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: We updated the draft according to the kind suggestions of the reviewers, other than the caption and visualizations, which we will work on later but wanted to make sure the writing and content address the reviewer's concerns. Thanks again.

Assigned Action Editor: ~William_T_Redman1

Submission Number: 7584