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

TL;DR

This paper develops a high-precision PINN framework for unbounded domains, enabling detailed study of singularities in PDEs with applications to complex fluid dynamics equations.

Contribution

It introduces a modular approach for high-precision PINNs in unbounded domains, improving solution accuracy for PDE singularity analysis.

Findings

Achieved high-precision solutions for 1D Burgers equation.

Obtained more accurate solutions for 2D Boussinesq equation with fewer training steps.

Discussed strategies for reaching machine precision in higher-dimensional PDE problems.

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.