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

• URL: http://arxiv.org/abs/2506.19243v1
• Date: Tue, 24 Jun 2025 02:01:44 GMT
• Title: High precision PINNs in unbounded domains: application to singularity formulation in PDEs
• Authors: Yixuan Wang, Ziming Liu, Zongyi Li, Anima Anandkumar, Thomas Y. Hou,
• Abstract summary: We study the choices of neural network ansatz, sampling strategy, and optimization algorithm.<n>For 1D Burgers equation, our framework can lead to a solution with very high precision.<n>For the 2D Boussinesq equation, we obtain a solution whose loss is $4$ digits smaller than that obtained in citewang2023asymptotic with fewer training steps.
• Score: 83.50980325611066
• License: http://creativecommons.org/licenses/by/4.0/
• 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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