这篇论文给DGM和PINN的用户吃了定心丸：对于一类半线性PDE，梯度下降训练真的能收敛到正确解，不是只找到局部最优。
这篇论文证明了对于一类半线性偏微分方程，使用梯度下降训练神经网络最小化PDE残差目标函数时，网络参数会全局收敛到PDE的解。研究针对Deep Galerkin Method和Physics Informed Neural Networks这两种主流方法，解决了长期存在的非凸目标函数可能只收敛到局部极小值的问题。论文结果覆盖了包含解及其一阶导数非线性的PDE类，为科学机器学习算法提供了数学基础。
Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs
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. For a class of semi-linear PDEs (nonlinear in the solution and its first derivative), we prove that neural networks trained with gradient descent to minimize the PDE residual objective function will converge to the PDE solution.