Project Grant 2601870

The National Science Foundation Directorate for Mathematical and Physical Sciences awarded Brown University $315,000 on August 1, 2026, under the Mathematical and Physical Sciences program (CFDA 47.049) to study the random geometry of empirical risk landscapes in deep neural networks.

The award funds collaborative research investigating mathematical principles that explain when neural networks generalize well and when training algorithms find good solutions. The project examines the geometry of high-dimensional manifolds formed by network parameters that achieve zero or near-zero training loss in the overparameterized regime, including manifold volume, metric structure, distance from random initialization, and relationship to test error. The research analyzes local geometry near empirical risk minimizers, such as curvature properties that predict generalization, and investigates which regions of the interpolator manifold are selected by training algorithms and how choices such as early stopping influence generalization. By connecting geometry, optimization, and generalization, the project builds a mathematical framework for understanding non-linear deep learning models to support development of more reliable, efficient, and interpretable artificial intelligence systems.

Work is performed in Providence, Rhode Island. The period of performance runs through July 31, 2029.