High precision PINNs in unbounded domains: application to singularity formulation in PDEs

TMLR Paper7584 Authors

Abstract: We investigate the high-precision training of Physics-Informed Neural Networks (PINNs) in unbounded domains, with a special focus on applications to singularity formulation in PDEs. We propose a modularized approach and study the choices of neural network ansatz, sampling strategy, and optimization algorithm. When combined with rigorous computer-assisted proofs and PDE analysis, the numerical solutions identified by PINNs, provided they are of high precision, can serve as a powerful tool for studying singularities in PDEs. For 1D Burgers equation, our framework can lead to a solution with very high precision, and for the 2D Boussinesq equation, which is directly related to the singularity formulation in 3D Euler and Navier-Stokes equations, we obtain a solution whose loss is 4 digits smaller than that obtained in \cite{wang2023asymptotic} with fewer training steps. We also discuss potential directions for pushing towards machine precision for higher-dimensional problems.

Submission Type: Regular submission (no more than 12 pages of main content)

Previous TMLR Submission Url: https://openreview.net/forum?id=CXqR1sqAAP

Changes Since Last Submission: This manuscript was previously desk-rejected twice due to blind and format issues We have addressed the issue by fixing them

Assigned Action Editor: ~William_T_Redman1

Submission Number: 7584