reasoning robustness in VLMs and evaluate how well these models perform under different variants of the same question, such as changes in visual numerical values or function graphs. While several vision-based math benchmarks have been developed to assess VLMs' problem-solving capabilities, these benchmarks contain only static sets of problems and cannot easily evaluate mathematical reasoning robustness. To fill this gap, we introduce DynaMath, a dynamic visual math benchmark designed for in-depth assessment of VLMs. DynaMath includes 501 high-quality, multi-topic seed questions, each represented as a Python program. Those programs are carefully designed and annotated to enable the automatic generation of a much larger set of concrete questions, including many different types of visual and textual variations. DynaMath allows us to evaluate the generalization ability of VLMs, by assessing their performance under varying input conditions of a seed question. We evaluated 14 SOTA VLMs with 5,010 generated concrete questions. Our results show that the worst-case model accuracy, defined as the percentage of correctly answered seed questions in all 10 variants, is significantly lower than the average-case accuracy. Our analysis emphasizes the need to study the robustness of VLMs' reasoning abilities, and DynaMath provides valuable insights to guide the development of more reliable models for mathematical reasoning.

Discovering governing equations from observational data remains a fundamental challenge in scientific modeling, particularly when the underlying mathematical structure is unknown. Traditional sparse identification methods like SINDy excel at discovering parsimonious models but require researchers to specify candidate basis functions a priori, a limitation that often leads to model failure when critical terms are omitted or when systems exhibit unconventional dynamics. Purely symbolic regression approaches offer unlimited flexibility but struggle with noise sensitivity and frequently produce overly complex, unstable equations. We present AutoSINDy, a hybrid Discovery-then-Solve framework that combines the exploratory power of symbolic regression with the robust sparsity-promoting capabilities of SINDy. Our method operates in three stages: (1) PySR-based symbolic regression discovers candidate functional forms from bootstrapped data chunks; (2) a curation pipeline decomposes, expands, and filters these expressions using collinearity analysis to construct a minimal yet comprehensive library; and (3) SINDy identifies sparse governing equations from this custom-tailored library. Extensive experiments across canonical nonlinear systems demonstrate that AutoSINDy consistently recovers ground-truth equations even under high observational noise, achieving a ground-truth recovery rate of 92.8% across all trials. Compared with standard SINDy using enriched libraries and standalone symbolic regression, AutoSINDy achieves higher predictive accuracy, superior generalization to unseen trajectories, and substantially lower symbolic complexity.

Large language models for mathematical reasoning are typically trained with outcome-based rewards, which credit only the final answer. In our experiments, we observe that this paradigm is highly susceptible to reward hacking, leading to a substantial overestimation of a model's reasoning ability. This is evidenced by a high incidence of false positives - solutions that reach the correct final answer through an unsound reasoning process. Through a systematic analysis with human verification, we establish a taxonomy of these failure modes, identifying patterns like Miracle Steps - abrupt jumps to a correct output without a valid preceding derivation. Probing experiments suggest a strong association between these Miracle Steps and memorization, where the model appears to recall the answer directly rather than deriving it. To mitigate this systemic issue, we introduce the Rubric Reward Model (RRM), a process-oriented reward function that evaluates the entire reasoning trajectory against problem-specific rubrics. The generative RRM provides fine-grained, calibrated rewards (0-1) that explicitly penalize logical flaws and encourage rigorous deduction. When integrated into a reinforcement learning pipeline, RRM-based training consistently outperforms outcome-only supervision across four math benchmarks. Notably, it boosts Verified Pass@1024 on AIME2024 from 26.7% to 62.6% and reduces the incidence of Miracle Steps by 71%. Our work demonstrates that rewarding the solution process is crucial for building models that are not only more accurate but also more reliable.

