PINNs一直跑不过传统方法，DSGNAR用二阶优化解决了问题，精度和速度都大幅提升，代码已开源，值得看看。
DSGNAR是一种针对物理信息神经网络（PINNs）训练中病态损失问题的二阶优化框架，通过双重草图高斯-牛顿与自适应正则化策略，在多个基准上取得突破。在双精度下，相对ℓ2误差可低至3×10^{-16}；在经典Burgers方程上比现有结果提升五个数量级，在高维Poisson问题上提升八个数量级。单精度下，Burgers方程可在10秒内达到ℓ2_rel=4.75×10^{-7}。该方法对不同架构、精度和超参数均表现鲁棒。
An Optimisation Framework for the Well-Conditioned Training of Physics-Informed Neural Networks
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