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：维度灾难随着维度的增加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·E·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（PDEs）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PDE真正扩展至高维度尚未实现。我们提出了一种新方法，用于扩展物理信息神经网络（PINNs）以求解任意高维PDE。该方法名为随机维度梯度下降（SDGD），它将PDE梯度分解为对应不同维度的子分量，并在每次PINNs训练迭代中随机采样这些维度子集。我们从理论上证明了所提方法的收敛性及其他理想性质。通过多种不同测试，我们展示了该方法能够高效求解许多闻名的高维PDE难题，包括汉密尔顿-雅可比-贝尔曼（HJB）方程和薛定谔方程，即使在数万维度的条件下，也能在单个GPU上利用PINNs无网格方法快速求解。值得注意的是，我们利用SDGD结合PINNs，在单个GPU上于12小时内求解了具有非平凡、各向异性且不可分离解的10万有效维度的非线性PDE。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，以将其扩展至任意高维PDE的求解。