Daily Papers

Multi-step symbolic reasoning is critical for advancing downstream performance on financial tasks. Yet, benchmarks for systematically evaluating this capability are lacking. Existing datasets like FinQA and ConvFinQA supervise only final numerical answers, without assessing intermediate reasoning steps. To address this, we introduce FinChain, the first symbolic benchmark designed for verifiable Chain-of- Thought (CoT) financial reasoning. Spanning 54 topics across 12 financial domains, Fin- Chain offers five parameterized templates per topic, each varying in reasoning complexity and domain expertise required. Each dataset instance includes an executable Python trace, enabling automatic generation of extensive training data and easy adaptation to other domains. We also introduce ChainEval, a new metric for automatic evaluation of both final answers and intermediate reasoning. Benchmarking 30 LLMs on our dataset, we find that even state-of-the-art models have considerable room for improvement in multi-step financial reasoning. All templates and evaluation metrics for FinChain are available at https: //github.com/mbzuai-nlp/finchain.

Large language models have made significant progress in mathematical reasoning, which serves as an important testbed for AI and could impact scientific research if further advanced. By scaling reasoning with reinforcement learning that rewards correct final answers, LLMs have improved from poor performance to saturating quantitative reasoning competitions like AIME and HMMT in one year. However, this approach faces fundamental limitations. Pursuing higher final answer accuracy doesn't address a key issue: correct answers don't guarantee correct reasoning. Moreover, many mathematical tasks like theorem proving require rigorous step-by-step derivation rather than numerical answers, making final answer rewards inapplicable. To push the limits of deep reasoning, we believe it is necessary to verify the comprehensiveness and rigor of mathematical reasoning. Self-verification is particularly important for scaling test-time compute, especially for open problems without known solutions. Towards self-verifiable mathematical reasoning, we investigate how to train an accurate and faithful LLM-based verifier for theorem proving. We then train a proof generator using the verifier as the reward model, and incentivize the generator to identify and resolve as many issues as possible in their own proofs before finalizing them. To maintain the generation-verification gap as the generator becomes stronger, we propose to scale verification compute to automatically label new hard-to-verify proofs, creating training data to further improve the verifier. Our resulting model, DeepSeekMath-V2, demonstrates strong theorem-proving capabilities, achieving gold-level scores on IMO 2025 and CMO 2024 and a near-perfect 118/120 on Putnam 2024 with scaled test-time compute.

Large language models (LLMs) are increasingly being applied to financial analysis, reporting, investment decision support, risk management, compliance, and professional training. However, robust evaluation of their domain competence in finance remains incomplete. Widely used open benchmarks such as FinQA, ConvFinQA, and TAT-QA have played an important role in advancing financial question answering and numerical reasoning, but they focus primarily on question answering over financial reports and do not provide an explicit hierarchy of professional difficulty. Broader resources, including FinanceBench, PIXIU, FinBen, and FLaME, expand the coverage of financial tasks, yet the problem of evaluating the transition from foundational knowledge to expert-level financial reasoning remains open. In this work, we present FINESSE-Bench, a suite of eight specialized benchmarks comprising 3,993 questions for hierarchical evaluation of financial competencies in LLMs. FINESSE-Bench combines exam-oriented datasets inspired by professional certifications (CFA-like Levels 1-3, CMT-like Level 2, and CFTe-like Level 1), applied trading task collections, and a Russian-language olympiad benchmark. This design enables evaluation of domain breadth, performance degradation as difficulty increases, the ability to solve computational tasks, and model behavior in specialized financial domains. We also describe a unified evaluation protocol covering multiple-choice questions, numerical answers, and short open-ended responses, together with an automated scoring scheme for freeform answers based on the LLM-as-judge paradigm. FINESSE-Bench is intended both as a complement to existing open financial benchmarks and as a tool for more substantive evaluation of professionally relevant financial competencies in large language models.

Recent math benchmarks for large language models (LLMs) such as MathArena indicate that state-of-the-art reasoning models achieve impressive performance on mathematical competitions like AIME, with the leading model, o3-mini, achieving scores comparable to top human competitors. However, these benchmarks evaluate models solely based on final numerical answers, neglecting rigorous reasoning and proof generation which are essential for real-world mathematical tasks. To address this, we introduce the first comprehensive evaluation of full-solution reasoning for challenging mathematical problems. Using expert human annotators, we evaluated several state-of-the-art reasoning models on the six problems from the 2025 USAMO within hours of their release. Our results reveal that all tested models struggled significantly, achieving less than 5% on average. Through detailed analysis of reasoning traces, we identify the most common failure modes and find several unwanted artifacts arising from the optimization strategies employed during model training. Overall, our results suggest that current LLMs are inadequate for rigorous mathematical reasoning tasks, highlighting the need for substantial improvements in reasoning and proof generation capabilities.

Large language models (LLMs) have exhibited their problem-solving abilities in mathematical reasoning. Solving realistic optimization (OPT) problems in application scenarios requires advanced and applied mathematics ability. However, current OPT benchmarks that merely solve linear programming are far from complex realistic situations. In this work, we propose OptiBench, a benchmark for End-to-end optimization problem-solving with human-readable inputs and outputs. OptiBench contains rich optimization problems, including linear and nonlinear programming with or without tabular data, which can comprehensively evaluate LLMs' solving ability. In our benchmark, LLMs are required to call a code solver to provide precise numerical answers. Furthermore, to alleviate the data scarcity for optimization problems, and to bridge the gap between open-source LLMs on a small scale (e.g., Llama-3-8b) and closed-source LLMs (e.g., GPT-4), we further propose a data synthesis method namely ReSocratic. Unlike general data synthesis methods that proceed from questions to answers, \ReSocratic first incrementally synthesizes formatted optimization demonstration with mathematical formulations step by step and then back-translates the generated demonstrations into questions. Based on this, we synthesize the ReSocratic-29k dataset. We further conduct supervised fine-tuning with ReSocratic-29k on multiple open-source models. Experimental results show that ReSocratic-29k significantly improves the performance of open-source models.

Visual question answering on document images that contain textual, visual, and layout information, called document VQA, has received much attention recently. Although many datasets have been proposed for developing document VQA systems, most of the existing datasets focus on understanding the content relationships within a single image and not across multiple images. In this study, we propose a new multi-image document VQA dataset, SlideVQA, containing 2.6k+ slide decks composed of 52k+ slide images and 14.5k questions about a slide deck. SlideVQA requires complex reasoning, including single-hop, multi-hop, and numerical reasoning, and also provides annotated arithmetic expressions of numerical answers for enhancing the ability of numerical reasoning. Moreover, we developed a new end-to-end document VQA model that treats evidence selection and question answering in a unified sequence-to-sequence format. Experiments on SlideVQA show that our model outperformed existing state-of-the-art QA models, but that it still has a large gap behind human performance. We believe that our dataset will facilitate research on document VQA.

We introduce MMCRICBENCH-3K, a benchmark for Visual Question Answering (VQA) on cricket scorecards, designed to evaluate large vision-language models (LVLMs) on complex numerical and cross-lingual reasoning over semi-structured tabular images. MMCRICBENCH-3K comprises 1,463 synthetically generated scorecard images from ODI, T20, and Test formats, accompanied by 1,500 English QA pairs. It includes two subsets: MMCRICBENCH-E-1.5K, featuring English scorecards, and MMCRICBENCH-H-1.5K, containing visually similar Hindi scorecards, with all questions and answers kept in English to enable controlled cross-script evaluation. The task demands reasoning over structured numerical data, multi-image context, and implicit domain knowledge. Empirical results show that even state-of-the-art LVLMs, such as GPT-4o and Qwen2.5VL, struggle on the English subset despite it being their primary training language and exhibit a further drop in performance on the Hindi subset. This reveals key limitations in structure-aware visual text understanding, numerical reasoning, and cross-lingual generalization. The dataset is publicly available via Hugging Face at https://huggingface.co/datasets/DIALab/MMCricBench, to promote LVLM research in this direction.

Numerical reasoning over natural language has been a long-standing goal for the research community. However, cutting-edge language models have proven difficult to reliably generalize to a broad range of numbers, although they have shown proficiency in reasoning over common and simple numbers. In this paper, we propose a novel method to elicit and exploit the numerical reasoning knowledge hidden in pre-trained language models using simple anchor numbers. Concretely, we first leverage simple numbers as anchors to probe the implicitly inferred arithmetic expressions from language models, and then explicitly apply the expressions on complex numbers to get corresponding answers. To inversely elicit arithmetic expressions, we transform and formulate the task as an analytically solvable linear system. Experimental results on several numerical reasoning benchmarks demonstrate that our approach significantly improves numerical reasoning capabilities of existing LMs. More importantly, our approach is training-free and simply works in the inference phase, making it highly portable and achieving consistent performance benefits across a variety of language models (GPT-3, T5, BART, etc) in all zero-shot, few-shot, and fine-tuning scenarios.

Mathematical reasoning in Large Language Models (LLMs) is often evaluated using benchmarks with limited numerical ranges, failing to reflect real-world problem-solving across diverse scales. Furthermore, most existing evaluation methods only compare model outputs to ground-truth answers, obscuring insights into reasoning processes. To address these limitations, we introduce GSM-Ranges, a dataset generator derived from GSM8K that systematically perturbs numerical values in math problems to assess model robustness across varying numerical scales. Additionally, we propose a novel grading methodology that distinguishes between logical and non-logical errors, offering a more precise evaluation of reasoning processes beyond computational accuracy. Our experiments with various models reveal a significant increase in logical error rates-up to 14 percentage points-as numerical complexity rises, demonstrating a general weakness in reasoning with out-of-distribution numerical values. Moreover, while models demonstrate high accuracy on standalone arithmetic tasks, their performance deteriorates substantially when computations are embedded within word problems. These findings provide a comprehensive evaluation of LLMs' mathematical reasoning capabilities and inform future research directions for improving numerical generalization in language models.

Large Language Models (LLMs) are increasingly being explored for clinical question answering and decision support, yet safe deployment critically requires reliable handling of patient measurements in heterogeneous clinical notes. Existing evaluations of LLMs for clinical numerical reasoning provide limited operation-level coverage, restricted primarily to arithmetic computation, and rarely assess the robustness of numerical understanding across clinical note formats. We introduce ClinicNumRobBench, a benchmark of 1,624 context-question instances with ground-truth answers that evaluates four main types of clinical numeracy: value retrieval, arithmetic computation, relational comparison, and aggregation. To stress-test robustness, ClinicNumRobBench presents longitudinal MIMIC-IV vital-sign records in three semantically equivalent representations, including a real-world note-style variant derived from the Open Patients dataset, and instantiates queries using 42 question templates. Experiments on 14 LLMs show that value retrieval is generally strong, with most models exceeding 85% accuracy, while relational comparison and aggregation remain challenging, with some models scoring below 15%. Fine-tuning on medical data can reduce numeracy relative to base models by over 30%, and performance drops under note-style variation indicate LLM sensitivity to format. ClinicNumRobBench offers a rigorous testbed for clinically reliable numerical reasoning. Code and data URL are available on https://github.com/MinhVuong2000/ClinicNumRobBench.

Do large language models (LLMs) solve reasoning tasks by learning robust generalizable algorithms, or do they memorize training data? To investigate this question, we use arithmetic reasoning as a representative task. Using causal analysis, we identify a subset of the model (a circuit) that explains most of the model's behavior for basic arithmetic logic and examine its functionality. By zooming in on the level of individual circuit neurons, we discover a sparse set of important neurons that implement simple heuristics. Each heuristic identifies a numerical input pattern and outputs corresponding answers. We hypothesize that the combination of these heuristic neurons is the mechanism used to produce correct arithmetic answers. To test this, we categorize each neuron into several heuristic types-such as neurons that activate when an operand falls within a certain range-and find that the unordered combination of these heuristic types is the mechanism that explains most of the model's accuracy on arithmetic prompts. Finally, we demonstrate that this mechanism appears as the main source of arithmetic accuracy early in training. Overall, our experimental results across several LLMs show that LLMs perform arithmetic using neither robust algorithms nor memorization; rather, they rely on a "bag of heuristics".

EasyMath is a compact benchmark for practical math reasoning in small language models. It covers thirteen categories, from basic arithmetic and order of operations to word problems, algebraic expressions, edge cases, and omits specialist topics. We tested 23 models (14M to 4B parameters) using exact, numerical, and symbolic checks on free-form answers in a zero-shot setting. Accuracy rises with size and training, chain-of-thought adds modest gains, and consistency improves at scale.

