热传导界面问题的一种神经网络算法

A neural network algorithm for heat conduction interface problems

摘要:

本文讨论了一种单隐层神经网络算法在数值求解热传导界面问题中的应用.该算法设定含有神经网络函数的近似解满足初边值条件和Dirichlet界面条件，并通过求解由原方程导出的关于神经网络权重的离散优化问题来训练近似解中的神经网络，以使近似解逼近真解.此外，本文还给出了一种基于随机梯度法思想的类随机梯度法来求解相应的离散优化问题.数值算例验证了算法的有效性.

Abstract:

This paper discusses a single hidden neural network algorithm for the heat conduction interface problems. The proposed algorithm sets the approximate solution to contain a neural network function and to meet the initial/boundary conditions and the Dirichlet interface condition, and trains the neural network by solving the discrete optimization problem derived from the original system so as to finally obtain the approximate solution. A new optimization algorithm based on the stochastic gradient method is also given to solve the corresponding discrete optimization problem. Numerical examples are provided to verify the effectiveness of the neural network algorithm.