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arXiv · 2506.19243

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

Yixuan Wang·Ziming Liu·Zongyi Li·Anima Anandkumar·Thomas Y. Hou
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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.

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Yixuan Wang, Ziming Liu, Zongyi Li, Anima Anandkumar, Thomas Y. Hou. 2025-06-24. High precision PINNs in unbounded domains: application to singularity formulation in PDEs. https://arxiv.org/abs/2506.19243

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