高精度神经网络求解无界域偏微分方程奇异点，精度超现有方法。
High precision PINNs in unbounded domains: application to singularity formulation in PDEs
- 模块化设计神经网络、采样与优化策略，提升无界域求解精度。
- 1D伯格斯方程解精度高，2D布西内斯克方程损失比前人低4位数。
- 适合研究流体方程奇点问题，为高维逼近提供新路径。
本文研究物理信息神经网络（PINNs）在无界域中的高精度训练，重点关注其在偏微分方程（PDE）奇点建模中的应用。提出一种模块化方法，系统分析神经网络形式、采样策略与优化算法的选择。结合严格的计算机辅助证明与PDE分析，若数值解精度足够高，PINNs可成为研究PDE奇点的强大工具。对1D伯格斯方程，框架可获得极高精度解；对2D布西内斯克方程（与3D欧拉和纳维-斯托克斯方程奇点密切相关），其损失值比文献[ Wang2023 ]中结果低4个数量级，且训练步数更少。此外，讨论了向更高维问题逼近至机器精度的潜在方向。
原文摘要 · Abstract (English)
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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