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Minimization Formulation for Neural Network Based Solution of Maxwell’s Equations in Frequency Domain

Existing neural network (NN) based methods minimize the least square error at sample points to solve Maxwell’s equations. In this paper, we propose to construct the loss function for the training of NN from a pertinent functional governing the solution of Maxwell’s equations. When minimizing the loss function, we find the stationary point of the functional, and hence the desired solution. Using this approach, the boundary conditions at the material interface are naturally satisfied during the minimization procedure, and therefore they do not need to be explicitly enforced in the loss function. Since Maxwell’s equations in frequency domain are complex valued and indefinite in general lossy and inhomogeneous problems, the functional developed here is different from that derived from a Ritz method for solving definite systems. Numerical experiments in both 2-and 3-D simulations have validated the proposed minimization formulation and its resultant NN-based solution. Comparisons with the least square error based NN method have also shown clear advantages of the proposed method.