AIMing: Discovery through Machine Learning in Partial Differential Equations

Recent advances in machine learning are demonstrating the potential to find solutions of challenging nonlinear partial differential equations which hold great significance in a variety of fields, including applied mathematics, physics, and engineering. Computer-assisted proofs are gaining momentum in this field, demonstrating their success in situations that have previously been out of reach using traditional approaches. By harnessing the synergy between mathematical principles and computational methods, the goal of this project is to unlock new mathematical insights, streamline computational processes, and push the boundaries of scientific exploration using novel machine learning approaches for understanding partial differential equations. Training of undergraduates, graduate students and postdocs is a fundamental component of this project, involving research questions that are rich and suitable for trainees. The investigator will also devote significant time to enhance and broaden participation in the mathematical community by organizing conferences, designing new courses and creating new programs, as well as contributing with open-source code to existing libraries such as mathlib or Arb. This project has two complementary research goals. Machine-learning approaches will be developed for solving long standing open problems in (broadly defined) nonlinear partial differential equations, including dispersive, elliptic, and geometric frameworks. In complement, the project also intends to advance the heuristics behind machine-learning numerics for such equations, guided by new mathematical knowledge. This involves discovering and verifying new solutions to longstanding conjectures while at the same time improving on machine learning algorithms. This project is divided in four tasks: the first involves the incompressible Euler equation, the second addresses elliptic partial differential equations; the third studies geometric partial differential equations and Calabi-Yau metrics, and the fourth involves the development of mathlib and Lean code. 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.