HARDMath2: A Benchmark for Applied Mathematics Built by Students as Part of a Graduate Class
Abstrak
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.
Penulis (45)
James V. Roggeveen
Erik Y. Wang
Will Flintoft
Peter Donets
Lucy S. Nathwani
Nickholas Gutierrez
David Ettel
Anton Marius Graf
Siddharth Dandavate
Arjun Nageswaran
Raglan Ward
Ava Williamson
Anne Mykland
Kacper K. Migacz
Yijun Wang
Egemen Bostan
Duy Thuc Nguyen
Zhe He
Marc L. Descoteaux
Felix Yeung
Shida Liu
Jorge García Ponce
Luke Zhu
Yuyang Chen
Ekaterina S. Ivshina
Miguel Fernandez
Minjae Kim
Kennan Gumbs
Matthew Scott Tan
Russell Yang
Mai Hoang
David Brown
Isabella A. Silveira
Lavon Sykes
Ahmed Roman
William Fredenberg
Yiming Chen
Lucas Martin
Yixing Tang
Kelly Werker Smith
Hongyu Liao
Logan G. Wilson
Alexander Dazhen Cai
Andrea Elizabeth Biju
Michael P. Brenner
Akses Cepat
- Tahun Terbit
- 2025
- Bahasa
- en
- Sumber Database
- arXiv
- Akses
- Open Access ✓