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 and the Schr\"{o}dinger equations in thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. For example, we solve nontrivial nonlinear PDEs (the HJB-Lin equation and the BSB 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：维度灾难会随着维度的增加，导致计算成本呈指数级增长，从而严重消耗计算资源。正如理查德·贝尔曼在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成功，但此类计算仍然代价高昂，且通用非线性偏微分方程在高维下的真正扩展从未实现。本文提出了一种新方法，用于扩展物理信息神经网络以求解任意高维偏微分方程。该新方法称为随机维度梯度下降（SDGD），它将偏微分方程的梯度分解为对应不同维度的片段，并在每次训练PINNs的迭代中随机采样这些维度片段的子集。我们从理论上证明了所提方法的收敛保证及其他期望性质。实验表明，所提方法使我们能够利用PINNs的无网格方法，在单个GPU上快速求解许多公认难解的高维偏微分方程，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与PINNs，在单个GPU上于6小时内求解了100,000维度的非线性偏微分方程（HJB-Lin方程和BSB方程）。由于SDGD是一种通用的PINNs训练方法，因此它可应用于任何当前及未来的PINNs变体，以将其扩展到任意高维偏微分方程。