Forging New Paradigms in Analysis and Partial Differential Equations: A Synergy of Artificial Intelligence and Computer-Assisted Proofs

This project advances the national interest by attacking long-standing problems in mathematics and, in doing so, by developing and validating new ways of using computers and artificial intelligence as rigorous and trustworthy tools for scientific discovery. The project also contains an extensive education and training component that brings students at every level, from high school through the postdoctoral stage, into active mathematical research, creates new courses and programs, and disseminates the resulting knowledge widely to specialists and to the general public. The research is organized into three connected directions: the first concerns the incompressible Euler equation and related models; the second concerns the generalized surface quasi-geostrophic equation and related active scalar models; and the third concerns applications to spectral geometry and mathematical physics, including eigenvalue problems. The methodological core of the project is the interplay between traditional and modern mathematics. Interval arithmetic and computer-assisted proofs are used to obtain rigorous, quantitative results in regimes where classical techniques do not succeed, for example to enclose singular solutions or to control the spectrum of complicated operators. Machine learning methods, in particular physics-informed neural networks, are used to discover new self-similar and even unstable blow-up profiles with very high precision, and emerging tools based on large language models are used to attack open problems at scale. These computational approaches are combined throughout with modern methods from partial differential equations and harmonic analysis. 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.