Large language models (LLMs) demonstrate strong general reasoning and language understanding, yet their performance degrades in domains governed by strict formal rules, precise terminology, and legally binding structure. Tax law exemplifies these challenges, as correct answers require exact statutory citation, structured legal argumentation, and numerical accuracy under rigid grading schemes. We algorithmically generate SteuerEx, the first open benchmark derived from authentic German university tax law examinations. SteuerEx comprises 115 expert-validated examination questions spanning six core tax law domains and multiple academic levels, and employs a statement-level, partial-credit evaluation framework that closely mirrors real examination practice. We further present SteuerLLM, a domain-adapted LLM for German tax law trained on a large-scale synthetic dataset generated from authentic examination material using a controlled retrieval-augmented pipeline. SteuerLLM (28B parameters) consistently outperforms general-purpose instruction-tuned models of comparable size and, in several cases, substantially larger systems, demonstrating that domain-specific data and architectural adaptation are more decisive than parameter scale for performance on realistic legal reasoning tasks. All benchmark data, training datasets, model weights, and evaluation code are released openly to support reproducible research in domain-specific legal artificial intelligence. A web-based demo of SteuerLLM is available at https://steuerllm.i5.ai.fau.de.

Large language models are increasingly applied to materials science, yet fundamental questions remain about their reliability and knowledge encoding. Evaluating 25 LLMs across four materials science tasks -- over 200 base and fine-tuned configurations -- we find that output modality fundamentally determines model behavior. For symbolic tasks, fine-tuning converges to consistent, verifiable answers with reduced response entropy, while for numerical tasks, fine-tuning improves prediction accuracy but models remain inconsistent across repeated inference runs, limiting their reliability as quantitative predictors. For numerical regression, we find that better performance can be obtained by extracting embeddings directly from intermediate transformer layers than from model text output, revealing an ``LLM head bottleneck,'' though this effect is property- and dataset-dependent. Finally, we present a longitudinal study of GPT model performance in materials science, tracking four models over 18 months and observing 9--43\% performance variation that poses reproducibility challenges for scientific applications.

Current Vision-Language Models (VLMs) fail at quantitative spatial reasoning because their architectures destroy pixel-level information required for counting and measurements. Vision encoders compress images through patch embeddings, reducing spatial indexing and losing the precise pixel-level tracking required for accurate counting. We present two contributions to address this fundamental limitation. First, we introduce SQuID (Satellite Quantitative Intelligence Dataset), a benchmark of 2,000 satellite image Question-Answer pairs with both numerical range and categorical answers, designed to evaluate quantitative spatial reasoning. The dataset spans three difficulty tiers with annotations automatically generated from human labels and their learned variability. Second, we propose QVLM (Quantitative Vision-Language Model), a code-generation architecture that maintains pixel precision by decoupling language understanding from visual analysis. Instead of encoding images into embeddings, QVLM generates executable code that first calls a segmentation model to obtain pixel-level masks, then operates directly on these masks, preserving spatial indexing throughout the reasoning process. Our experiments show that QVLM using GPT-5 as coder achieves 42.0% accuracy on SQuID compared to 28.1% for a VLM prompted with image-question pairs. Our work reveals that, for quantitative spatial reasoning, architectural decoupling enables better accuracy on quantitative tasks.

While Large Language Models (LLMs) have shown impressive capabilities in numerous Natural Language Processing (NLP) tasks, they still struggle with financial question answering (QA), particularly when numerical reasoning is required. Recently, LLM-based multi-agent frameworks have demonstrated remarkable effectiveness in multi-step reasoning, which is crucial for financial QA tasks as it involves extracting relevant information from tables and text and then performing numerical reasoning on the extracted data to infer answers. In this study, we propose a multi-agent framework incorporating a critic agent that reflects on the reasoning steps and final answers for each question. Additionally, we enhance our system by adding multiple critic agents, each focusing on a specific aspect of the answer. Our results indicate that this framework significantly improves performance compared to single-agent reasoning, with an average performance increase of 15% for the LLaMA3-8B model and 5% for the LLaMA3-70B model. Furthermore, our framework performs on par with, and in some cases surpasses, larger single-agent LLMs such as LLaMA3.1-405B and GPT-4o-mini, though it falls slightly short compared to Claude-3.5 Sonnet. Overall, our framework presents an effective solution to enhance open-source LLMs for financial QA tasks, offering a cost-effective alternative to larger models like Claude-3.5 Sonnet.

Geometric Problem Solving (GPS) remains at the heart of enhancing mathematical reasoning in large language models because it requires the combination of diagrammatic understanding, symbolic manipulation and logical inference. In existing literature, researchers have chiefly focused on synchronising the diagram descriptions with text literals and solving the problem. In this vein, they have either taken a neural, symbolic or neuro-symbolic approach. But this solves only the first two of the requirements, namely diagrammatic understanding and symbolic manipulation, while leaving logical inference underdeveloped. The logical inference is often limited to one chain-of-thought (CoT). To address this weakness in hitherto existing models, this paper proposes MARS-GPS, that generates multiple parallel reasoning rollouts augmented with Python code execution for numerical verification, ranks them using token-level entropy as a confidence signal, and aggregates answers through a multi-stage voting and self-verification pipeline. Empirical results show that MARS-GPS with 8 parallel rollouts achieves 88.8% on Geometry3K, a nearly +11% improvement over the prior state-of-the-art, with accuracy scaling consistently as the number of rollouts increases from 1 to 16 (+6.0% on ablation subset). We provide our code and data in an anonymous repository: https://anonymous.4open.science/r/MARS-GPS-DE55.

The creation of high-quality datasets to improve Large Language Model (LLM) reasoning remains a significant challenge, as current methods often suffer from generating low-quality/incorrect answers and limited information richness from available data sources. To address this, we propose AgenticMath, a novel agentic pipeline for generating high-quality mathematical question-answer pairs to enhance the supervised fine-tuning of LLMs. Our method operates through four stages: (1) Seed Question Filter that selects questions with high information richness, complexity, and clarity; (2) an Agentic Question Rephrase step that employs a multi-agent system to generate diverse, logically consistent paraphrases; (3) an Answer Augment step where rewrite answers using chain-of-thought reasoning to enhance numerical and logical correctness, without reliance on human-provided labels; and (4) a final Question and Answer Evaluation that retains only the most superior pairs. Extensive experiments demonstrate that, fine-tuning 3B-8B parameter LLMs on AgenticMath generated datasets (comprising only 30-60K math samples) achieves competitive or superior performance on diverse in domain and out-of-domain mathematical reasoning benchmarks compared to baselines trained on much more data (e.g., 400K or 2.3M samples). Our work demonstrates that targeted, high-quality data generation is a more efficient path to improving mathematical reasoning in LLMs than large-scale, low-quality alternatives.

Financial reports offer critical insights into a company's operations, yet their extensive length typically spanning 30 40 pages poses challenges for swift decision making in dynamic markets. To address this, we leveraged finetuned Large Language Models (LLMs) to distill key indicators and operational metrics from these reports basis questions from the user. We devised a method to locate critical data, and leverage the FinQA dataset to fine-tune both Llama-2 7B and T5 models for customized question answering. We achieved results comparable to baseline on the final numerical answer, a competitive accuracy in numerical reasoning and calculation.

The ability to understand and work with numbers (numeracy) is critical for many complex reasoning tasks. Currently, most NLP models treat numbers in text in the same way as other tokens---they embed them as distributed vectors. Is this enough to capture numeracy? We begin by investigating the numerical reasoning capabilities of a state-of-the-art question answering model on the DROP dataset. We find this model excels on questions that require numerical reasoning, i.e., it already captures numeracy. To understand how this capability emerges, we probe token embedding methods (e.g., BERT, GloVe) on synthetic list maximum, number decoding, and addition tasks. A surprising degree of numeracy is naturally present in standard embeddings. For example, GloVe and word2vec accurately encode magnitude for numbers up to 1,000. Furthermore, character-level embeddings are even more precise---ELMo captures numeracy the best for all pre-trained methods---but BERT, which uses sub-word units, is less exact.

Language models struggle with handling numerical data and performing arithmetic operations. We hypothesize that this limitation can be partially attributed to non-intuitive textual numbers representation. When a digit is read or generated by a causal language model it does not know its place value (e.g. thousands vs. hundreds) until the entire number is processed. To address this issue, we propose a simple adjustment to how numbers are represented by including the count of digits before each number. For instance, instead of "42", we suggest using "{2:42}" as the new format. This approach, which we term NumeroLogic, offers an added advantage in number generation by serving as a Chain of Thought (CoT). By requiring the model to consider the number of digits first, it enhances the reasoning process before generating the actual number. We use arithmetic tasks to demonstrate the effectiveness of the NumeroLogic formatting. We further demonstrate NumeroLogic applicability to general natural language modeling, improving language understanding performance in the MMLU benchmark.

Large Language Models (LLMs) have demonstrated impressive capabilities in natural language processing tasks, such as text generation and semantic understanding. However, their performance on numerical reasoning tasks, such as basic arithmetic, numerical retrieval, and magnitude comparison, remains surprisingly poor. This gap arises from their reliance on surface-level statistical patterns rather than understanding numbers as continuous magnitudes. Existing benchmarks primarily focus on either linguistic competence or structured mathematical problem-solving, neglecting fundamental numerical reasoning required in real-world scenarios. To bridge this gap, we propose NumericBench, a comprehensive benchmark to evaluate six fundamental numerical capabilities: number recognition, arithmetic operations, contextual retrieval, comparison, summary, and logical reasoning. NumericBench includes datasets ranging from synthetic number lists to the crawled real-world data, addressing challenges like long contexts, noise, and multi-step reasoning. Extensive experiments on state-of-the-art LLMs, including GPT-4 and DeepSeek, reveal persistent weaknesses in numerical reasoning, highlighting the urgent need to improve numerically-aware language modeling. The benchmark is released in: https://github.com/TreeAI-Lab/NumericBench.

While question answering over knowledge bases (KBQA) has shown progress in addressing factoid questions, KBQA with numerical reasoning remains relatively unexplored. In this paper, we focus on the complex numerical reasoning in KBQA and propose a new task, NR-KBQA, which necessitates the ability to perform both multi-hop reasoning and numerical reasoning. We design a logic form in Python format called PyQL to represent the reasoning process of numerical reasoning questions. To facilitate the development of NR-KBQA, we present a large dataset called MarkQA, which is automatically constructed from a small set of seeds. Each question in MarkQA is equipped with its corresponding SPARQL query, alongside the step-by-step reasoning process in the QDMR format and PyQL program. Experimental results of some state-of-the-art QA methods on the MarkQA show that complex numerical reasoning in KBQA faces great challenges.

Although state-of-the-art LLMs can solve math problems, we find that they make errors on numerical comparisons with mixed notation: "Which is larger, 5.7 times 10^2 or 580?" This raises a fundamental question: Do LLMs even know how big these numbers are? We probe the hidden states of several smaller open-source LLMs. A single linear projection of an appropriate hidden layer encodes the log-magnitudes of both kinds of numerals, allowing us to recover the numbers with relative error of about 2.3% (on restricted synthetic text) or 19.06% (on scientific papers). Furthermore, the hidden state after reading a pair of numerals encodes their ranking, with a linear classifier achieving over 90% accuracy. Yet surprisingly, when explicitly asked to rank the same pairs of numerals, these LLMs achieve only 50-70% accuracy, with worse performance for models whose probes are less effective. Finally, we show that incorporating the classifier probe's log-loss as an auxiliary objective during finetuning brings an additional 3.22% improvement in verbalized accuracy over base models, demonstrating that improving models' internal magnitude representations can enhance their numerical reasoning capabilities.

Existing auto-regressive pre-trained language models (PLMs) like T5 and BART, have been well applied to table question answering by UNIFIEDSKG and TAPEX, respectively, and demonstrated state-of-the-art results on multiple benchmarks. However, auto-regressive PLMs are challenged by recent emerging numerical reasoning datasets, such as TAT-QA, due to the error-prone implicit calculation. In this paper, we present TaCube, to pre-compute aggregation/arithmetic results for the table in advance, so that they are handy and readily available for PLMs to answer numerical reasoning questions. TaCube systematically and comprehensively covers a collection of computational operations over table segments. By simply concatenating TaCube to the input sequence of PLMs, it shows significant experimental effectiveness. TaCube promotes the F1 score from 49.6% to 66.2% on TAT-QA and achieves new state-of-the-art results on WikiTQ (59.6% denotation accuracy). TaCube's improvements on numerical reasoning cases are even more notable: on TAT-QA, TaCube promotes the exact match accuracy of BART-large by 39.6% on sum, 52.5% on average, 36.6% on substraction, and 22.2% on division. We believe that TaCube is a general and portable pre-computation solution that can be potentially integrated to various numerical reasoning frameworks

We demonstrate that a neural network pre-trained on text and fine-tuned on code solves mathematics course problems, explains solutions, and generates new questions at a human level. We automatically synthesize programs using few-shot learning and OpenAI's Codex transformer and execute them to solve course problems at 81% automatic accuracy. We curate a new dataset of questions from MIT's largest mathematics courses (Single Variable and Multivariable Calculus, Differential Equations, Introduction to Probability and Statistics, Linear Algebra, and Mathematics for Computer Science) and Columbia University's Computational Linear Algebra. We solve questions from a MATH dataset (on Prealgebra, Algebra, Counting and Probability, Intermediate Algebra, Number Theory, and Precalculus), the latest benchmark of advanced mathematics problems designed to assess mathematical reasoning. We randomly sample questions and generate solutions with multiple modalities, including numbers, equations, and plots. The latest GPT-3 language model pre-trained on text automatically solves only 18.8% of these university questions using zero-shot learning and 30.8% using few-shot learning and the most recent chain of thought prompting. In contrast, program synthesis with few-shot learning using Codex fine-tuned on code generates programs that automatically solve 81% of these questions. Our approach improves the previous state-of-the-art automatic solution accuracy on the benchmark topics from 8.8% to 81.1%. We perform a survey to evaluate the quality and difficulty of generated questions. This work is the first to automatically solve university-level mathematics course questions at a human level and the first work to explain and generate university-level mathematics course questions at scale, a milestone for higher education.

