Numerically solving high-dimensional partial differential equations (PDEs) is
a major challenge. Conventional methods, such as finite difference methods, are
unable to solve high-dimensional PDEs due to the curse-of-dimensionality. A
variety of deep learning methods have been recently developed to try and solve
high-dimensional PDEs by approximating the solution using a neural network. In
this paper, we prove global convergence for one of the commonly-used deep
learning algorithms for solving PDEs, the Deep Galerkin Method (DGM). DGM
trains a neural network approximator to solve the PDE using stochastic gradient
descent. We prove that, as the number of hidden units in the single-layer
network goes to infinity (i.e., in the ``wide network limit"), the trained
neural network converges to the solution of an infinite-dimensional linear
ordinary differential equation (ODE). The PDE residual of the limiting
approximator converges to zero as the training time $\rightarrow \infty$. Under
mild assumptions, this convergence also implies that the neural network
approximator converges to the solution of the PDE. A closely related class of
deep learning methods for PDEs is Physics Informed Neural Networks (PINNs).
Using the same mathematical techniques, we can prove a similar global
convergence result for the PINN neural network approximators. Both proofs
require analyzing a kernel function in the limit ODE governing the evolution of
the limit neural network approximator. A key technical challenge is that the
kernel function, which is a composition of the PDE operator and the neural
tangent kernel (NTK) operator, lacks a spectral gap, therefore requiring a
careful analysis of its properties.

此研究论文探讨了如何利用深度学习的方法解决高维偏微分方程的问题，并证明了 Deep Galerkin Method 和 Physics Informed Neural Networks 神经网络逼近器的全局收敛性，随着隐藏单元数的增加，这些神经网络收敛于一个无限维线性常微分方程的解。

用于求解偏微分方程的 Deep Galerkin 和 PINNs 方法的全局收敛性

Global Convergence of Deep Galerkin and PINNs Methods for Solving  Partial Differential Equations

High-dimensional PDEs have been a longstanding computational challenge. We
propose to solve high-dimensional PDEs by approximating the solution with a
deep neural network which is trained to satisfy the differential operator,
initial condition, and boundary conditions. Our algorithm is meshfree, which is
key since meshes become infeasible in higher dimensions. Instead of forming a
mesh, the neural network is trained on batches of randomly sampled time and
space points. The algorithm is tested on a class of high-dimensional free
boundary PDEs, which we are able to accurately solve in up to $200$ dimensions.
The algorithm is also tested on a high-dimensional Hamilton-Jacobi-Bellman PDE
and Burgers' equation. The deep learning algorithm approximates the general
solution to the Burgers' equation for a continuum of different boundary
conditions and physical conditions (which can be viewed as a high-dimensional
space). We call the algorithm a "Deep Galerkin Method (DGM)" since it is
similar in spirit to Galerkin methods, with the solution approximated by a
neural network instead of a linear combination of basis functions. In addition,
we prove a theorem regarding the approximation power of neural networks for a
class of quasilinear parabolic PDEs.

提出了一种名为 Deep Galerkin Method（DGM）的算法，它使用深度神经网络来解决高维偏微分方程问题。相较于形成网格，该算法不依赖于网格，而是通过对随机采样的时间和空间点的批量训练来实现。它在一类高维自由边界的偏微分方程、高维哈密顿 - 雅各比 - 贝尔曼（Hamilton-Jacobi-Bellman）偏微分方程和 Burgers 方程上得到了测试，并且在高维空间中能够准确地近似各种边界条件和物理条件下的 Burgers 方程的一般解。此外，论文还证明了神经网络在一类拟线性抛物型偏微分方程上的逼近能力。