Mai P. Hoang

James V. Roggeveen, Erik Y. Wang, Will Flintoft et al.

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.

Paulina Tomaszewska, Elżbieta Sienkiewicz, Mai P. Hoang et al.

We propose 'Deep spatial context' (DSCon) method, which serves for investigation of the attention-based vision models using the concept of spatial context. It was inspired by histopathologists, however, the method can be applied to various domains. The DSCon allows for a quantitative measure of the spatial context's role using three Spatial Context Measures: $SCM_{features}$, $SCM_{targets}$, $SCM_{residuals}$ to distinguish whether the spatial context is observable within the features of neighboring regions, their target values (attention scores) or residuals, respectively. It is achieved by integrating spatial regression into the pipeline. The DSCon helps to verify research questions. The experiments reveal that spatial relationships are much bigger in the case of the classification of tumor lesions than normal tissues. Moreover, it turns out that the larger the size of the neighborhood taken into account within spatial regression, the less valuable contextual information is. Furthermore, it is observed that the spatial context measure is the largest when considered within the feature space as opposed to the targets and residuals.