提出新优化框架，让物理信息神经网络更准更快求解微分方程。
An Optimisation Framework for the Well-Conditioned Training of Physics-Informed Neural Networks
- 用双重随机化高斯-牛顿法，解决损失函数病态问题。
- 在双精度下误差低至3×10⁻¹⁶，比现有方法提升五到八个数量级。
- 单精度下10秒内达逼近舍入误差极限，适合高维与复杂方程求解。
物理信息神经网络（PINNs）是求解偏微分方程的有前景方法，但其精度难以超越传统求解器。根本原因在于优化过程中的严重病态损失景观。本文提出可扩展的二阶优化框架DSGNAR：双重随机化高斯-牛顿自适应比例法，有效应对病态性，实现前所未有的精度与速度。在非线性、混沌、多尺度、高维及纳维-斯托克斯等各类问题上，该框架显著优于当前最优：双精度下相对ℓ₂误差低至3×10⁻¹⁶，经典Burgers方程误差降低五个数量级，高维泊松问题提升八个数量级，且运行更快。单精度下，10秒内即达Burgers方程ℓ₂ᵣₑₗ = 4.75×10⁻⁷，接近舍入误差极限。该框架对网络结构、精度和初始超参数均具鲁棒性。代码已开源。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers. The obstacle is increasingly understood to be one of optimisation, owing to the severely ill-conditioned loss landscape. We present $\textbf{DSGNAR}$: Doubly-Sketched Gauss-Newton with Adaptive Ratio, a scalable second-order optimisation framework that confronts this ill-conditioning and, in doing so, obtains unprecedented accuracy and speed. $\textbf{DSGNAR}$ couples a doubly-sketched Gauss-Newton model with a novel strategy that carefully controls both regularisation and step length. Across a suite of problems spanning nonlinear, chaotic, multi-scale, high-dimensional, and Navier-Stokes, the framework greatly improves on the state of the art: able to attain relative $\ell_2$ errors as low as $3\times10^{-16}$ in double precision, improve contemporary results by five orders of magnitude on the canonical Burgers' equation, and as much as eight orders on a high-dimensional Poisson problem, while remaining markedly faster. We further show that, in single precision, solutions at the limit of round-off error can be obtained very quickly: Burgers' equation to $\ell_2^{\text{rel}} = 4.75 \times 10^{-7}$ in under ten seconds. The framework is also robust to the choice of architecture, arithmetic precision, and initial hyperparameters. The code is available at https://www.github.com/wephy/physics-informed-neural-networks
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