Neural networks have a reputation for being better at solving statistical or approximate problems than at performing calculations or working with symbolic data. In this paper, we show that they can be surprisingly good at more elaborated tasks in mathematics, such as symbolic integration and solving differential equations. We propose a syntax for representing mathematical problems, and methods for generating large datasets that can be used to train sequence-to-sequence models. We achieve results that outperform commercial Computer Algebra Systems such as Matlab or Mathematica.

Large language models (LLMs) can solve an increasing number of complex reasoning tasks while making surprising mistakes in basic numerical understanding and processing (such as 9.11 > 9.9). The latter ability is essential for tackling complex arithmetic and mathematical problems and serves as a foundation for most reasoning tasks, but previous work paid little attention to it or only discussed several restricted tasks (like integer addition). In this paper, we comprehensively investigate the numerical understanding and processing ability (NUPA) of LLMs. Firstly, we introduce a benchmark covering four common numerical representations and 17 distinct numerical tasks in four major categories, resulting in 41 meaningful combinations in total. These tasks are derived from primary and secondary education curricula, encompassing nearly all everyday numerical understanding and processing scenarios, and the rules of these tasks are very simple and clear. Through the benchmark, we find that current LLMs fail frequently in many of the tasks. To study the problem, we train small models with existing and potential techniques for enhancing NUPA (such as tokenizers, PEs, and number formats), comprehensively evaluating their effectiveness using our testbed. We also finetune practical-scale LLMs on our proposed NUPA tasks and find that 1) naive finetuning can improve NUPA a lot on many but not all tasks, and 2) surprisingly, techniques designed to enhance NUPA prove ineffective for finetuning pretrained models. We further explore the impact of chain-of-thought techniques on NUPA. Our work provides a more detailed and comprehensive understanding of NUPA in LLMs. Our benchmark and code are released at https://github.com/GraphPKU/number_cookbook.

As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.

Mathematical capabilities were previously believed to emerge in common language models only at a very large scale or require extensive math-related pre-training. This paper shows that the LLaMA-2 7B model with common pre-training already exhibits strong mathematical abilities, as evidenced by its impressive accuracy of 97.7% and 72.0% on the GSM8K and MATH benchmarks, respectively, when selecting the best response from 256 random generations. The primary issue with the current base model is the difficulty in consistently eliciting its inherent mathematical capabilities. Notably, the accuracy for the first answer drops to 49.5% and 7.9% on the GSM8K and MATH benchmarks, respectively. We find that simply scaling up the SFT data can significantly enhance the reliability of generating correct answers. However, the potential for extensive scaling is constrained by the scarcity of publicly available math questions. To overcome this limitation, we employ synthetic data, which proves to be nearly as effective as real data and shows no clear saturation when scaled up to approximately one million samples. This straightforward approach achieves an accuracy of 82.6% on GSM8K and 40.6% on MATH using LLaMA-2 7B models, surpassing previous models by 14.2% and 20.8%, respectively. We also provide insights into scaling behaviors across different reasoning complexities and error types.

Automated math word problem solvers based on neural networks have successfully managed to obtain 70-80\% accuracy in solving arithmetic word problems. However, it has been shown that these solvers may rely on superficial patterns to obtain their equations. In order to determine what information math word problem solvers use to generate solutions, we remove parts of the input and measure the model's performance on the perturbed dataset. Our results show that the model is not sensitive to the removal of many words from the input and can still manage to find a correct answer when given a nonsense question. This indicates that automatic solvers do not follow the semantic logic of math word problems, and may be overfitting to the presence of specific words.

Embedding words in high-dimensional vector spaces has proven valuable in many natural language applications. In this work, we investigate whether similarly-trained embeddings of integers can capture concepts that are useful for mathematical applications. We probe the integer embeddings for mathematical knowledge, apply them to a set of numerical reasoning tasks, and show that by learning the representations from mathematical sequence data, we can substantially improve over number embeddings learned from English text corpora.

Recently, the Natural Language Inference (NLI) task has been studied for semi-structured tables that do not have a strict format. Although neural approaches have achieved high performance in various types of NLI, including NLI between semi-structured tables and texts, they still have difficulty in performing a numerical type of inference, such as counting. To handle a numerical type of inference, we propose a logical inference system for reasoning between semi-structured tables and texts. We use logical representations as meaning representations for tables and texts and use model checking to handle a numerical type of inference between texts and tables. To evaluate the extent to which our system can perform inference with numerical comparatives, we make an evaluation protocol that focuses on numerical understanding between semi-structured tables and texts in English. We show that our system can more robustly perform inference between tables and texts that requires numerical understanding compared with current neural approaches.

Can Multimodal Large Language Models (MLLMs) develop an intuitive number sense similar to humans? Targeting this problem, we introduce Visual Number Benchmark (VisNumBench) to evaluate the number sense abilities of MLLMs across a wide range of visual numerical tasks. VisNumBench consists of about 1,900 multiple-choice question-answer pairs derived from both synthetic and real-world visual data, covering seven visual numerical attributes and four types of visual numerical estimation tasks. Our experiments on VisNumBench led to the following key findings: (i) The 17 MLLMs we tested, including open-source models such as Qwen2.5-VL and InternVL2.5, as well as proprietary models like GPT-4o and Gemini 2.0 Flash, perform significantly below human levels in number sense-related tasks. (ii) Multimodal mathematical models and multimodal chain-of-thought (CoT) models did not exhibit significant improvements in number sense abilities. (iii) Stronger MLLMs with larger parameter sizes and broader general abilities demonstrate modest gains in number sense abilities. We believe VisNumBench will serve as a valuable resource for the research community, encouraging further advancements in enhancing MLLMs' number sense abilities. All benchmark resources, including code and datasets, will be publicly available at https://wwwtttjjj.github.io/VisNumBench/.

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.

Advancement in Large Language Models (LLMs) reasoning capabilities enables them to solve scientific problems with enhanced efficacy. Thereby, a high-quality benchmark for comprehensive and appropriate assessment holds significance, while existing ones either confront the risk of data contamination or lack involved disciplines. To be specific, due to the data source overlap of LLMs training and static benchmark, the keys or number pattern of answers inadvertently memorized (i.e. data contamination), leading to systematic overestimation of their reasoning capabilities, especially numerical reasoning. We propose SciDA, a multidisciplinary benchmark that consists exclusively of over 1k Olympic-level numerical computation problems, allowing randomized numerical initializations for each inference round to avoid reliance on fixed numerical patterns. We conduct a series of experiments with both closed-source and open-source top-performing LLMs, and it is observed that the performance of LLMs drop significantly under random numerical initialization. Thus, we provide truthful and unbiased assessments of the numerical reasoning capabilities of LLMs. The data is available at https://huggingface.co/datasets/m-a-p/SciDA

Large Language Models (LLMs) have demonstrated exceptional capabilities in various natural language tasks, often achieving performances that surpass those of humans. Despite these advancements, the domain of mathematics presents a distinctive challenge, primarily due to its specialized structure and the precision it demands. In this study, we adopted a two-step approach for investigating the proficiency of LLMs in answering mathematical questions. First, we employ the most effective LLMs, as identified by their performance on math question-answer benchmarks, to generate answers to 78 questions from the Math Stack Exchange (MSE). Second, a case analysis is conducted on the LLM that showed the highest performance, focusing on the quality and accuracy of its answers through manual evaluation. We found that GPT-4 performs best (nDCG of 0.48 and P@10 of 0.37) amongst existing LLMs fine-tuned for answering mathematics questions and outperforms the current best approach on ArqMATH3 Task1, considering P@10. Our Case analysis indicates that while the GPT-4 can generate relevant responses in certain instances, it does not consistently answer all questions accurately. This paper explores the current limitations of LLMs in navigating complex mathematical problem-solving. Through case analysis, we shed light on the gaps in LLM capabilities within mathematics, thereby setting the stage for future research and advancements in AI-driven mathematical reasoning. We make our code and findings publicly available for research: https://github.com/gipplab/LLM-Investig-MathStackExchange

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!

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.

We introduce a large-scale dataset of math word problems and an interpretable neural math problem solver that learns to map problems to operation programs. Due to annotation challenges, current datasets in this domain have been either relatively small in scale or did not offer precise operational annotations over diverse problem types. We introduce a new representation language to model precise operation programs corresponding to each math problem that aim to improve both the performance and the interpretability of the learned models. Using this representation language, our new dataset, MathQA, significantly enhances the AQuA dataset with fully-specified operational programs. We additionally introduce a neural sequence-to-program model enhanced with automatic problem categorization. Our experiments show improvements over competitive baselines in our MathQA as well as the AQuA dataset. The results are still significantly lower than human performance indicating that the dataset poses new challenges for future research. Our dataset is available at: https://math-qa.github.io/math-QA/

Recent advances in language models have demonstrated their capability to solve mathematical reasoning problems, achieving near-perfect accuracy on grade-school level math benchmarks like GSM8K. In this paper, we formally study how language models solve these problems. We design a series of controlled experiments to address several fundamental questions: (1) Can language models truly develop reasoning skills, or do they simply memorize templates? (2) What is the model's hidden (mental) reasoning process? (3) Do models solve math questions using skills similar to or different from humans? (4) Do models trained on GSM8K-like datasets develop reasoning skills beyond those necessary for solving GSM8K problems? (5) What mental process causes models to make reasoning mistakes? (6) How large or deep must a model be to effectively solve GSM8K-level math questions? Our study uncovers many hidden mechanisms by which language models solve mathematical questions, providing insights that extend beyond current understandings of LLMs.

The ability to precisely derive mathematical objects is a core requirement for downstream STEM applications, including mathematics, physics, and chemistry, where reasoning must culminate in formally structured expressions. Yet, current LM evaluations of mathematical and scientific reasoning rely heavily on simplified answer formats such as numerical values or multiple choice options due to the convenience of automated assessment. In this paper we provide three contributions for improving reasoning over mathematical objects: (i) we build and release training data and benchmarks for deriving mathematical objects, the Principia suite; (ii) we provide training recipes with strong LLM-judges and verifiers, where we show that on-policy judge training boosts performance; (iii) we show how on-policy training can also be used to scale test-time compute via aggregation. We find that strong LMs such as Qwen3-235B and o3 struggle on Principia, while our training recipes can bring significant improvements over different LLM backbones, while simultaneously improving results on existing numerical and MCQA tasks, demonstrating cross-format generalization of reasoning abilities.

Standard Neural Networks can learn mathematical operations, but they do not extrapolate. Extrapolation means that the model can apply to larger numbers, well beyond those observed during training. Recent architectures tackle arithmetic operations and can extrapolate; however, the equally important problem of quantitative reasoning remains unaddressed. In this work, we propose a novel architectural element, the Neural Status Register (NSR), for quantitative reasoning over numbers. Our NSR relaxes the discrete bit logic of physical status registers to continuous numbers and allows end-to-end learning with gradient descent. Experiments show that the NSR achieves solutions that extrapolate to numbers many orders of magnitude larger than those in the training set. We successfully train the NSR on number comparisons, piecewise discontinuous functions, counting in sequences, recurrently finding minimums, finding shortest paths in graphs, and comparing digits in images.

