High precision PINNs in unbounded domains: application to singularity formulation in PDEs
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 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.
Keywords
Cite
@article{arxiv.2506.19243,
title = {High precision PINNs in unbounded domains: application to singularity formulation in PDEs},
author = {Yixuan Wang and Ziming Liu and Zongyi Li and Anima Anandkumar and Thomas Y. Hou},
journal= {arXiv preprint arXiv:2506.19243},
year = {2025}
}