NSF-FRG Collaboration: Singularities in Incompressible Flows: Computer Assisted Proofs and Physics-Informed Neural Networks

Whether three-dimensional incompressible flows develop singularities in finite time and whether (weak) solutions of Navier-Stokes equations are unique, are two of the most important problems in mathematical fluid dynamics. Any progress towards resolving these problems would have significant implications for the entire field. This project integrates theoretical proofs, numerical analysis, and machine learning for understanding singularities in fluids. Recent investigations by the PIs demonstrate that intersection between mathematical proofs and deep learning offers an exciting avenue for understanding how singularity occurs in fluids. Together, the five PIs encompass strengths in several areas such as mathematical analysis, numerical simulation, or computer-assisted proofs. In addition, the project will foster collaborations and increased interactions between the researchers at several leading research universities in the US, utilizing tools developed in one field to advance another, and promote learning and training of students and postdoctoral researchers with a goal of broadening the participation of researchers from underrepresented groups in the mathematical sciences.

The PIs will focus on three specific projects: (1) non-uniqueness of the Leray-Hopf solutions of the Navier Stokes equations in 3 dimensions, (2) formation of singularities for solutions of the three-dimensional Euler equations, and (3) optimization of physics-informed neural networks (PINN). Students, postdoctoral fellows, and visitors will be actively involved in these collaborations. To promote these exchanges research workshops will be organized once a year at the PIs institutions. These meetings will have two main objectives: a training objective, involving lectures to disseminate current ideas and progress; and an annual meeting of the PIs to review the progress and plan future steps. The PIs will also organize a summer school at Princeton University, aimed at graduate students and advanced undergraduate students. The summer school will have a scientific component, including minicourses on the mathematics of fluids, and a mentorship component, including a round table discussion regarding careers in mathematics and a women in mathematics panel.

Tristan Buckmaster, Professor of Mathematics, New York University

Javier Gómez-Serrano, Associate Professor of Mathematics, Brown University

Alexandru Ionescu, Professor of Mathematics, Princeton University

Hao Jia, Associate Professor of Mathematics, University of Minnesota

Ching-Yao Lai, Assistant Professor of Geophysics, Stanford University

Gonzalo Cao-Labora, Graduate Student, Massachusetts Institute of Technology

Gerard Castro-López, Undergraduate Student, Universitat Politecnica de Catalunya (Spain)

Shan Chen, Graduate Student, University of Minnesota

Hyungjun Choi, Graduate Student, Princeton University

Charlie Cowen-Breen, Graduate Student, Massachusetts Institute of Technology

Tristan Léger, Postdoctoral Associate, New York University

Jungkyoung Na, Graduate Student, Brown University

Stan Palasek, Postdoctoral Fellow, Princeton University and IAS

Noah Stevenson, Graduate Student, Princeton University

Bruno Vergara, Tamarkin Assistant Professor, Brown University

Yongji Wang, Postdoctoral Associate, New York University

Wang, Y. and Lai, C.Y., 2024. Multi-stage neural networks: Function approximator of machine precision. Journal of Computational Physics, p.112865.
G. Cao-Labora, J. Gómez-Serrano, J. Shi, and G. Staffilani, Non-radial implosion for compressible Euler and Navier-Stokes in $\T^3$ and $\R^3$. Preprint (2023), arXiv:2310:05325.

NSF-FRG Conference at University of Minnesota, April 12-14, 2024

NSF-FRG Conference at Princeton University, March 14-16, 2025

NSF-FRG Summer School at Princeton University, June 2-13, 2025