标题： 偏微分方程的神经网络元法 显示英文标题

标题： Neural Network Element Method for Partial Differential Equations

摘要： 在本文中，基于有限元网格和神经网络的结合，设计了一种新型的神经网络单元空间及相应的机器学习方法，用于求解偏微分方程。 有限元网格的应用使神经网络单元空间能够在复杂几何域上直接满足边界值条件。 使用神经网络使得近似解的精度可以达到神经网络近似的高水平，即使对于具有奇异性的问题也是如此。 我们还提供了所提出方法的误差分析以方便理解。 本文提出的数值方法为神经网络基础的机器学习算法解决来自工程应用的更广泛问题提供了途径。 显示英文摘要

摘要： In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The application of finite element mesh makes the neural network element space satisfy the boundary value conditions directly on the complex geometric domains. The use of neural networks allows the accuracy of the approximate solution to reach the high level of neural network approximation even for the problems with singularities. We also provide the error analysis of the proposed method for the understanding. The proposed numerical method in this paper provides the way to enable neural network-based machine learning algorithms to solve a broader range of problems arising from engineering applications.