Application Of Machine Learning Algorithm To Forward And Inverse Heat Equation

In solving the forward and inverse problems of heat conduction equation,traditional numerical methods need to discretize the definite solution in time and space.The computational complexity of the problem depends on the order and dimension of heat conduction equation,and the boundary condition is complex.In recent years,the use of deep neural network to solve partial differential equations has been widely concerned.This paper mainly attempts to construct a deep neural network model based on physical inform to solve the heat conduction equation,hoping to provide an efficient algorithm for the numerical solution of the heat conduction equation.At the same time,the boundary inverse problem of the heat conduction equation and the inverse problem of the heat conduction equation with parameters are studied.In this paper,we first review the main idea and process of solving partial differential equation with multilayer feedforward neural networks and Physics-informed neural networks,and the selection criteria of activation function and the method of parameter optimization are discussed.Then,according to the heat conduction equation,a deep neural network is established to approximate the solution of the equation,and its partial derivative is obtained.The Physical-informed neural network is brought,into the heat conduction equation,and the loss function is constructed according to the initial and boundary points and the equation.The L-BFGS algorithm is used to optimize the training network,and the optimal parameters of the network a.re obtained.Numerical experiments satisfying Dirichlet boundary conditions and Neumann boundary conditions demonstrate the effectiveness of the algorithm.Then,the coupled model of thermal protective clothing with three layers of fabric is solved to obtain the changes of skin temperature under high temperature environment.And then inverse problem of the heat conduction equation is solved by using the feature that the network can predict the value of any space-time position.Finally,the parametric inverse problem of the heat conduction equation is solved by using the Physics-informed neural networks,and the parameters of the equation are obtained by training the network with the observed data The effectiveness of the proposed method is verified by two numerical experiments based on the given observation data with and without noise.The results show that the Physics-informed neural networks can obtain accurate results without spatiotemporal discretization when solving the heat conduction equation.