Large language models often default to step-by-step computation even when efficient numerical shortcuts are available. This raises a basic question: do they exhibit number sense in a human-like behavioral sense, i.e., the ability to recognize numerical structure, apply shortcuts when appropriate, and avoid them when they are not? We introduce SenseMath, a controlled benchmark for evaluating structure-sensitive numerical reasoning in LLMs. SenseMath contains 4,800 items spanning eight shortcut categories and four digit scales, with matched strong-shortcut, weak-shortcut, and control variants. It supports three evaluation settings of increasing cognitive demand: Shortcut Use (whether models can apply shortcuts on shortcut-amenable problems); Applicability Judgment (whether they can recognize when a shortcut is appropriate or misleading); and Problem Generation (whether they can generate new problem items that correctly admit a given type of shortcut). Our evaluation across five LLMs, ranging from GPT-4o-mini to Llama-3.1-8B, shows a consistent pattern: when explicitly prompted, models readily adopt shortcut strategies and achieve substantial accuracy gains on shortcut-amenable items (up to 15%), yet under standard chain-of-thought prompting they spontaneously employ such strategies in fewer than 40% of cases, even when they demonstrably possess the requisite capability. Moreover, this competence is confined to the Use level; models systematically over-generalise shortcuts to problems where they do not apply, and fail to generate valid shortcut-bearing problems from scratch. Together, these results suggest that current LLMs exhibit procedural shortcut fluency without the structural understanding of when and why shortcuts work that underlies human number sense.

Given the ubiquitous nature of numbers in text, reasoning with numbers to perform simple calculations is an important skill of AI systems. While many datasets and models have been developed to this end, state-of-the-art AI systems are brittle; failing to perform the underlying mathematical reasoning when they appear in a slightly different scenario. Drawing inspiration from GLUE that was proposed in the context of natural language understanding, we propose NumGLUE, a multi-task benchmark that evaluates the performance of AI systems on eight different tasks, that at their core require simple arithmetic understanding. We show that this benchmark is far from being solved with neural models including state-of-the-art large-scale language models performing significantly worse than humans (lower by 46.4%). Further, NumGLUE promotes sharing knowledge across tasks, especially those with limited training data as evidenced by the superior performance (average gain of 3.4% on each task) when a model is jointly trained on all the tasks as opposed to task-specific modeling. Finally, we hope that NumGLUE will encourage systems that perform robust and general arithmetic reasoning within language, a first step towards being able to perform more complex mathematical reasoning.

Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to train approximations to multiple differential equations simultaneously and are thus a general purpose solver that can be adapted to downstream tasks. Current PDE foundation models focus on either learning general solution operators and/or the governing system of equations, and thus only handle numerical or symbolic modalities. However, real-world applications may require more flexible data modalities, e.g. text analysis or descriptive outputs. To address this gap, we propose a novel multimodal deep learning approach that leverages a transformer-based architecture to approximate solution operators for a wide variety of ODEs and PDEs. Our method integrates numerical inputs, such as equation parameters and initial conditions, with text descriptions of physical processes or system dynamics. This enables our model to handle settings where symbolic representations may be incomplete or unavailable. In addition to providing accurate numerical predictions, our approach generates interpretable scientific text descriptions, offering deeper insights into the underlying dynamics and solution properties. The numerical experiments show that our model provides accurate solutions for in-distribution data (with average relative error less than 3.3%) and out-of-distribution data (average relative error less than 7.8%) together with precise text descriptions (with correct descriptions generated 100% of times). In certain tests, the model is also shown to be capable of extrapolating solutions in time.

We provide here a dataset for tasks related to natural language understanding and natural language inference. The dataset contains logical puzzles in natural language from three domains: comparing puzzles, knighs and knaves, and zebra puzzles. Each puzzle is associated with the entire set of atomic questions that can be generated based on the relations and individuals occurring in the text. For each question we provide the correct answer: entailment, contradiction or ambiguity. The answer's correctness is verified against theorem provers. Good puzzles have two properties: (i) each piece of information is necessary and (ii) no unnecessary information is provided. These properties make puzzles interesting candidates for machine comprehension tasks.

Long-form numerical reasoning in financial analysis aims to generate a reasoning program to calculate the correct answer for a given question. Previous work followed a retriever-generator framework, where the retriever selects key facts from a long-form document, and the generator generates a reasoning program based on retrieved facts. However, they treated all facts equally without considering the different contributions of facts with and without numbers. Meanwhile, the program consistency were ignored under supervised training, resulting in lower training accuracy and diversity. To solve these problems, we proposed APOLLO to improve the long-form numerical reasoning framework. For the retriever, we adopt a number-aware negative sampling strategy to enable the retriever to be more discriminative on key numerical facts. For the generator, we design consistency-based reinforcement learning and target program augmentation strategy based on the consistency of program execution results. Experimental results on the FinQA and ConvFinQA leaderboard verify the effectiveness of our proposed method, achieving the new state-of-the-art.

Temporal question answering is an established method for evaluating temporal reasoning in large language models. Expected answers are often numeric (e.g., dates or durations), yet model responses are evaluated like regular text with exact match (EM), unable to distinguish small from large errors. In this investigative work, we frame temporal question answering as a numerical estimation task to assess the shortcomings of EM. We introduce TempAnswerQA, a benchmark distilled from Test of Time and TempTabQA, where all questions require a numerical, temporal answer, allowing us to evaluate models beyond EM. We use the forecasting metrics symmetric mean absolute percentage error (sMAPE) and mean absolute scaled error (MASE). With sMAPE, we find that error size and EM are decoupled. Models with low EM still have low sMAPE (both ~20%), and some models have high sMAPE despite high EM. Scaling errors by the deviation of the ground truth data with MASE reshuffles model rankings compared to EM, revealing gaps in models' understanding of temporal domain knowledge, especially when trained with synthetic data. Lastly, the models' most frequent error is to deviate by only pm1 from the ground truth. sMAPE and MASE, unlike EM, adequately weight these errors. Our findings underscore the need for specialised metrics for temporal QA tasks. Code and data are available on https://github.com/aauss/temporal-answer-qa.

Recent advancements in language models have led to significant improvements in mathematical reasoning across various benchmarks. However, most of these benchmarks rely on automatic evaluation methods that only compare final answers using heuristics, without verifying the underlying reasoning steps. This limitation results in false positive solutions, where models may produce correct final answers but with flawed deduction paths. In this paper, we systematically examine the prevalence of false positive solutions in mathematical problem solving for language models. We analyze the characteristics and extent of this issue across different open-source models, datasets of varying difficulty levels, and decoding strategies. Specifically, we explore how false positives influence the inference time scaling behavior of language models. Our experimental results reveal that: (1) false positive solutions persist across different models, datasets, and decoding methods, (2) sampling-based inference time scaling methods do not alleviate the problem, and (3) the pass@N evaluation metric is more susceptible to false positives, suggesting a significantly lower scaling ceiling than what automatic evaluations indicate. Additionally, we analyze specific instances of false positives and discuss potential limitations in self-improvement techniques and synthetic data generation under such conditions. Our data and code are publicly available at https://github.com/Wloner0809/False-Positives-in-Math.

Numerical reasoning over text (NRoT) presents unique challenges that are not well addressed by existing pre-training objectives. We explore five sequential training schedules that adapt a pre-trained T5 model for NRoT. Our final model is adapted from T5, but further pre-trained on three datasets designed to strengthen skills necessary for NRoT and general reading comprehension before being fine-tuned on the Discrete Reasoning over Text (DROP) dataset. The training improves DROP's adjusted F1 performance (a numeracy-focused score) from 45.90 to 70.83. Our model closes in on GenBERT (72.4), a custom BERT-Base model using the same datasets with significantly more parameters. We show that training the T5 multitasking framework with multiple numerical reasoning datasets of increasing difficulty, good performance on DROP can be achieved without manually engineering partitioned functionality between distributed and symbol modules.

Mathematical reasoning---a core ability within human intelligence---presents some unique challenges as a domain: we do not come to understand and solve mathematical problems primarily on the back of experience and evidence, but on the basis of inferring, learning, and exploiting laws, axioms, and symbol manipulation rules. In this paper, we present a new challenge for the evaluation (and eventually the design) of neural architectures and similar system, developing a task suite of mathematics problems involving sequential questions and answers in a free-form textual input/output format. The structured nature of the mathematics domain, covering arithmetic, algebra, probability and calculus, enables the construction of training and test splits designed to clearly illuminate the capabilities and failure-modes of different architectures, as well as evaluate their ability to compose and relate knowledge and learned processes. Having described the data generation process and its potential future expansions, we conduct a comprehensive analysis of models from two broad classes of the most powerful sequence-to-sequence architectures and find notable differences in their ability to resolve mathematical problems and generalize their knowledge.

Large Language Models (LLMs) do not differentially represent numbers, which are pervasive in text. In contrast, neuroscience research has identified distinct neural representations for numbers and words. In this work, we investigate how well popular LLMs capture the magnitudes of numbers (e.g., that 4 < 5) from a behavioral lens. Prior research on the representational capabilities of LLMs evaluates whether they show human-level performance, for instance, high overall accuracy on standard benchmarks. Here, we ask a different question, one inspired by cognitive science: How closely do the number representations of LLMscorrespond to those of human language users, who typically demonstrate the distance, size, and ratio effects? We depend on a linking hypothesis to map the similarities among the model embeddings of number words and digits to human response times. The results reveal surprisingly human-like representations across language models of different architectures, despite the absence of the neural circuitry that directly supports these representations in the human brain. This research shows the utility of understanding LLMs using behavioral benchmarks and points the way to future work on the number representations of LLMs and their cognitive plausibility.

The sheer volume of financial statements makes it difficult for humans to access and analyze a business's financials. Robust numerical reasoning likewise faces unique challenges in this domain. In this work, we focus on answering deep questions over financial data, aiming to automate the analysis of a large corpus of financial documents. In contrast to existing tasks on general domain, the finance domain includes complex numerical reasoning and understanding of heterogeneous representations. To facilitate analytical progress, we propose a new large-scale dataset, FinQA, with Question-Answering pairs over Financial reports, written by financial experts. We also annotate the gold reasoning programs to ensure full explainability. We further introduce baselines and conduct comprehensive experiments in our dataset. The results demonstrate that popular, large, pre-trained models fall far short of expert humans in acquiring finance knowledge and in complex multi-step numerical reasoning on that knowledge. Our dataset -- the first of its kind -- should therefore enable significant, new community research into complex application domains. The dataset and code are publicly availablehttps://github.com/czyssrs/FinQA.

With the recent advance in large pre-trained language models, researchers have achieved record performances in NLP tasks that mostly focus on language pattern matching. The community is experiencing the shift of the challenge from how to model language to the imitation of complex reasoning abilities like human beings. In this work, we investigate the application domain of finance that involves real-world, complex numerical reasoning. We propose a new large-scale dataset, ConvFinQA, aiming to study the chain of numerical reasoning in conversational question answering. Our dataset poses great challenge in modeling long-range, complex numerical reasoning paths in real-world conversations. We conduct comprehensive experiments and analyses with both the neural symbolic methods and the prompting-based methods, to provide insights into the reasoning mechanisms of these two divisions. We believe our new dataset should serve as a valuable resource to push forward the exploration of real-world, complex reasoning tasks as the next research focus. Our dataset and code is publicly available at https://github.com/czyssrs/ConvFinQA.

Large language models (LLMs) have pushed the limits of natural language understanding and exhibited excellent problem-solving ability. Despite the great success, most existing open-source LLMs (\eg, LLaMA-2) are still far away from satisfactory for solving mathematical problem due to the complex reasoning procedures. To bridge this gap, we propose MetaMath, a fine-tuned language model that specializes in mathematical reasoning. Specifically, we start by bootstrapping mathematical questions by rewriting the question from multiple perspectives without extra knowledge, which results in a new dataset called {MetaMathQA}. Then we fine-tune the LLaMA-2 models on MetaMathQA. Experimental results on two popular benchmarks (\ie, GSM8K and MATH) for mathematical reasoning demonstrate that MetaMath outperforms a suite of open-source LLMs by a significant margin. Our MetaMath-7B model achieves 66.4% on GSM8K and 19.4% on MATH, exceeding the state-of-the-art models of the same size by 11.5% and 8.7%. Particularly, {MetaMath-70B} achieves an accuracy of 82.3% on {GSM8K}, slightly better than {GPT-3.5-Turbo}. We release the {MetaMathQA} dataset, the {MetaMath} models with different model sizes and the training code for public use.

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.

Language models are not accurate in numerical problems. Their architecture does not allow for anything less than a probabilistic next word. This paper introduces ComputeGPT: an approach of creating a chat model able to answer computational problems through running on-demand code. ComputeGPT converts each question to relevant code, runs the code, and returns the computed answer as part of the chat. We combine this approach with a local browser-based Python interpretation and fine-tuned prompts in order to achieve state-of-the-art efficiency on numerical problems and provide a suitable front-end and safe environment for the code to be executed in.

