HARDMath2: A Benchmark for Applied Mathematics Built by Students as Part of a Graduate Class

TL;DR

HARDMath2 is a new benchmark dataset of 211 applied mathematics problems created by students and instructors, designed to evaluate and improve the mathematical reasoning capabilities of large language models in approximation-based problems.

Contribution

This paper introduces HARDMath2, a collaboratively developed dataset of graduate-level applied math problems, and demonstrates its effectiveness in highlighting current LLM limitations and fostering student understanding.

Findings

Leading models struggle with many HARDMath2 problems.

Student interaction enhances problem difficulty and understanding.

Benchmark reveals gaps in current LLM mathematical reasoning.

Abstract

Large language models (LLMs) have shown remarkable progress in mathematical problem-solving, but evaluation has largely focused on problems that have exact analytical solutions or involve formal proofs, often overlooking approximation-based problems ubiquitous in applied science and engineering. To fill this gap, we build on prior work and present HARDMath2, a dataset of 211 original problems covering the core topics in an introductory graduate applied math class, including boundary-layer analysis, WKB methods, asymptotic solutions of nonlinear partial differential equations, and the asymptotics of oscillatory integrals. This dataset was designed and verified by the students and instructors of a core graduate applied mathematics course at Harvard. We build the dataset through a novel collaborative environment that challenges students to write and refine difficult problems consistent with the class syllabus, peer-validate solutions, test different models, and automatically check LLM-generated solutions against their own answers and numerical ground truths. Evaluation results show that leading frontier models still struggle with many of the problems in the dataset, highlighting a gap in the mathematical reasoning skills of current LLMs. Importantly, students identified strategies to create increasingly difficult problems by interacting with the models and exploiting common failure modes. This back-and-forth with the models not only resulted in a richer and more challenging benchmark but also led to qualitative improvements in the students' understanding of the course material, which is increasingly important as we enter an age where state-of-the-art language models can solve many challenging problems across a wide domain of fields.