神经网络求解一类椭圆型微分方程及其反问题

Neural network for solving elliptic differential equations and inverse problems

根据神经网络方法把椭圆型方程正问题转化为无约束的优化问题,应用反向传播和基于梯度下降的Adam算法对参数进行优化,得到优化目标泛函的最小值,从而实现椭圆型微分方程正问题数值求解。在正问题有效求解的基础上,研究了两类椭圆型微分方程反问题:参数识别反问题和边界反问题。在构造反问题的损失函数中,除了方程的L 2范数外,增加了基于边界观测数据的L 2误差。提出了一类微分方程边界条件自动满足的神经网络构建算法。从数值模拟结果来看,所提出的神经网络算法对求解椭圆型微分方程及其反问题是行之有效的。

According to the neural network method,the forward elliptic differential equation is transformed into an unconstrained optimization problem.The parameters are optimized by back propagation and Adam algorithm based on the gradient descent method,and the minimum value of the optimization objective functional is obtained.Based on the effective solution to the forward problem,two kinds of inverse problems of elliptic equations,i.e.,inverse parameter identification and inverse boundary problem,are investigated..The loss function of the inverse problem is defined by adding the L 2 norm of the equation and the L 2 norm error based on the boundary observation data.A neural network algorithm is proposed,which satisfies the boundary conditions of the partial differential equations.Numerical simulation results show that the proposed neural network algorithm is effective for solving the elliptic differential equations and their inverse problems.