The recently released Google Gemini class of models are the first to comprehensively report results that rival the OpenAI GPT series across a wide variety of tasks. In this paper, we do an in-depth exploration of Gemini's language abilities, making two contributions. First, we provide a third-party, objective comparison of the abilities of the OpenAI GPT and Google Gemini models with reproducible code and fully transparent results. Second, we take a closer look at the results, identifying areas where one of the two model classes excels. We perform this analysis over 10 datasets testing a variety of language abilities, including reasoning, answering knowledge-based questions, solving math problems, translating between languages, generating code, and acting as instruction-following agents. From this analysis, we find that Gemini Pro achieves accuracy that is close but slightly inferior to the corresponding GPT 3.5 Turbo on all tasks that we benchmarked. We further provide explanations for some of this under-performance, including failures in mathematical reasoning with many digits, sensitivity to multiple-choice answer ordering, aggressive content filtering, and others. We also identify areas where Gemini demonstrates comparably high performance, including generation into non-English languages, and handling longer and more complex reasoning chains. Code and data for reproduction can be found at https://github.com/neulab/gemini-benchmark

Pretrained language models have shown superior performance on many natural language processing tasks, yet they still struggle at multi-step formal reasoning tasks like grade school math problems. One key challenge of finetuning them to solve such math reasoning problems is that many existing datasets only contain one reference solution for each problem, despite the fact that there are often alternative solutions resembling different reasoning paths to the final answer. This way, the finetuned models are biased towards the limited reference solutions, which limits their generalization to unseen examples. To mitigate this issue, we propose to let the model perform sampling during training and learn from both self-sampled fully-correct solutions, which yield the correct answer upon execution, and partially-correct solutions, whose intermediate state matches an intermediate state of a known correct solution. We show that our use of self-sampled correct and partially-correct solutions can benefit learning and help guide the sampling process, leading to more efficient exploration of the solution space. Additionally, we explore various training objectives to support learning from multiple solutions per example and find they greatly affect the performance. Experiments on two math reasoning datasets show the effectiveness of our method compared to learning from a single reference solution with MLE, where we improve PASS@100 from 35.5% to 44.5% for GSM8K, and 27.6% to 36.2% PASS@80 for MathQA. Such improvements are also consistent across different model sizes. Our code is available at https://github.com/microsoft/TraceCodegen.

The game show Lucky 13 differs from other television game shows in that contestants are required to place a bet on their own knowledge of trivia by selecting a range that contains the number of questions that they answered correctly. We present a model for this game show using binomial random variables and generate tables outlining the optimal range the player should select based on maximization of two different utility functions. After analyzing the decisions made by some actual contestants on this show, we present a numerical simulation for how many questions an average player is expected to answer correctly based on question categories observed for two sample contestants.

Neural networks can learn to represent and manipulate numerical information, but they seldom generalize well outside of the range of numerical values encountered during training. To encourage more systematic numerical extrapolation, we propose an architecture that represents numerical quantities as linear activations which are manipulated using primitive arithmetic operators, controlled by learned gates. We call this module a neural arithmetic logic unit (NALU), by analogy to the arithmetic logic unit in traditional processors. Experiments show that NALU-enhanced neural networks can learn to track time, perform arithmetic over images of numbers, translate numerical language into real-valued scalars, execute computer code, and count objects in images. In contrast to conventional architectures, we obtain substantially better generalization both inside and outside of the range of numerical values encountered during training, often extrapolating orders of magnitude beyond trained numerical ranges.

This paper investigates whether large language models (LLMs) utilize numerical attributes encoded in a low-dimensional subspace of the embedding space when answering logical comparison questions (e.g., Was Cristiano born before Messi?). We first identified these subspaces using partial least squares regression, which effectively encodes the numerical attributes associated with the entities in comparison prompts. Further, we demonstrate causality by intervening in these subspaces to manipulate hidden states, thereby altering the LLM's comparison outcomes. Experimental results show that our findings hold for different numerical attributes, indicating that LLMs utilize the linearly encoded information for numerical reasoning.

Large pre-trained language models (LMs) are known to encode substantial amounts of linguistic information. However, high-level reasoning skills, such as numerical reasoning, are difficult to learn from a language-modeling objective only. Consequently, existing models for numerical reasoning have used specialized architectures with limited flexibility. In this work, we show that numerical reasoning is amenable to automatic data generation, and thus one can inject this skill into pre-trained LMs, by generating large amounts of data, and training in a multi-task setup. We show that pre-training our model, GenBERT, on this data, dramatically improves performance on DROP (49.3 rightarrow 72.3 F1), reaching performance that matches state-of-the-art models of comparable size, while using a simple and general-purpose encoder-decoder architecture. Moreover, GenBERT generalizes well to math word problem datasets, while maintaining high performance on standard RC tasks. Our approach provides a general recipe for injecting skills into large pre-trained LMs, whenever the skill is amenable to automatic data augmentation.

We present the Chinese Elementary School Math Word Problems (CMATH) dataset, comprising 1.7k elementary school-level math word problems with detailed annotations, source from actual Chinese workbooks and exams. This dataset aims to provide a benchmark tool for assessing the following question: to what grade level of elementary school math do the abilities of popular large language models (LLMs) correspond? We evaluate a variety of popular LLMs, including both commercial and open-source options, and discover that only GPT-4 achieves success (accuracy geq 60\%) across all six elementary school grades, while other models falter at different grade levels. Furthermore, we assess the robustness of several top-performing LLMs by augmenting the original problems in the CMATH dataset with distracting information. Our findings reveal that GPT-4 is able to maintains robustness, while other model fail. We anticipate that our study will expose limitations in LLMs' arithmetic and reasoning capabilities, and promote their ongoing development and advancement.

We present ORCA (Omni Research on Calculation in AI) Benchmark - a novel benchmark that evaluates large language models (LLMs) on multi-domain, real-life quantitative reasoning using verified outputs from Omni's calculator engine. In 500 natural-language tasks across domains such as finance, physics, health, and statistics, the five state-of-the-art systems (ChatGPT-5, Gemini~2.5~Flash, Claude~Sonnet~4.5, Grok~4, and DeepSeek~V3.2) achieved only 45--63,% accuracy, with errors mainly related to rounding (35,%) and calculation mistakes (33,%). Results in specific domains indicate strengths in mathematics and engineering, but weaknesses in physics and natural sciences. Correlation analysis (r approx 0.40--0.65) shows that the models often fail together but differ in the types of errors they make, highlighting their partial complementarity rather than redundancy. Unlike standard math datasets, ORCA evaluates step-by-step reasoning, numerical precision, and domain generalization across real problems from finance, physics, health, and statistics.

Solving algebraic word problems requires executing a series of arithmetic operations---a program---to obtain a final answer. However, since programs can be arbitrarily complicated, inducing them directly from question-answer pairs is a formidable challenge. To make this task more feasible, we solve these problems by generating answer rationales, sequences of natural language and human-readable mathematical expressions that derive the final answer through a series of small steps. Although rationales do not explicitly specify programs, they provide a scaffolding for their structure via intermediate milestones. To evaluate our approach, we have created a new 100,000-sample dataset of questions, answers and rationales. Experimental results show that indirect supervision of program learning via answer rationales is a promising strategy for inducing arithmetic programs.

Math reasoning is becoming an ever increasing area of focus as we scale large language models. However, even the previously-toughest evals like MATH are now close to saturated by frontier models (90.0% for o1-mini and 86.5% for Gemini 1.5 Pro). We introduce HARP, Human Annotated Reasoning Problems (for Math), consisting of 5,409 problems from the US national math competitions (A(J)HSME, AMC, AIME, USA(J)MO). Of these, 4,780 have answers that are automatically check-able (with libraries such as SymPy). These problems range six difficulty levels, with frontier models performing relatively poorly on the hardest bracket of 197 problems (average accuracy 41.1% for o1-mini, and 9.6% for Gemini 1.5 Pro). Our dataset also features multiple choices (for 4,110 problems) and an average of two human-written, ground-truth solutions per problem, offering new avenues of research that we explore briefly. We report evaluations for many frontier models and share some interesting analyses, such as demonstrating that frontier models across families intrinsically scale their inference-time compute for more difficult problems. Finally, we open source all code used for dataset construction (including scraping) and all code for evaluation (including answer checking) to enable future research at: https://github.com/aadityasingh/HARP.

This paper investigates the mathematical reasoning capabilities of large language models (LLMs) using 50 newly constructed high-school-level word problems. Unlike prior studies that focus solely on answer correctness, we rigorously analyze both final answers and solution steps to identify reasoning failures. Evaluating eight state-of-the-art models - including Mixtral, Llama, Gemini, GPT-4o, and OpenAI's o1 variants - we find that while newer models (e.g., o3-mini, deepseek-r1) achieve higher accuracy, all models exhibit errors in spatial reasoning, strategic planning, and arithmetic, sometimes producing correct answers through flawed logic. Common failure modes include unwarranted assumptions, over-reliance on numerical patterns, and difficulty translating physical intuition into mathematical steps. Manual analysis reveals that models struggle with problems requiring multi-step deduction or real-world knowledge, despite possessing broad mathematical knowledge. Our results underscore the importance of evaluating reasoning processes, not just answers, and caution against overestimating LLMs' problem-solving proficiency. The study highlights persistent gaps in LLMs' generalization abilities, emphasizing the need for targeted improvements in structured reasoning and constraint handling.

In an era where symbolic mathematical equations are indispensable for modeling complex natural phenomena, scientific inquiry often involves collecting observations and translating them into mathematical expressions. Recently, deep learning has emerged as a powerful tool for extracting insights from data. However, existing models typically specialize in either numeric or symbolic domains, and are usually trained in a supervised manner tailored to specific tasks. This approach neglects the substantial benefits that could arise from a task-agnostic unified understanding between symbolic equations and their numeric counterparts. To bridge the gap, we introduce SNIP, a Symbolic-Numeric Integrated Pre-training, which employs joint contrastive learning between symbolic and numeric domains, enhancing their mutual similarities in the pre-trained embeddings. By performing latent space analysis, we observe that SNIP provides cross-domain insights into the representations, revealing that symbolic supervision enhances the embeddings of numeric data and vice versa. We evaluate SNIP across diverse tasks, including symbolic-to-numeric mathematical property prediction and numeric-to-symbolic equation discovery, commonly known as symbolic regression. Results show that SNIP effectively transfers to various tasks, consistently outperforming fully supervised baselines and competing strongly with established task-specific methods, especially in few-shot learning scenarios where available data is limited.

Previous studies have typically assumed that large language models are unable to accurately perform arithmetic operations, particularly multiplication of >8 digits, and operations involving decimals and fractions, without the use of calculator tools. This paper aims to challenge this misconception. With sufficient training data, a 2 billion-parameter language model can accurately perform multi-digit arithmetic operations with almost 100% accuracy without data leakage, significantly surpassing GPT-4 (whose multi-digit multiplication accuracy is only 4.3%). We also demonstrate that our MathGLM, fine-tuned from GLM-10B on a dataset with additional multi-step arithmetic operations and math problems described in text, achieves similar performance to GPT-4 on a 5,000-samples Chinese math problem test set.

Transformers, central to the successes in modern Natural Language Processing, often falter on arithmetic tasks despite their vast capabilities --which paradoxically include remarkable coding abilities. We observe that a crucial challenge is their naive reliance on positional information to solve arithmetic problems with a small number of digits, leading to poor performance on larger numbers. Herein, we delve deeper into the role of positional encoding, and propose several ways to fix the issue, either by modifying the positional encoding directly, or by modifying the representation of the arithmetic task to leverage standard positional encoding differently. We investigate the value of these modifications for three tasks: (i) classical multiplication, (ii) length extrapolation in addition, and (iii) addition in natural language context. For (i) we train a small model on a small dataset (100M parameters and 300k samples) with remarkable aptitude in (direct, no scratchpad) 15 digits multiplication and essentially perfect up to 12 digits, while usual training in this context would give a model failing at 4 digits multiplication. In the experiments on addition, we use a mere 120k samples to demonstrate: for (ii) extrapolation from 10 digits to testing on 12 digits numbers while usual training would have no extrapolation, and for (iii) almost perfect accuracy up to 5 digits while usual training would be correct only up to 3 digits (which is essentially memorization with a training set of 120k samples).

Automatic math problem solving has recently attracted increasing attention as a long-standing AI benchmark. In this paper, we focus on solving geometric problems, which requires a comprehensive understanding of textual descriptions, visual diagrams, and theorem knowledge. However, the existing methods were highly dependent on handcraft rules and were merely evaluated on small-scale datasets. Therefore, we propose a Geometric Question Answering dataset GeoQA, containing 4,998 geometric problems with corresponding annotated programs, which illustrate the solving process of the given problems. Compared with another publicly available dataset GeoS, GeoQA is 25 times larger, in which the program annotations can provide a practical testbed for future research on explicit and explainable numerical reasoning. Moreover, we introduce a Neural Geometric Solver (NGS) to address geometric problems by comprehensively parsing multimodal information and generating interpretable programs. We further add multiple self-supervised auxiliary tasks on NGS to enhance cross-modal semantic representation. Extensive experiments on GeoQA validate the effectiveness of our proposed NGS and auxiliary tasks. However, the results are still significantly lower than human performance, which leaves large room for future research. Our benchmark and code are released at https://github.com/chen-judge/GeoQA .

