Collaborative Research: RI: Medium: Machine learning for PDEs, and with PDEs

Partial differential equations (PDEs) are a ubiquitous modeling and analysis tool in both pure and applied mathematics and are used in biology, chemistry, quantum mechanics, and many other areas. In the recent few years, the synergy between PDEs and machine learning has dramatically strengthened. On one hand, machine learning methods, specifically neural networks, have been shown to be very useful for improving the process of solving PDEs---at both the level of representing the solutions of individual PDEs, and by capturing the mapping from a PDE to a solution. On the other hand, with the advent of diffusion models as the dominant approach to generative AI, stable, efficient and parallelizable solvers for PDEs are ever more important for training large-scale AI systems. This project will build mathematical foundations for several key questions pertaining to both the use of machine learning for PDE solving, and the use of PDEs as a tool for generative modeling. It will explore issues around the representational power of different neural architectures, their inductive biases, their statistical complexity, and their numerical stability. It will also aim to further elucidate the relative tradeoffs of different PDE-based generative models. The investigators will leverage their joint expertise in mathematical foundations of PDEs and generative modeling, as well as numerical aspects of optimization to fruitfully mine the rich overlap between PDEs and machine learning. 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.