Large Language Models have achieved remarkable success in natural language processing tasks, with Reinforcement Learning playing a key role in adapting them to specific applications. However, obtaining ground truth answers for training LLMs in mathematical problem-solving is often challenging, costly, and sometimes unfeasible. This research delves into the utilization of format and length as surrogate signals to train LLMs for mathematical problem-solving, bypassing the need for traditional ground truth answers.Our study shows that a reward function centered on format correctness alone can yield performance improvements comparable to the standard GRPO algorithm in early phases. Recognizing the limitations of format-only rewards in the later phases, we incorporate length-based rewards. The resulting GRPO approach, leveraging format-length surrogate signals, not only matches but surpasses the performance of the standard GRPO algorithm relying on ground truth answers in certain scenarios, achieving 40.0\% accuracy on AIME2024 with a 7B base model. Through systematic exploration and experimentation, this research not only offers a practical solution for training LLMs to solve mathematical problems and reducing the dependence on extensive ground truth data collection, but also reveals the essence of why our label-free approach succeeds: base model is like an excellent student who has already mastered mathematical and logical reasoning skills, but performs poorly on the test paper, it simply needs to develop good answering habits to achieve outstanding results in exams , in other words, to unlock the capabilities it already possesses.

Over the last decades, deep neural networks based-models became the dominant paradigm in machine learning. Further, the use of artificial neural networks in symbolic learning has been seen as increasingly relevant recently. To study the capabilities of neural networks in the symbolic AI domain, researchers have explored the ability of deep neural networks to learn mathematical constructions, such as addition and multiplication, logic inference, such as theorem provers, and even the execution of computer programs. The latter is known to be too complex a task for neural networks. Therefore, the results were not always successful, and often required the introduction of biased elements in the learning process, in addition to restricting the scope of possible programs to be executed. In this work, we will analyze the ability of neural networks to learn how to execute programs as a whole. To do so, we propose a different approach. Instead of using an imperative programming language, with complex structures, we use the Lambda Calculus (λ-Calculus), a simple, but Turing-Complete mathematical formalism, which serves as the basis for modern functional programming languages and is at the heart of computability theory. We will introduce the use of integrated neural learning and lambda calculi formalization. Finally, we explore execution of a program in λ-Calculus is based on reductions, we will show that it is enough to learn how to perform these reductions so that we can execute any program. Keywords: Machine Learning, Lambda Calculus, Neurosymbolic AI, Neural Networks, Transformer Model, Sequence-to-Sequence Models, Computational Models

Neural networks have proven to be a highly effective tool for solving complex problems in many areas of life. Recently, their importance and practical usability have further been reinforced with the advent of deep learning. One of the important conditions for the success of neural networks is the choice of an appropriate activation function introducing non-linearity into the model. Many types of these functions have been proposed in the literature in the past, but there is no single comprehensive source containing their exhaustive overview. The absence of this overview, even in our experience, leads to redundancy and the unintentional rediscovery of already existing activation functions. To bridge this gap, our paper presents an extensive survey involving 400 activation functions, which is several times larger in scale than previous surveys. Our comprehensive compilation also references these surveys; however, its main goal is to provide the most comprehensive overview and systematization of previously published activation functions with links to their original sources. The secondary aim is to update the current understanding of this family of functions.

Large Language Models (LLMs) have made notable progress in automated theorem proving, yet existing formal benchmarks remain limited in both mathematical coverage and difficulty. Most are concentrated in areas that are easier to formalize, such as algebra and elementary number theory, and provide limited coverage of subfields that require deeper reasoning, including mathematical analysis. To address this gap, we introduce MA-ProofBench, to the best of our knowledge, the first formal theorem-proving benchmark dedicated to Mathematical Analysis. The benchmark contains 200 formalized theorems covering 6 core topics and 27 subcategories, including measure and integration theory, complex analysis, and functional analysis. The problems are divided into two difficulty levels, an undergraduate level (Level I, 100 problems) and a Ph.D. qualifying level (Level II, 100 problems), to evaluate how well LLMs perform formal reasoning at different mathematical depths. Each problem is constructed through a human-led, LLM-assisted formalization pipeline followed by independent expert review, ensuring that the formal statements remain faithful to the original mathematics. We evaluate a range of recent general-purpose reasoning models and formal theorem provers on MA-ProofBench. However, most models perform poorly: even the best-performing model, GPT-5.5, achieves only 16% Pass@8 on Level I and 5% on Level II, while most models stay close to 0% on Level II. Further analysis identifies Mathlib hallucinations and incomplete proofs as the two dominant failure modes, while an evaluation on the natural-language version of the benchmark exposes a clear gap between informal and formal reasoning. MA-ProofBench is intended to serve as a reliable reference for tracking progress in formal mathematical reasoning in advanced domains.