Large language models (LLMs) have demonstrated remarkable capabilities in problem-solving. However, their proficiency in solving mathematical problems remains inadequate. We propose MathScale, a simple and scalable method to create high-quality mathematical reasoning data using frontier LLMs (e.g., {\tt GPT-3.5}). Inspired by the cognitive mechanism in human mathematical learning, it first extracts topics and knowledge points from seed math questions and then build a concept graph, which is subsequently used to generate new math questions. MathScale exhibits effective scalability along the size axis of the math dataset that we generate. As a result, we create a mathematical reasoning dataset (MathScaleQA) containing two million math question-answer pairs. To evaluate mathematical reasoning abilities of LLMs comprehensively, we construct {\sc MwpBench}, a benchmark of Math Word Problems, which is a collection of ten datasets (including GSM8K and MATH) covering K-12, college, and competition level math problems. We apply MathScaleQA to fine-tune open-source LLMs (e.g., LLaMA-2 and Mistral), resulting in significantly improved capabilities in mathematical reasoning. Evaluated on {\sc MwpBench}, MathScale-7B achieves state-of-the-art performance across all datasets, surpassing its best peers of equivalent size by 42.9\% in micro average accuracy and 43.7\% in macro average accuracy, respectively.

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.

Large Language Models (LLMs) have exhibited exceptional performance across a broad range of tasks and domains. However, they still encounter difficulties in solving mathematical problems due to the rigorous and logical nature of mathematics. Previous studies have employed techniques such as supervised fine-tuning (SFT), prompt engineering, and search-based methods to improve the mathematical problem-solving abilities of LLMs. Despite these efforts, their performance remains suboptimal and demands substantial computational resources. To address this issue, we propose a novel approach, BEATS, to enhance mathematical problem-solving abilities. Our method leverages newly designed prompts that guide the model to iteratively rewrite, advance by one step, and generate answers based on previous steps. Additionally, we introduce a new back-verification technique that uses LLMs to validate the correctness of the generated answers. Furthermore, we employ a pruning tree search to optimize search time while achieving strong performance. Notably, our method improves Qwen2-7b-Instruct's score from 36.94 to 61.52, outperforming GPT4's 42.5 on the MATH benchmark.

In this paper we look at the ability of recent large language models (LLMs) at solving mathematical problems in combinatorics. We compare models LLaMA-2, LLaMA-3.1, GPT-4, and Mixtral against each other and against human pupils and undergraduates with prior experience in mathematical olympiads. To facilitate these comparisons we introduce the Combi-Puzzles dataset, which contains 125 problem variants based on 25 combinatorial reasoning problems. Each problem is presented in one of five distinct forms, created by systematically manipulating the problem statements through adversarial additions, numeric parameter changes, and linguistic obfuscation. Our variations preserve the mathematical core and are designed to measure the generalisability of LLM problem-solving abilities, while also increasing confidence that problems are submitted to LLMs in forms that have not been seen as training instances. We found that a model based on GPT-4 outperformed all other models in producing correct responses, and performed significantly better in the mathematical variation of the problems than humans. We also found that modifications to problem statements significantly impact the LLM's performance, while human performance remains unaffected.

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.

Advanced applied mathematics problems are underrepresented in existing Large Language Model (LLM) benchmark datasets. To address this, we introduce HARDMath, a dataset inspired by a graduate course on asymptotic methods, featuring challenging applied mathematics problems that require analytical approximation techniques. These problems demand a combination of mathematical reasoning, computational tools, and subjective judgment, making them difficult for LLMs. Our framework auto-generates a large number of problems with solutions validated against numerical ground truths. We evaluate both open- and closed-source LLMs on HARDMath-mini, a sub-sampled test set of 366 problems, as well as on 40 word problems formulated in applied science contexts. Even leading closed-source models like GPT-4 achieve only 43.8% overall accuracy with few-shot Chain-of-Thought prompting, and all models demonstrate significantly lower performance compared to results on existing mathematics benchmark datasets. We additionally conduct a detailed error analysis to gain insights into the failure cases of LLMs. These results demonstrate limitations of current LLM performance on advanced graduate-level applied math problems and underscore the importance of datasets like HARDMath to advance mathematical abilities of LLMs.

In this paper, we examine the solvability of a functional equation in a Lipschitz space. As an application, we use our result to determine the existence and uniqueness of solutions to an equation describing a specific type of choice behavior model for the learning process of the paradise fish. Finally, we present some concrete examples where, using numerical techniques, we obtain approximations to the solution of the functional equation. As the straightforward Picard's iteration can be very expensive, we show that an analytical suboptimal least-squares approximation can be chosen in practice, resulting in very good accuracy.

Though Large Language Models (LLMs) have shown remarkable abilities in mathematics reasoning, they are still struggling with performing numeric operations accurately, such as addition and multiplication. Numbers can be tokenized into tokens in various ways by different LLMs and affect the numeric operations performance. Currently, there are two representatives: 1) Tokenize into 1-digit, and 2) Tokenize into 1sim 3 digit. The difference is roughly equivalent to using different numeral systems (namely base 10 or base 10^{3}). In light of this, we study the scaling behavior of different numeral systems in the context of transformer-based large language models. We empirically show that a base 10 system is consistently more data-efficient than a base 10^{2} or 10^{3} system across training data scale, model sizes under from-scratch training settings, while different number systems have very similar fine-tuning performances. We attribute this to higher token frequencies of a base 10 system. Additionally, we reveal extrapolation behavior patterns on addition and multiplication. We identify that base 100 and base 1000 systems struggle on token-level discernment and token-level operations. We also sheds light on the mechanism learnt by the models.

With recent dramatic increases in AI system capabilities, there has been growing interest in utilizing machine learning for reasoning-heavy, quantitative tasks, particularly mathematics. While there are many resources capturing mathematics at the high-school, undergraduate, and graduate level, there are far fewer resources available that align with the level of difficulty and open endedness encountered by professional mathematicians working on open problems. To address this, we introduce a new collection of datasets, the Algebraic Combinatorics Dataset Repository (ACD Repo), representing either foundational results or open problems in algebraic combinatorics, a subfield of mathematics that studies discrete structures arising from abstract algebra. Further differentiating our dataset collection is the fact that it aims at the conjecturing process. Each dataset includes an open-ended research-level question and a large collection of examples (up to 10M in some cases) from which conjectures should be generated. We describe all nine datasets, the different ways machine learning models can be applied to them (e.g., training with narrow models followed by interpretability analysis or program synthesis with LLMs), and discuss some of the challenges involved in designing datasets like these.

Recent LLMs have demonstrated remarkable performance in solving exam-like math word problems. However, the degree to which these numerical reasoning skills are effective in real-world scenarios, particularly in expert domains, is still largely unexplored. This paper introduces DocMath-Eval, a comprehensive benchmark specifically designed to evaluate the numerical reasoning and problem-solving capabilities of LLMs in the context of understanding and analyzing financial documents containing both text and tables. We evaluate a wide spectrum of 19 LLMs, including those specialized in coding and finance. We also incorporate different prompting strategies (i.e., Chain-of-Thoughts and Program-of-Thoughts) to comprehensively assess the capabilities and limitations of existing LLMs in DocMath-Eval. We found that, although the current best-performing system (i.e., GPT-4), can perform well on simple problems such as calculating the rate of increase in a financial metric within a short document context, it significantly lags behind human experts in more complex problems grounded in longer contexts. We believe DocMath-Eval can be used as a valuable benchmark to evaluate LLMs' capabilities to solve challenging numerical reasoning problems in expert domains. We will release the benchmark and code at https://github.com/yale-nlp/DocMath-Eval.

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 introduce IndiaFinBench, to our knowledge the first publicly available evaluation benchmark for assessing large language model (LLM) performance on Indian financial regulatory text. Existing financial NLP benchmarks draw exclusively from Western financial corpora (SEC filings, US earnings reports, and English-language financial news), leaving a significant gap in coverage of non-Western regulatory frameworks. IndiaFinBench addresses this gap with 406 expert-annotated question-answer pairs drawn from 192 documents sourced from the Securities and Exchange Board of India (SEBI) and the Reserve Bank of India (RBI), spanning four task types: regulatory interpretation (174 items), numerical reasoning (92 items), contradiction detection (62 items), and temporal reasoning (78 items). Annotation quality is validated through a model-based secondary pass (kappa=0.918 on contradiction detection) and a 60-item human inter-annotator agreement evaluation (kappa=0.611; 76.7% overall agreement). We evaluate twelve models under zero-shot conditions, with accuracy ranging from 70.4% (Gemma 4 E4B) to 89.7% (Gemini 2.5 Flash). All models substantially outperform a non-specialist human baseline of 60.0%. Numerical reasoning is the most discriminative task, with a 35.9 percentage-point spread across models. Bootstrap significance testing (10,000 resamples) reveals three statistically distinct performance tiers. The dataset, evaluation code, and all model outputs are available at https://github.com/rajveerpall/IndiaFinBench

Recent advances in large language models (LLMs) and multimodal LLMs (MLLMs) have led to strong reasoning ability across a wide range of tasks. However, their ability to perform mathematical reasoning from spoken input remains underexplored. Prior studies on speech modality have mostly focused on factual speech understanding or simple audio reasoning tasks, providing limited insight into logical step-by-step reasoning, such as that required for mathematical problem solving. To address this gap, we introduce Spoken Math Question Answering (Spoken-MQA), a new benchmark designed to evaluate the mathematical reasoning capabilities of speech-based models, including both cascade models (ASR + LLMs) and end-to-end speech LLMs. Spoken-MQA covers a diverse set of math problems, including pure arithmetic, single-step and multi-step contextual reasoning, and knowledge-oriented reasoning problems, all presented in unambiguous natural spoken language. Through extensive experiments, we find that: (1) while some speech LLMs perform competitively on contextual reasoning tasks involving basic arithmetic, they still struggle with direct arithmetic problems; (2) current LLMs exhibit a strong bias toward symbolic mathematical expressions written in LaTex and have difficulty interpreting verbalized mathematical expressions; and (3) mathematical knowledge reasoning abilities are significantly degraded in current speech LLMs.

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.

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.

Our work presents a novel angle for evaluating language models' (LMs) mathematical abilities, by investigating whether they can discern skills and concepts enabled by math content. We contribute two datasets: one consisting of 385 fine-grained descriptions of K-12 math skills and concepts, or standards, from Achieve the Core (ATC), and another of 9.9K problems labeled with these standards (MathFish). Working with experienced teachers, we find that LMs struggle to tag and verify standards linked to problems, and instead predict labels that are close to ground truth, but differ in subtle ways. We also show that LMs often generate problems that do not fully align with standards described in prompts. Finally, we categorize problems in GSM8k using math standards, allowing us to better understand why some problems are more difficult to solve for models than others.

Understanding and creating mathematics using natural mathematical language - the mixture of symbolic and natural language used by humans - is a challenging and important problem for driving progress in machine learning. As a step in this direction, we develop NaturalProofs, a multi-domain corpus of mathematical statements and their proofs, written in natural mathematical language. NaturalProofs unifies broad coverage, deep coverage, and low-resource mathematical sources, allowing for evaluating both in-distribution and zero-shot generalization. Using NaturalProofs, we benchmark strong neural methods on mathematical reference retrieval and generation tasks which test a system's ability to determine key results that appear in a proof. Large-scale sequence models show promise compared to classical information retrieval methods, yet their performance and out-of-domain generalization leave substantial room for improvement. NaturalProofs opens many avenues for research on challenging mathematical tasks.

Reading comprehension has recently seen rapid progress, with systems matching humans on the most popular datasets for the task. However, a large body of work has highlighted the brittleness of these systems, showing that there is much work left to be done. We introduce a new English reading comprehension benchmark, DROP, which requires Discrete Reasoning Over the content of Paragraphs. In this crowdsourced, adversarially-created, 96k-question benchmark, a system must resolve references in a question, perhaps to multiple input positions, and perform discrete operations over them (such as addition, counting, or sorting). These operations require a much more comprehensive understanding of the content of paragraphs than what was necessary for prior datasets. We apply state-of-the-art methods from both the reading comprehension and semantic parsing literature on this dataset and show that the best systems only achieve 32.7% F1 on our generalized accuracy metric, while expert human performance is 96.0%. We additionally present a new model that combines reading comprehension methods with simple numerical reasoning to achieve 47.0% F1.

This paper introduces a new suite of question answering datasets for Norwegian; NorOpenBookQA, NorCommonSenseQA, NorTruthfulQA, and NRK-Quiz-QA. The data covers a wide range of skills and knowledge domains, including world knowledge, commonsense reasoning, truthfulness, and knowledge about Norway. Covering both of the written standards of Norwegian - Bokm{\aa}l and Nynorsk - our datasets comprise over 10k question-answer pairs, created by native speakers. We detail our dataset creation approach and present the results of evaluating 11 language models (LMs) in zero- and few-shot regimes. Most LMs perform better in Bokm{\aa}l than Nynorsk, struggle most with commonsense reasoning, and are often untruthful in generating answers to questions. All our datasets and annotation materials are publicly available.

