The formal language for mathematical expressions to which our certification algorithm is applied is specified by the grammar depicted in Figure 1. The language is rich enough to cover all the examples in the main paper and this supplement. In this grammar, number is a placeholder for an arbitrary floating point number, variable is a placeholder for variable names starting with a Latin character and function is a placeholder for the supported elementary differentiable functions like exp,log and sum. Here, is used for transposition and a preceding . Here are some examples from the language (the fist example uses a transposition and the fifth and seventh example use elementwise operations): 2-norm Xw y 2: (X*w-y)'*(X*w-y) logistic log(1+exp(x)): log(1+exp(x)) 1 quadratic x2: x^2 relative entropy xlog(x/y): x*log(x/y), x>0, y>0 logistic regression Our implementation of the Hessian approach works on vectorized and normalized expression DAGs (directed acyclic graphs) for Hessians that contain every subexpression exactly once.

Symbolic regression, the task of predicting the mathematical expression of a function from the observation of its values, is a difficult task which usually involves a two-step procedure: predicting the skeleton of the expression up to the choice of numerical constants, then fitting the constants by optimizing a non-convex loss function. The dominant approach is genetic programming, which evolves candidates by iterating this subroutine a large number of times. Neural networks have recently been tasked to predict the correct skeleton in a single try, but remain much less powerful.In this paper, we challenge this two-step procedure, and task a Transformer to directly predict the full mathematical expression, constants included. One can subsequently refine the predicted constants by feeding them to the non-convex optimizer as an informed initialization. We present ablations to show that this end-to-end approach yields better results, sometimes even without the refinement step. We evaluate our model on problems from the SRBench benchmark and show that our model approaches the performance of state-of-the-art genetic programming with several orders of magnitude faster inference.

Personalized mathematics education is growing rapidly, creating a strong demand for large sets of similar practice problems. Yet existing studies on mathematics problem generation have focused on data augmentation for training neural language models rather than on direct educational deployment. To bridge this gap, we define a new task, Isomorphic Math Problem Generation (IMPG), designed to produce structurally consistent variants of source problems. Subsequently, we explored LLM-based frameworks for automatic IMPG through successive refinements, and established Computational Blueprints for Isomorphic Twins (CBIT). With meta-level generation and template-based selective variation, CBIT achieves high mathematical correctness and structural consistency while reducing the cost of generation. Empirical results across refinements demonstrate that CBIT is superior on generation accuracy and cost-effectiveness at scale. Most importantly, CBIT-generated problems exhibited an error rate 17.8% lower than expert-authored items, with deployment to 6,732 learners on a commercial education platform yielding 186,870 interactions.

Symbolic regression (SR) models complex systems by discovering mathematical expressions that capture underlying relationships in observed data. However, most SR methods prioritize minimizing prediction error over identifying the governing equations, often producing overly complex or inaccurate expressions. To address this, we present a decomposable SR method that generates interpretable multivariate expressions leveraging transformer models, genetic algorithms (GAs), and genetic programming (GP). In particular, our explainable SR method distills a trained ``opaque'' regression model into mathematical expressions that serve as explanations of its computed function. Our method employs a Multi-Set Transformer to generate multiple univariate symbolic skeletons that characterize how each variable influences the opaque model's response. We then evaluate the generated skeletons' performance using a GA-based approach to select a subset of high-quality candidates before incrementally merging them via a GP-based cascade procedure that preserves their original skeleton structure. The final multivariate skeletons undergo coefficient optimization via a GA. We evaluated our method on problems with controlled and varying degrees of noise, demonstrating lower or comparable interpolation and extrapolation errors compared to two GP-based methods, three neural SR methods, and a hybrid approach. Unlike them, our approach consistently learned expressions that matched the original mathematical structure.

Mathematical expressions (MEs) have complex two-dimensional structures in which symbols can be present at any nested depth like superscripts, subscripts, above, below etc. As MEs are represented using LaTeX format, several text retrieval methods based on string matching, vector space models etc., have also been applied for ME retrieval problem in the literature. As these methods are based on syntactic similarity, recently deep learning approaches based on embedding have been used for semantic similarity. In our present work, we have focused on the retrieval of mathematical expressions using deep learning approaches. In our approach, semantic features are extracted from the MEs using a deep recurrent neural network (DRNN) and these features have been used for matching and retrieval. We have trained the network for a classification task which determines the complexity of an ME. ME complexity has been quantified in terms of its nested depth. Based on the nested depth, we have considered three complexity classes of MEs: Simple, Medium and Complex. After training the network, outputs just before the the final fully connected layer are extracted for all the MEs. These outputs form the semantic features of MEs and are stored in a database. For a given ME query, its semantic features are computed using the trained DRNN and matched against the semantic feature database. Matching is performed based on the standard euclidean distance and top 'k' nearest matches are retrieved, where 'k' is a user-defined parameter. Our approach has been illustrated on a database of 829 MEs.

