证明了DGM和PINN能收敛到非线性PDE的真实解
Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs
- 用梯度下降训练神经网络最小化PDE残差
- 网络宽度与训练时间趋于无穷时，收敛到真实解
- 为深度学习求解偏微分方程提供理论支撑
深度伽辽金法（DGM）和物理信息神经网络（PINNs）已成为科学机器学习中求解偏微分方程（PDEs）的常用方法。这些方法通过（随机）梯度下降训练神经网络，使其逼近PDE解，目标是最小化神经网络的PDE残差。由于PDE残差目标函数是非凸的，训练后的神经网络理论上可能仅收敛到局部极小值（而非PDE的真实解）。因此，这些算法的数学基础长期存在疑问。本文研究一类包含解及其一阶导数非线性的半线性PDEs，证明当网络宽度和训练时间趋于无穷时，通过梯度下降最小化残差的神经网络将收敛至PDE的真实解。
原文摘要 · Abstract (English)
The Deep Galerkin Method (DGM) and Physics Informed Neural Networks (PINNs) have become widely-used methods for solving partial differential equations (PDEs) in the rapidly growing field of scientific machine learning. In these methods, a neural network is trained to approximate the PDE solution by using (stochastic) gradient descent to minimize the PDE residual of the neural network. Due to the non-convexity of the PDE residual objective function, the trained neural network may, in principle, only converge to a local minimizer of the objective function (which would not be a solution of the PDE). Therefore, there is a longstanding question regarding the mathematical foundations of these algorithms, and it is highly valuable to establish that the trained neural network will converge to the PDE solution. In this paper, we consider a class of semilinear PDEs with nonlinearities in the solution and its first derivative. For this class of PDEs, we prove that neural networks trained with gradient descent to minimize the PDE residual objective function will converge to the PDE solution as the network width and training time $\rightarrow \infty$.
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