Recent progress in Multi-modal Large Language Models (MLLMs) has enabled step-by-step multi-modal mathematical reasoning by performing visual operations based on the textual instructions. A promising approach uses code as an intermediate representation to precisely express and manipulate the images in the reasoning steps. However, existing evaluations focus mainly on text-only reasoning outputs, leaving the MLLM's ability to perform accurate visual operations via code largely unexplored. This work takes a first step toward addressing that gap by evaluating MLLM's code-based capabilities in multi-modal mathematical reasoning.Specifically, our framework focuses on two key evaluation aspects: (1) Multi-modal Code Generation (MCG) evaluates the model's ability to accurately understand and construct visualizations from scratch. (2) Multi-modal Code Editing (MCE) assesses the model's capacity for fine-grained operations, which include three types: Deletion, Modification and Annotation. To evaluate the above tasks, we incorporate a dataset that covers the five most popular types of mathematical figures, including geometric diagrams, function plots, and three types of statistical charts, to provide a comprehensive and effective measurement of existing MLLMs. Our experimental evaluation involves nine mainstream MLLMs, and the results reveal that existing models still lag significantly behind human performance in performing fine-grained visual operations.

Motivated by W. P. Thurston, we ask: What is the shape of a meromorphic function on a simply connected Riemann surface Ω_z? We consider Speiser functions, i.e. meromorphic functions on a simply connected Riemann surface, that have a finite number q at least 2 of singular (critical or asymptotic) values. As a first result, we make precise the correspondence between: Speiser functions w(z), Speiser Riemann surfaces R_w(z), Speiser q-tessellation, and analytic Speiser graphs of index q. As the second main result, we characterize tessellations with alternating colors (equivalently abstract pre-Speiser graphs) that are realized by Speiser functions on Ω_z. The characterization is in terms of the q-regular extension problem of bipartite planar graphs. As third main results, the Speiser Riemann surface R_w(z) can be constructed by isometric glueing of a finite number of types of sheets, where each sheet is a maximal domain of single-valuedness of the inverse of w(z). Furthermore, a unique decomposition of R_w(z) into maximal logarithmic towers and a soul is provided. Using vector fields we recognize that logarithmic towers come in two flavors: exponential or h-tangent blocks, directly related to the exponential or the hyperbolic tangent functions on the upper half plane. The surface R_w(z) of a finite Speiser function is characterized by surgery of a rational block and a finite number of exponential or h-tangent blocks.

Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case the wave function is constructed such that the phase factor is the same as the Hamilton's principle function "S". To demonstrate our proposed approach two examples are investigated in details.

Bessel functions are critical in scientific computing for applications such as machine learning, protein structure modeling, and robotics. However, currently, available routines lack precision or fail for certain input ranges, such as when the order v is large, and GPU-specific implementations are limited. We address the precision limitations of current numerical implementations while dramatically improving the runtime. We propose two novel algorithms for computing the logarithm of modified Bessel functions of the first and second kinds by computing intermediate values on a logarithmic scale. Our algorithms are robust and never have issues with underflows or overflows while having relative errors on the order of machine precision, even for inputs where existing libraries fail. In C++/CUDA, our algorithms have median and maximum speedups of 45x and 6150x for GPU and 17x and 3403x for CPU, respectively, over the ranges of inputs and third-party libraries tested. Compared to SciPy, the algorithms have median and maximum speedups of 77x and 300x for GPU and 35x and 98x for CPU, respectively, over the tested inputs. The ability to robustly compute a solution and the low relative errors allow us to fit von Mises-Fisher, vMF, distributions to high-dimensional neural network features. This is, e.g., relevant for uncertainty quantification in metric learning. We obtain image feature data by processing CIFAR10 training images with the convolutional layers of a pre-trained ResNet50. We successfully fit vMF distributions to 2048-, 8192-, and 32768-dimensional image feature data using our algorithms. Our approach provides fast and accurate results while existing implementations in SciPy and mpmath fail to fit successfully. Our approach is readily implementable on GPUs, and we provide a fast open-source implementation alongside this paper.

