用于求解非线性偏微分方程的DGM和PINN算法的全局收敛性
Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs
- Mathematical Institute, University of Oxford（牛津大学数学研究所）
- Department of Mathematics and Statistics, Boston University（波士顿大学数学与统计系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究一类半线性偏微分方程，证明用梯度下降训练来最小化PDE残差目标函数的神经网络可收敛到PDE解，为DGM和PINN算法求解此类方程的数学基础提供了依据。
AI中文摘要：
深度伽辽金方法（DGM）和物理信息神经网络（PINNs）在科学机器学习快速发展的领域中已成为求解偏微分方程（PDEs）的广泛使用方法。在这些方法中，通过使用（随机）梯度下降来最小化神经网络的PDE残差，训练神经网络以逼近PDE解。由于PDE残差目标函数的非凸性，训练后的神经网络原则上可能仅收敛到目标函数的局部极小值（这不是PDE的解）。因此，关于这些算法的数学基础存在一个长期问题，确定训练后的神经网络将收敛到PDE解非常有价值。对于一类半线性PDEs（解及其一阶导数为非线性），我们证明了用梯度下降训练以最小化PDE残差目标函数的神经网络将收敛到PDE解。
英文摘要：
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 $\rightarrow \infty$.