Mathematical reasoning serves as a cornerstone for assessing the fundamental cognitive capabilities of human intelligence. In recent times, there has been a notable surge in the development of Large Language Models (LLMs) geared towards the automated resolution of mathematical problems. However, the landscape of mathematical problem types is vast and varied, with LLM-oriented techniques undergoing evaluation across diverse datasets and settings. This diversity makes it challenging to discern the true advancements and obstacles within this burgeoning field. This survey endeavors to address four pivotal dimensions: i) a comprehensive exploration of the various mathematical problems and their corresponding datasets that have been investigated; ii) an examination of the spectrum of LLM-oriented techniques that have been proposed for mathematical problem-solving; iii) an overview of factors and concerns affecting LLMs in solving math; and iv) an elucidation of the persisting challenges within this domain. To the best of our knowledge, this survey stands as one of the first extensive examinations of the landscape of LLMs in the realm of mathematics, providing a holistic perspective on the current state, accomplishments, and future challenges in this rapidly evolving field.

Long-form answers, consisting of multiple sentences, can provide nuanced and comprehensive answers to a broader set of questions. To better understand this complex and understudied task, we study the functional structure of long-form answers collected from three datasets, ELI5, WebGPT and Natural Questions. Our main goal is to understand how humans organize information to craft complex answers. We develop an ontology of six sentence-level functional roles for long-form answers, and annotate 3.9k sentences in 640 answer paragraphs. Different answer collection methods manifest in different discourse structures. We further analyze model-generated answers -- finding that annotators agree less with each other when annotating model-generated answers compared to annotating human-written answers. Our annotated data enables training a strong classifier that can be used for automatic analysis. We hope our work can inspire future research on discourse-level modeling and evaluation of long-form QA systems.

The poor performance of transformers on arithmetic tasks seems to stem in large part from their inability to keep track of the exact position of each digit inside of a large span of digits. We mend this problem by adding an embedding to each digit that encodes its position relative to the start of the number. In addition to the boost these embeddings provide on their own, we show that this fix enables architectural modifications such as input injection and recurrent layers to improve performance even further. With positions resolved, we can study the logical extrapolation ability of transformers. Can they solve arithmetic problems that are larger and more complex than those in their training data? We find that training on only 20 digit numbers with a single GPU for one day, we can reach state-of-the-art performance, achieving up to 99% accuracy on 100 digit addition problems. Finally, we show that these gains in numeracy also unlock improvements on other multi-step reasoning tasks including sorting and multiplication.

The recently released GPT-4 Code Interpreter has demonstrated remarkable proficiency in solving challenging math problems, primarily attributed to its ability to seamlessly reason with natural language, generate code, execute code, and continue reasoning based on the execution output. In this paper, we present a method to fine-tune open-source language models, enabling them to use code for modeling and deriving math equations and, consequently, enhancing their mathematical reasoning abilities. We propose a method of generating novel and high-quality datasets with math problems and their code-based solutions, referred to as MathCodeInstruct. Each solution interleaves natural language, code, and execution results. We also introduce a customized supervised fine-tuning and inference approach. This approach yields the MathCoder models, a family of models capable of generating code-based solutions for solving challenging math problems. Impressively, the MathCoder models achieve state-of-the-art scores among open-source LLMs on the MATH (45.2%) and GSM8K (83.9%) datasets, substantially outperforming other open-source alternatives. Notably, the MathCoder model not only surpasses ChatGPT-3.5 and PaLM-2 on GSM8K and MATH but also outperforms GPT-4 on the competition-level MATH dataset. The dataset and models will be released at https://github.com/mathllm/MathCoder.

Stochastic rounding (SR) is a probabilistic method used to round numbers to floating-point and fixed-point representations. In length n summation, the worst-case error of SR grows as n with high probability, unlike for standard modes, like round-to-nearest (RN), which grows as n. For this reason, the former is increasingly employed in large-scale, low-precision computations as an RN alternative. Additionally, SR alleviates stagnation, whereby relatively small summands are completely rounded off and do not contribute to the sum. We provide an update to [Croci et al., Roy. Soc. Open Sci. 9.3 (2022), pp. 1-25], a survey which discusses the development and use of SR between 1949 and 2022, citing over 100 references. Since then, there has been a surge of new research, and this update covers almost four years of further progress in applying, analysing, and implementing SR. Our main focus is limited-precision stochastic rounding, a new variant that fixes the precision of the random numbers used. We provide insights into industrial and numerical analysis activities surrounding SR, highlighting the next possible steps in making this rounding mode more widely available in hardware.

Theorem proving in natural mathematical language - the mixture of symbolic and natural language used by humans - plays a central role in mathematical advances and education, and tests aspects of reasoning that are core to intelligence. Yet it has remained underexplored with modern generative models. We study large-scale language models on two new generation tasks: suggesting the next step in a mathematical proof, and full proof generation. We develop NaturalProver, a language model that generates proofs by conditioning on background references (e.g. theorems and definitions that are either retrieved or human-provided), and optionally enforces their presence with constrained decoding. On theorems from the NaturalProofs benchmark, NaturalProver improves the quality of next-step suggestions and generated proofs over fine-tuned GPT-3, according to human evaluations from university-level mathematics students. NaturalProver is capable of proving some theorems that require short (2-6 step) proofs, and providing next-step suggestions that are rated as correct and useful over 40% of the time, which is to our knowledge the first demonstration of these capabilities using neural language models.

Mathematics is often perceived as a complex subject by students, leading to high failure rates in exams. To improve Mathematics skills, it is important to provide sample questions for students to practice problem-solving. Manually creating Math Word Problems (MWPs) is time consuming for tutors, because they have to type in natural language while adhering to grammar and spelling rules of the language. Existing Deep Learning techniques for MWP generation either require a tutor to provide the initial portion of the MWP, and/or additional information such as an equation. In this paper, we present an MWP generation system based on Large Language Models (LLMs) that overcome the need for additional input - the only input to our system is the number of MWPs needed, the grade and the type of question (e.g. addition, subtraction). Unlike the existing LLM-based solutions for MWP generation, we carried out an extensive set of experiments involving different LLMs, prompting strategies, techniques to improve the diversity of questions, as well as techniques that employ human feedback to improve LLM performance. Human and automated evaluations confirmed that the generated MWPs are high in quality, with minimal spelling and grammar issues. However, LLMs still struggle to generate questions that adhere to the specified grade and question type requirements.

The mathematical capabilities of AI systems are complex and multifaceted. Most existing research has predominantly focused on the correctness of AI-generated solutions to mathematical problems. In this work, we argue that beyond producing correct answers, AI systems should also be capable of, or assist humans in, developing novel solutions to mathematical challenges. This study explores the creative potential of Large Language Models (LLMs) in mathematical reasoning, an aspect that has received limited attention in prior research. We introduce a novel framework and benchmark, CreativeMath, which encompasses problems ranging from middle school curricula to Olympic-level competitions, designed to assess LLMs' ability to propose innovative solutions after some known solutions have been provided. Our experiments demonstrate that, while LLMs perform well on standard mathematical tasks, their capacity for creative problem-solving varies considerably. Notably, the Gemini-1.5-Pro model outperformed other LLMs in generating novel solutions. This research opens a new frontier in evaluating AI creativity, shedding light on both the strengths and limitations of LLMs in fostering mathematical innovation, and setting the stage for future developments in AI-assisted mathematical discovery.

Large language models (LLMs) have achieved impressive success on many benchmarks for mathematical reasoning. However, there is growing concern that some of this performance actually reflects dataset contamination, where data closely resembling benchmark questions leaks into the training data, instead of true reasoning ability. To investigate this claim rigorously, we commission Grade School Math 1000 (GSM1k). GSM1k is designed to mirror the style and complexity of the established GSM8k benchmark, the gold standard for measuring elementary mathematical reasoning. We ensure that the two benchmarks are comparable across important metrics such as human solve rates, number of steps in solution, answer magnitude, and more. When evaluating leading open- and closed-source LLMs on GSM1k, we observe accuracy drops of up to 13%, with several families of models (e.g., Phi and Mistral) showing evidence of systematic overfitting across almost all model sizes. At the same time, many models, especially those on the frontier, (e.g., Gemini/GPT/Claude) show minimal signs of overfitting. Further analysis suggests a positive relationship (Spearman's r^2=0.32) between a model's probability of generating an example from GSM8k and its performance gap between GSM8k and GSM1k, suggesting that many models may have partially memorized GSM8k.

Mathematical questioning is crucial for assessing students problem-solving skills. Since manually creating such questions requires substantial effort, automatic methods have been explored. Existing state-of-the-art models rely on fine-tuning strategies and struggle to generate questions that heavily involve multiple steps of logical and arithmetic reasoning. Meanwhile, large language models(LLMs) such as ChatGPT have excelled in many NLP tasks involving logical and arithmetic reasoning. Nonetheless, their applications in generating educational questions are underutilized, especially in the field of mathematics. To bridge this gap, we take the first step to conduct an in-depth analysis of ChatGPT in generating pre-university math questions. Our analysis is categorized into two main settings: context-aware and context-unaware. In the context-aware setting, we evaluate ChatGPT on existing math question-answering benchmarks covering elementary, secondary, and ternary classes. In the context-unaware setting, we evaluate ChatGPT in generating math questions for each lesson from pre-university math curriculums that we crawl. Our crawling results in TopicMath, a comprehensive and novel collection of pre-university math curriculums collected from 121 math topics and 428 lessons from elementary, secondary, and tertiary classes. Through this analysis, we aim to provide insight into the potential of ChatGPT as a math questioner.

In precision-oriented tasks like answer ranking, it is more important to rank many relevant answers highly than to retrieve all relevant answers. It follows that a good ranking strategy would be to learn how to identify the easiest correct answers first (i.e., assign a high ranking score to answers that have characteristics that usually indicate relevance, and a low ranking score to those with characteristics that do not), before incorporating more complex logic to handle difficult cases (e.g., semantic matching or reasoning). In this work, we apply this idea to the training of neural answer rankers using curriculum learning. We propose several heuristics to estimate the difficulty of a given training sample. We show that the proposed heuristics can be used to build a training curriculum that down-weights difficult samples early in the training process. As the training process progresses, our approach gradually shifts to weighting all samples equally, regardless of difficulty. We present a comprehensive evaluation of our proposed idea on three answer ranking datasets. Results show that our approach leads to superior performance of two leading neural ranking architectures, namely BERT and ConvKNRM, using both pointwise and pairwise losses. When applied to a BERT-based ranker, our method yields up to a 4% improvement in MRR and a 9% improvement in P@1 (compared to the model trained without a curriculum). This results in models that can achieve comparable performance to more expensive state-of-the-art techniques.

Small-scale models offer various computational advantages, and yet to which extent size is critical for problem-solving abilities remains an open question. Specifically for solving grade school math, the smallest model size so far required to break the 80\% barrier on the GSM8K benchmark remains to be 34B. Our work studies how high-quality datasets may be the key for small language models to acquire mathematical reasoning. We introduce TinyGSM, a synthetic dataset of 12.3M grade school math problems paired with Python solutions, generated fully by GPT-3.5. After finetuning on TinyGSM, we find that a duo of a 1.3B generation model and a 1.3B verifier model can achieve 81.5\% accuracy, outperforming existing models that are orders of magnitude larger. This also rivals the performance of the GPT-3.5 ``teacher'' model (77.4\%), from which our model's training data is generated. Our approach is simple and has two key components: 1) the high-quality dataset TinyGSM, 2) the use of a verifier, which selects the final outputs from multiple candidate generations.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

We consider in this work quantities that can be obtained as limits of powers of parametrized matrices, for instance the inverse matrix or the logarithm of the determinant. Under the assumption of affine dependence in the parameters, we use the Empirical Interpolation Method (EIM) to derive an approximation for powers of these matrices, from which we derive a nonintrusive approximation for the aforementioned limits. We derive upper bounds of the error made by the obtained formula. Finally, numerical comparisons with classical intrusive and nonintrusive approximation techniques are provided: in the considered test-cases, our algorithm performs well compared to the nonintrusive ones.