We investigate the mathematical capabilities of ChatGPT by testing it on publicly available datasets, as well as hand-crafted ones, and measuring its performance against other models trained on a mathematical corpus, such as Minerva. We also test whether ChatGPT can be a useful assistant to professional mathematicians by emulating various use cases that come up in the daily professional activities of mathematicians (question answering, theorem searching). In contrast to formal mathematics, where large databases of formal proofs are available (e.g., the Lean Mathematical Library), current datasets of natural-language mathematics, used to benchmark language models, only cover elementary mathematics. We address this issue by introducing a new dataset: GHOSTS. It is the first natural-language dataset made and curated by working researchers in mathematics that (1) aims to cover graduate-level mathematics and (2) provides a holistic overview of the mathematical capabilities of language models. We benchmark ChatGPT on GHOSTS and evaluate performance against fine-grained criteria. We make this new dataset publicly available to assist a community-driven comparison of ChatGPT with (future) large language models in terms of advanced mathematical comprehension. We conclude that contrary to many positive reports in the media (a potential case of selection bias), ChatGPT's mathematical abilities are significantly below those of an average mathematics graduate student. Our results show that ChatGPT often understands the question but fails to provide correct solutions. Hence, if your goal is to use it to pass a university exam, you would be better off copying from your average peer!

Signal Processing (SP) and Machine Learning (ML) rely on good math and coding knowledge, in particular, linear algebra, probability, trigonometry, and complex numbers. A good grasp of these relies on scalar algebra learned in middle school. The ability to understand and use scalar algebra well, in turn, relies on a good foundation in basic arithmetic. Because of various systemic barriers, many students are not able to build a strong foundation in arithmetic in elementary school. This leads them to struggle with algebra and everything after that. Since math learning is cumulative, the gap between those without a strong early foundation and everyone else keeps increasing over the school years and becomes difficult to fill in college. In this article we discuss how SP faculty, students, and professionals can play an important role in starting, and participating in, university-run, or other, out-of-school math support programs to supplement students' learning. Two example programs run by the authors, CyMath at Iowa State and Algebra by 7th Grade (Ab7G) at Purdue, and one run by the Actuarial Foundation, are described. We conclude with providing some simple zero-cost suggestions for public schools that, if adopted, could benefit a much larger number of students than what out-of-school programs can reach.

Mathematical reasoning is a fundamental aspect of human intelligence and is applicable in various fields, including science, engineering, finance, and everyday life. The development of artificial intelligence (AI) systems capable of solving math problems and proving theorems has garnered significant interest in the fields of machine learning and natural language processing. For example, mathematics serves as a testbed for aspects of reasoning that are challenging for powerful deep learning models, driving new algorithmic and modeling advances. On the other hand, recent advances in large-scale neural language models have opened up new benchmarks and opportunities to use deep learning for mathematical reasoning. In this survey paper, we review the key tasks, datasets, and methods at the intersection of mathematical reasoning and deep learning over the past decade. We also evaluate existing benchmarks and methods, and discuss future research directions in this domain.

We show that for infinitely many primes p, there exist dual functions of order k over F_p^n that cannot be approximated in L_infty-distance by polynomial phase functions of degree k-1. This answers in the negative a natural finite-field analog of a problem of Frantzikinakis on L_infty-approximations of dual functions over N (a.k.a. multiple correlation sequences) by nilsequences.

