基于深度神经网络的复杂区域偏微分方程求解

基于深度神经网络求解复杂区域的椭圆型偏微分方程,通过实现深度前馈人工神经网络,构造合适的损失函数和神经网络求解策略,并且提出针对椭圆型偏微分方程的精确、有效的策略和数值方法.该方法只需要在边界和内部上分别选取少量样本点作为训练集,经过迭代学习神经网络的参数使其逼近椭圆型偏微分方程的解.与传统数值方法相比,本方法具有无网格特点,无需生成计算网格,便于处理任意复杂区域问题.数值算例表明此方法可以求解具有复杂区域的微分方程问题且具有较好的数值精度.

Solving partial differential equation with complex geometries based on deep neural network

Based on the deep neural network, this paper solved elliptic partial differential equations in complex regions. By realizing the deep feedforward artificial neural network, the appropriate loss function and neural network solution strategy was constructred, and the accurate and effective strategies as well as numerical methods for elliptic partial differential equations were put forward. This method simply needed to select a small number of sample points on the boundary and inside region as the training set, and the parameters of the neural network were learned iteratively to approximate the solution of the elliptic partial differential equation. Compared with the traditional numerical method, this method is mesh free and does not need to generate computational grid, so it is easy to deal with any complex computational region. Numerical examples show that this method can solve differential equation problems with complex regions and has good numerical accuracy.