The curse-of-dimensionality taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional PDEs, as Richard E. Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerically partial differential equations (PDEs) in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. We develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs into pieces corresponding to different dimensions and randomly samples a subset of these dimensional pieces in each iteration of training PINNs. We prove theoretically the convergence and other desired properties of the proposed method. We demonstrate in various diverse tests that the proposed method can solve many notoriously hard high-dimensional PDEs, including the Hamilton-Jacobi-Bellman (HJB) and the Schr\"{o}dinger equations in tens of thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. Notably, we solve nonlinear PDEs with nontrivial, anisotropic, and inseparable solutions in 100,000 effective dimensions in 12 hours on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, it can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.

翻译：维度灾难随着维度增加会导致计算成本呈指数级增长，严重消耗计算资源。正如理查德·贝尔曼60多年前首次指出的那样，这给高维偏微分方程的求解带来了巨大挑战。尽管近年来在高维偏微分方程数值求解方面取得了一些进展，但此类计算成本极高，且通用非线性偏微分方程在真正意义上实现高维扩展始终未能达成。我们开发了一种新方法，用于扩展物理信息神经网络以求解任意高维偏微分方程。该方法名为随机维度梯度下降，它将偏微分方程的梯度分解为对应不同维度的片段，并在每次训练物理信息神经网络的迭代中随机采样这些维度片段的一个子集。我们从理论上证明了该方法的收敛性及其他理想特性。通过多项多样化测试，我们证明该方法能够快速求解众多公认难以处理的高维偏微分方程，包括在单个图形处理器上使用物理信息神经网络无网格方法处理数万维度的哈密顿-雅可比-贝尔曼方程和薛定谔方程。值得注意的是，我们使用基于随机维度梯度下降的物理信息神经网络，在单个图形处理器上仅用12小时便求解了具有非平凡、各向异性且不可分离解的非线性偏微分方程，有效维度高达10万。由于随机维度梯度下降是物理信息神经网络的一种通用训练方法，它可应用于当前及未来任意变体的物理信息神经网络，使其能够扩展至任意高维偏微分方程。