We establish a unified framework for understanding the results on the Weil quadratic form obtained by Yoshida (1992), Bombieri (2001, 2003), Connes--Consani (2023), and Connes--Consani--Moscovici (2025+) from the perspective of the screw function introduced in Suzuki (2023). An advantage of the approach via the screw function is that it provides a method to study the Weil quadratic form, which is originally defined in terms of distributions, by means of continuous functions. Based on this framework, we formulate a conjecture stating that a self-adjoint operator whose eigenvalues are the imaginary parts of the nontrivial zeros of the Riemann zeta function can be obtained as the limit, as a to infty, of self-adjoint operators arising from nonlocal realizations of the first-order differential operator on the finite interval [-a,a]. All these results are obtained without assuming the Riemann Hypothesis. This conjecture may be compared with the limit formula for the Riemann zeta function expressed in terms of zeta-regularized products proposed by Connes, Consani, and Moscovici, and it sheds new light on the spectral-theoretic interpretation of the nontrivial zeros of the Riemann zeta function.

We made a comparative analysis of numerical methods for multidimensional optimization. The main parameter is a number of computations of the test function to reach necessary accuracy, as it is computationally "slow". For complex functions, analytic differentiation by many parameters can cause problems associated with a significant complication of the program and thus slowing its operation. For comparison, we used the methods: "brute force" (or minimization on a regular grid), Monte Carlo, steepest descent, conjugate gradients, Brent's method (golden section search), parabolic interpolation etc. The Monte-Carlo method was applied to the eclipsing binary system AM Leo.

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs

Many intellectual endeavors require mathematical problem solving, but this skill remains beyond the capabilities of computers. To measure this ability in machine learning models, we introduce MATH, a new dataset of 12,500 challenging competition mathematics problems. Each problem in MATH has a full step-by-step solution which can be used to teach models to generate answer derivations and explanations. To facilitate future research and increase accuracy on MATH, we also contribute a large auxiliary pretraining dataset which helps teach models the fundamentals of mathematics. Even though we are able to increase accuracy on MATH, our results show that accuracy remains relatively low, even with enormous Transformer models. Moreover, we find that simply increasing budgets and model parameter counts will be impractical for achieving strong mathematical reasoning if scaling trends continue. While scaling Transformers is automatically solving most other text-based tasks, scaling is not currently solving MATH. To have more traction on mathematical problem solving we will likely need new algorithmic advancements from the broader research community.

Generating functions, which are widely used in combinatorics and probability theory, encode function values into the coefficients of a polynomial. In this paper, we explore their use as a tractable probabilistic model, and propose probabilistic generating circuits (PGCs) for their efficient representation. PGCs are strictly more expressive efficient than many existing tractable probabilistic models, including determinantal point processes (DPPs), probabilistic circuits (PCs) such as sum-product networks, and tractable graphical models. We contend that PGCs are not just a theoretical framework that unifies vastly different existing models, but also show great potential in modeling realistic data. We exhibit a simple class of PGCs that are not trivially subsumed by simple combinations of PCs and DPPs, and obtain competitive performance on a suite of density estimation benchmarks. We also highlight PGCs' connection to the theory of strongly Rayleigh distributions.

Deep learning has been wildly successful in practice and most state-of-the-art machine learning methods are based on neural networks. Lacking, however, is a rigorous mathematical theory that adequately explains the amazing performance of deep neural networks. In this article, we present a relatively new mathematical framework that provides the beginning of a deeper understanding of deep learning. This framework precisely characterizes the functional properties of neural networks that are trained to fit to data. The key mathematical tools which support this framework include transform-domain sparse regularization, the Radon transform of computed tomography, and approximation theory, which are all techniques deeply rooted in signal processing. This framework explains the effect of weight decay regularization in neural network training, the use of skip connections and low-rank weight matrices in network architectures, the role of sparsity in neural networks, and explains why neural networks can perform well in high-dimensional problems.