The curse-of-dimensionality (CoD) taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional PDEs as Richard 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. In this paper, 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 samples randomly a subset of these dimensional pieces in each iteration of training PINNs. We theoretically prove the convergence guarantee and other desired properties of the proposed method. We experimentally demonstrate that the proposed method allows us to solve many notoriously hard high-dimensional PDEs, including the Hamilton-Jacobi-Bellman (HJB) and the Schr\"{o}dinger equations in thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. For instance, we solve nontrivial nonlinear PDEs (one HJB equation and one Black-Scholes equation) in 100,000 dimensions in 6 hours on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, SDGD 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（curse-of-dimensionality，CoD）随着维度的增加，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，这给求解高维偏微分方程带来了巨大挑战。尽管近年来在高维偏微分方程的数值求解方面取得了一些进展，但此类计算成本高昂，且通用非线性偏微分方程在高维下的真正扩展从未实现。本文开发了一种新方法，通过扩展物理信息神经网络（PINNs）来求解任意高维偏微分方程。这种新方法被称为随机维度梯度下降（Stochastic Dimension Gradient Descent，SDGD），它将偏微分方程的梯度分解为对应不同维度的片段，并在训练PINNs的每次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所提方法的收敛保证及其他理想性质。实验表明，该方法使我们能够利用PINNs的无网格方法，在单个GPU上极快速地求解许多公认困难的高维偏微分方程，包括数千维度的哈密顿-雅可比-贝尔曼方程和薛定谔方程。例如，我们使用SDGD结合PINNs，在单个GPU上6小时内求解了100,000维的非平凡非线性偏微分方程（一个HJB方程和一个布莱克-斯科尔斯方程）。由于SDGD是PINNs的通用训练方法，它可应用于当前及未来任何PINNs变体，以将其扩展到任意高维偏微分方程。