Structure-Informed Machine Learning: Harnessing and Extending Group Symmetries

The large amounts of data and computation required of modern artificial intelligence (AI) systems continues to make efficiency, reliability, and uncertainty quantification fundamental scientific challenges. This project develops new mathematical foundations for machine learning methods that incorporate intrinsic symmetries and other forms for structure found in many real-world datasets. The mathematics developed in this project aims to provide foundational principles for designing AI systems that can better exploit structure in data, improve data efficiency, and more reliably quantify uncertainty. By strengthening the mathematical foundations of generative artificial intelligence, the project aims to advance the progress of science and support applications ranging from image analysis, to molecular design, to video generation, with potential long-term benefits for health, technology, and economic prosperity. The project will also contribute to workforce development through research opportunities and training for undergraduate and graduate students as well as the development of new coursework in mathematical machine learning and deep generative models. This project will establish mathematical foundations and novel algorithms for structure-informed machine learning, with an emphasis on group symmetries in generative models and equivariant neural networks. A central goal is to develop rigorous uncertainty quantification and sample-complexity guarantees for symmetry-informed diffusion and flow-based models, extending from fixed-dimensional settings to generative models for datasets containing samples of varying dimensionality. The project will also investigate the implicit bias induced by gradient-based training of group-equivariant neural networks and will compare the effects of built-in equivariance and data augmentation. In addition, the project will study how symmetry-based learning principles can be extended beyond classical group symmetries to more general forms of structured data. The research will draw on mathematical tools from probability and stochastic analysis, harmonic analysis, optimization, partial differential equations, and numerical analysis. The resulting theory and algorithms will provide mathematical guidance for the design of data-efficient and uncertainty-aware AI systems, advancing both the foundations of machine learning and the broader understanding of how structure can be leveraged in artificial intelligence. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.