Deep learning-based partial differential equation(PDE) solvers have received much attention in the past few years. Methods of this category can solve a wide range of PDEs with high accuracy, typically by transforming the problems into highly nonlinear optimization problems of neural network parameters. This work reviews several deep learning solvers proposed a few years ago, including PINN, WAN, DRM, and VPINN. Numerical results are provided to make comparisons amongst them and address the importance of loss formulation and the optimization method. A rigorous error analysis for PINN is also presented. Finally, we discuss the current limitations and bottlenecks of these methods.

翻译：基于深度学习的偏微分方程求解器在过去几年中受到广泛关注。此类方法通常通过将问题转化为神经网络参数的高度非线性优化问题，能够以高精度求解各类偏微分方程。本文综述了几年前提出的若干深度学习求解器，包括PINN、WAN、DRM和VPINN。通过数值实验结果对这些方法进行比较，并阐述了损失函数构建与优化方法的重要性。文中同时给出了PINN的严格误差分析。最后，我们讨论了这些方法当前存在的局限性与瓶颈。