The aim of this article is to provide a firm mathematical foundation for the application of deep gradient flow methods (DGFMs) for the solution of (high-dimensional) partial differential equations (PDEs). We decompose the generalization error of DGFMs into an approximation and a training error. We first show that the solution of PDEs that satisfy reasonable and verifiable assumptions can be approximated by neural networks, thus the approximation error tends to zero as the number of neurons tends to infinity. Then, we derive the gradient flow that the training process follows in the ``wide network limit'' and analyze the limit of this flow as the training time tends to infinity. These results combined show that the generalization error of DGFMs tends to zero as the number of neurons and the training time tend to infinity.

翻译：本文旨在为应用深度梯度流方法求解（高维）偏微分方程提供坚实的数学基础。我们将深度梯度流方法的泛化误差分解为近似误差与训练误差。首先证明，满足合理且可验证假设的偏微分方程解可由神经网络逼近，因此当神经元数量趋于无穷时，近似误差趋于零。随后，推导出训练过程在“宽网络极限”下遵循的梯度流，并分析该流在训练时间趋于无穷时的极限。综合这些结果表明，当神经元数量与训练时间趋于无穷时，深度梯度流方法的泛化误差趋于零。