Automatically assessing handwritten mathematical solutions is an important problem in educational technology with practical applications, but it remains a significant challenge due to the diverse formats, unstructured layouts, and symbolic complexity of student work. To address this challenge, we introduce VEHME-a Vision-Language Model for Evaluating Handwritten Mathematics Expressions-designed to assess open-form handwritten math responses with high accuracy and interpretable reasoning traces. VEHME integrates a two-phase training pipeline: (i) supervised fine-tuning using structured reasoning data, and (ii) reinforcement learning that aligns model outputs with multi-dimensional grading objectives, including correctness, reasoning depth, and error localization. To enhance spatial understanding, we propose an Expression-Aware Visual Prompting Module, trained on our synthesized multi-line math expressions dataset to robustly guide attention in visually heterogeneous inputs. Evaluated on AIHub and FERMAT datasets, VEHME achieves state-of-the-art performance among open-source models and approaches the accuracy of proprietary systems, demonstrating its potential as a scalable and accessible tool for automated math assessment. Our training and experiment code is publicly available at our GitHub repository.

Here, we (1) provide the grammar for the formal language of mathematical expressions to which our certification algorithm is applied, (2) we provide more algorithmic details about our implementation of the Hessian approach, (3) we show that our implementation of the Hessian approach can also certify the remaining differentiable CVX atoms with vector input, which we could not discuss in the main paper because of space constraints, and (4) we provide more examples of differentiable functions that can be certified by the Hessian approach but are missing from CVX's DCP implementation. The formal language for mathematical expressions to which our certification algorithm is applied is specified by the grammar depicted in Figure 1. The language is rich enough to cover all the examples in the main paper and this supplement. In this grammar, number is a placeholder for an arbitrary floating point number, variable is a placeholder for variable names starting with a Latin character and function is a placeholder for the supported elementary differentiable functions like exp, log and sum . Our implementation of the Hessian approach works on vectorized and normalized expression DAGs (directed acyclic graphs) for Hessians that contain every subexpression exactly once.

The understanding and modeling of complex physical phenomena through dynamical systems has historically driven scientific progress, as it provides the tools for predicting the behavior of different systems under diverse conditions through time. The discovery of dynamical systems has been indispensable in engineering, as it allows for the analysis and prediction of complex behaviors for computational modeling, diagnostics, prognostics, and control of engineered systems. Joining recent efforts that harness the power of symbolic regression in this domain, we propose a novel framework for the end-to-end discovery of ordinary differential equations (ODEs), termed Grammar-based ODE Discovery Engine (GODE). The proposed methodology combines formal grammars with dimensionality reduction and stochastic search for efficiently navigating high-dimensional combinatorial spaces. Grammars allow us to seed domain knowledge and structure for both constraining, as well as, exploring the space of candidate expressions. GODE proves to be more sample- and parameter-efficient than state-of-the-art transformer-based models and to discover more accurate and parsimonious ODE expressions than both genetic programming- and other grammar-based methods for more complex inference tasks, such as the discovery of structural dynamics. Thus, we introduce a tool that could play a catalytic role in dynamics discovery tasks, including modeling, system identification, and monitoring tasks.

Large language models (LLMs) excel at general mathematical reasoning but fail catastrophically on specialized technical mathematics. In wireless communications, where problems require precise manipulation of information-theoretic bounds, optimization constraints, and signal processing formulations, even state-of-the-art models struggle to achieve competent performance. We present WirelessMathLM, demonstrating that compact models (0.5B-7B parameters) can match or exceed much larger models through domain-specific reinforcement learning with verifiable rewards. Our key insight is that wireless mathematics problems possess a unique property--verifiable correctness--that enables effective reinforcement learning without human feedback. We construct WirelessMathBench-XL, a comprehensive benchmark of 4,027 problems from 970 papers. Using Group Relative Policy Optimization (GRPO) with binary verification rewards, we train models directly from base checkpoints without supervised warm-start. Our 7B model achieves 39.5% accuracy on WirelessMathBench-XL, approaching GPT-4o (40.4%) while using about 100 times fewer parameters than DeepSeek-R1 (671B, 57.4%). Remarkably, GRPO training nearly doubles performance across all model scales (0.5B +11%, 3B +103%, 7B +81%), with positive transfer to general mathematics benchmarks--our models gain +8.4 points on average across MATH, Minerva-Math, OlympiadBench, AMC, and AIME without any training on these tasks.

Chain-of-Thought (CoT) prompting enhances the math reasoning capability of large language models (LLMs) to a large margin. However, the mechanism underlying such improvements remains unexplored. In this paper, we present \textbf{SalaMAnder} (\textbf{S}h\textbf{a}p\textbf{l}ey-b\textbf{a}sed \textbf{M}athematical Expression \textbf{A}ttribution a\textbf{nd} M\textbf{e}t\textbf{r}ic), a theoretically grounded methodology as well as a mathematically rigorous evaluation metric for quantifying component-level contributions in few-shot CoT reasoning. Concretely, we leverage the Shapley value for mathematical expression attribution and develop an efficient stratified sampling algorithm that significantly reduces the computational complexity. Besides, we develop the \textbf{CoSP} (\textbf{C}ardinality \textbf{o}f \textbf{S}hapley \textbf{P}ositives) metric through covariance analysis. Comprehensive validation across popular LLM models and diverse mathematical benchmarks demonstrates that the CoSP metric within our SalaMAnder framework exhibits a robust monotonic correlation with model performance, not only providing theoretical explanations for the empirical success of existing few-shot CoT but also establishing mathematically rigorous principles for prompt construction optimization. Furthermore, we verify the reliability of the explanation, based on which we unify the insights of previous work.