Motivated by recent advances in operational weather forecasting, we study the efficacy of low-precision arithmetic for climate simulations. We develop a framework to measure rounding error in a climate model which provides a stress-test for a low-precision version of the model, and we apply our method to a variety of models including the Lorenz system; a shallow water approximation for flow over a ridge; and a coarse resolution global atmospheric model with simplified parameterisations (SPEEDY). Although double precision (52 significant bits) is standard across operational climate models, in our experiments we find that single precision (23 sbits) is more than enough and that as low as half precision (10 sbits) is often sufficient. For example, SPEEDY can be run with 12 sbits across the entire code with negligible rounding error and this can be lowered to 10 sbits if very minor errors are accepted, amounting to less than 0.1 mm/6hr for the average grid-point precipitation, for example. Our test is based on the Wasserstein metric and this provides stringent non-parametric bounds on rounding error accounting for annual means as well as extreme weather events. In addition, by testing models using both round-to-nearest (RN) and stochastic rounding (SR) we find that SR can mitigate rounding error across a range of applications. Thus our results also provide evidence that SR could be relevant to next-generation climate models. While many studies have shown that low-precision arithmetic can be suitable on short-term weather forecasting timescales, our results give the first evidence that a similar low precision level can be suitable for climate.

Medical reasoning models (MRMs) achieve superior performance on medical benchmarks compared to medical LLMs; however, high accuracy alone is insufficient for practical deployment. One of such requirements for real-world application is robustness to varying output constraints. Specifically, posing the same medical question while requesting different answer formats should not affect the underlying correctness of the response. We investigate this phenomenon in this paper, focusing on MRMs. To quantify this behavior, we propose the metric answer-format robustness: the ability to reliably generate correct outputs across varying specified formats. We examine three representative formats: multiple-choice, open-ended question-answering, and ranked lists. Across 15 proprietary and open-weight models, we observe substantial variation in format robustness (35-100%). Furthermore, we conduct controlled fine-tuning experiments on a shared backbone with matched training data to isolate the effects of the fine-tuning paradigm. We find that supervised fine-tuning yields more stable behavior across formats, whereas reinforcement fine-tuning often exhibits higher cross-format brittleness, with the degree of instability strongly dependent on reward design. Overall, answer-format robustness in MRMs is trainable yet brittle and requires careful evaluation for practical medical use.

Advancements in LLMs have significantly expanded their capabilities across various domains. However, mathematical reasoning remains a challenging area, prompting the development of math-specific LLMs. These models typically follow a two-stage training paradigm: pre-training with math-related corpora and post-training with problem datasets for SFT. Despite these efforts, the improvements in mathematical reasoning achieved through continued pre-training (CPT) are often less significant compared to those obtained via SFT. This study addresses this discrepancy by exploring alternative strategies during the pre-training phase, focusing on the use of problem-solving data over general mathematical corpora. We investigate three primary research questions: (1) Can problem-solving data enhance the model's mathematical reasoning capabilities more effectively than general mathematical corpora during CPT? (2) Are synthetic data from the same source equally effective, and which synthesis methods are most efficient? (3) How do the capabilities developed from the same problem-solving data differ between the CPT and SFT stages, and what factors contribute to these differences? Our findings indicate that problem-solving data significantly enhances the model's mathematical capabilities compared to general mathematical corpora. We also identify effective data synthesis methods, demonstrating that the tutorship amplification synthesis method achieves the best performance. Furthermore, while SFT facilitates instruction-following abilities, it underperforms compared to CPT with the same data, which can be partially attributed to its poor learning capacity for hard multi-step problem-solving data. These insights provide valuable guidance for optimizing the mathematical reasoning capabilities of LLMs, culminating in our development of a powerful mathematical base model called JiuZhang-8B.

How can we perform computations over natural language representations to solve tasks that require symbolic and numeric reasoning? We propose natural language embedded programs (NLEP) as a unifying framework for addressing math/symbolic reasoning, natural language understanding, and instruction following tasks. Our approach prompts a language model to generate full Python programs that define functions over data structures which contain natural language representations of structured knowledge. A Python interpreter then executes the generated code and prints the output. Despite using a task-general prompt, we find that this approach can improve upon strong baselines across a range of different tasks including math and symbolic reasoning, text classification, question answering, and instruction following. We further find the generated programs are often interpretable and enable post-hoc verification of the intermediate reasoning steps.

Math word problems are critical K-8 educational tools, but writing them is time-consuming and requires domain expertise. We suggest that language models can support K-8 math education by automatically generating problems at scale. To be educational, generated problems must be 1) solvable, 2) accurate, and 3) appropriate. Existing datasets are unlabeled for these criteria, making them ill-suited for training problem generators. We introduce MATHWELL, a Llama-2 (70B) model iteratively finetuned to generate K-8 math word problems using data from expert annotation. Using MATHWELL, we generate the largest English word problem dataset to date, containing 20,490 problems. 3,484 are scored by domain experts who find MATHWELL has a 40% higher share of problems that have executable solutions and meet all criteria than alternatives, with 74% of its problems with executable solutions being solvable, accurate, and appropriate.

Can AI make progress on important, unsolved mathematical problems? Large language models are now capable of sophisticated mathematical and scientific reasoning, but whether they can perform novel research is still widely debated and underexplored. We introduce HorizonMath, a benchmark of over 100 predominantly unsolved problems spanning 8 domains in computational and applied mathematics, paired with an open-source evaluation framework for automated verification. Our benchmark targets a class of problems where discovery is hard, requiring meaningful mathematical insight, but verification is computationally efficient and simple. Because these solutions are unknown, HorizonMath is immune to data contamination, and most state-of-the-art models score near 0%. Existing research-level benchmarks instead rely on formal proof verification or manual review, both of which are expensive to scale. Using this platform, we find two problems for which GPT 5.4 Pro proposes solutions that improve on the best-known published results, representing potential novel contributions (pending expert review). We release HorizonMath as an open challenge and a growing community resource, where correct solutions to problems in the unsolved problem classes could constitute novel results in the mathematical literature.

Recent advancements in Large Language Models (LLMs) have sparked interest in their formal reasoning capabilities, particularly in mathematics. The GSM8K benchmark is widely used to assess the mathematical reasoning of models on grade-school-level questions. While the performance of LLMs on GSM8K has significantly improved in recent years, it remains unclear whether their mathematical reasoning capabilities have genuinely advanced, raising questions about the reliability of the reported metrics. To address these concerns, we conduct a large-scale study on several SOTA open and closed models. To overcome the limitations of existing evaluations, we introduce GSM-Symbolic, an improved benchmark created from symbolic templates that allow for the generation of a diverse set of questions. GSM-Symbolic enables more controllable evaluations, providing key insights and more reliable metrics for measuring the reasoning capabilities of models.Our findings reveal that LLMs exhibit noticeable variance when responding to different instantiations of the same question. Specifically, the performance of all models declines when only the numerical values in the question are altered in the GSM-Symbolic benchmark. Furthermore, we investigate the fragility of mathematical reasoning in these models and show that their performance significantly deteriorates as the number of clauses in a question increases. We hypothesize that this decline is because current LLMs cannot perform genuine logical reasoning; they replicate reasoning steps from their training data. Adding a single clause that seems relevant to the question causes significant performance drops (up to 65%) across all state-of-the-art models, even though the clause doesn't contribute to the reasoning chain needed for the final answer. Overall, our work offers a more nuanced understanding of LLMs' capabilities and limitations in mathematical reasoning.

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.

Large language models (LLMs) have achieved impressive performance across various mathematical reasoning benchmarks. However, there are increasing debates regarding whether these models truly understand and apply mathematical knowledge or merely rely on shortcuts for mathematical reasoning. One essential and frequently occurring evidence is that when the math questions are slightly changed, LLMs can behave incorrectly. This motivates us to evaluate the robustness of LLMs' math reasoning capability by testing a wide range of question variations. We introduce the adversarial grade school math (\datasetname) dataset, an extension of GSM8K augmented with various mathematical perturbations. Our experiments on 25 LLMs and 4 prompting techniques show that while LLMs exhibit different levels of math reasoning abilities, their performances are far from robust. In particular, even for problems that have been solved in GSM8K, LLMs can make mistakes when new statements are added or the question targets are altered. We also explore whether more robust performance can be achieved by composing existing prompting methods, in which we try an iterative method that generates and verifies each intermediate thought based on its reasoning goal and calculation result. Code and data are available at https://github.com/qtli/GSM-Plus.

Mathematical problem solving remains a challenging test of reasoning for large language and multimodal models, yet existing benchmarks are limited in size, language coverage, and task diversity. We introduce MathNet, a high-quality, large-scale, multimodal, and multilingual dataset of Olympiad-level math problems together with a benchmark for evaluating mathematical reasoning in generative models and mathematical retrieval in embedding-based systems. MathNet spans 47 countries, 17 languages, and two decades of competitions, comprising 30,676 expert-authored problems with solutions across diverse domains. In addition to the core dataset, we construct a retrieval benchmark consisting of mathematically equivalent and structurally similar problem pairs curated by human experts. MathNet supports three tasks: (i) Problem Solving, (ii) Math-Aware Retrieval, and (iii) Retrieval-Augmented Problem Solving. Experimental results show that even state-of-the-art reasoning models (78.4% for Gemini-3.1-Pro and 69.3% for GPT-5) remain challenged, while embedding models struggle to retrieve equivalent problems. We further show that retrieval-augmented generation performance is highly sensitive to retrieval quality; for example, DeepSeek-V3.2-Speciale achieves gains of up to 12%, obtaining the highest scores on the benchmark. MathNet provides the largest high-quality Olympiad dataset together with the first benchmark for evaluating mathematical problem retrieval, and we publicly release both the dataset and benchmark at https://mathnet.mit.edu.

To thoroughly assess the mathematical reasoning abilities of Large Language Models (LLMs), we need to carefully curate evaluation datasets covering diverse mathematical concepts and mathematical problems at different difficulty levels. In pursuit of this objective, we propose FineMath in this paper, a fine-grained mathematical evaluation benchmark dataset for assessing Chinese LLMs. FineMath is created to cover the major key mathematical concepts taught in elementary school math, which are further divided into 17 categories of math word problems, enabling in-depth analysis of mathematical reasoning abilities of LLMs. All the 17 categories of math word problems are manually annotated with their difficulty levels according to the number of reasoning steps required to solve these problems. We conduct extensive experiments on a wide range of LLMs on FineMath and find that there is still considerable room for improvements in terms of mathematical reasoning capability of Chinese LLMs. We also carry out an in-depth analysis on the evaluation process and methods that have been overlooked previously. These two factors significantly influence the model results and our understanding of their mathematical reasoning capabilities. The dataset will be publicly available soon.

This paper introduces a novel training methodology that enables a small Transformer model to generalize the addition of two-digit numbers to numbers with unseen lengths of digits. The proposed approach employs an autoregressive generation technique, processing from right to left, which mimics a common manual method for adding large numbers. To the best of my knowledge, this methodology has not been previously explored in the literature. All results are reproducible, and the corresponding R code is available at: https://github.com/AGPatriota/ALGA-R/.

We introduce a large language model (LLM) based approach to answer complex questions requiring multi-hop numerical reasoning over financial reports. While LLMs have exhibited remarkable performance on various natural language and reasoning tasks, complex reasoning problems often rely on few-shot prompts that require carefully crafted examples. In contrast, our approach uses novel zero-shot prompts that guide the LLM to encode the required reasoning into a Python program or a domain specific language. The generated program is then executed by a program interpreter, thus mitigating the limitations of LLM in performing accurate arithmetic calculations. We evaluate the proposed approach on three financial datasets using some of the recently developed generative pretrained transformer (GPT) models and perform comparisons with various zero-shot baselines. The experimental results demonstrate that our approach significantly improves the accuracy for all the LLMs over their respective baselines. We provide a detailed analysis of the results, generating insights to support our findings. The success of our approach demonstrates the enormous potential to extract complex domain specific numerical reasoning by designing zero-shot prompts to effectively exploit the knowledge embedded in LLMs.

Mathematical reasoning continues to be a critical challenge in large language model (LLM) development with significant interest. However, most of the cutting-edge progress in mathematical reasoning with LLMs has become closed-source due to lack of access to training data. This lack of data access limits researchers from understanding the impact of different choices for synthesizing and utilizing the data. With the goal of creating a high-quality finetuning (SFT) dataset for math reasoning, we conduct careful ablation experiments on data synthesis using the recently released Llama3.1 family of models. Our experiments show that: (a) solution format matters, with excessively verbose solutions proving detrimental to SFT performance, (b) data generated by a strong teacher outperforms on-policy data generated by a weak student model, (c) SFT is robust to low-quality solutions, allowing for imprecise data filtering, and (d) question diversity is crucial for achieving data scaling gains. Based on these insights, we create the OpenMathInstruct-2 dataset, which consists of 14M question-solution pairs (approx 600K unique questions), making it nearly eight times larger than the previous largest open-source math reasoning dataset. Finetuning the Llama-3.1-8B-Base using OpenMathInstruct-2 outperforms Llama3.1-8B-Instruct on MATH by an absolute 15.9\% (51.9\% rightarrow 67.8\%). Finally, to accelerate the open-source efforts, we release the code, the finetuned models, and the OpenMathInstruct-2 dataset under a commercially permissive license.