Neural network for solving elliptic differential equations and inverse problems
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 L2 norm of the equation and the L2 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.
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