An efficient neural network method with plane wave activation functions for solving Helmholtz equation

Abstract

In this paper, we propose a plane wave activation based neural network (PWNN) to solve the Helmholtz equation with constant coefficients and relatively large wave number k efficiently. Since the complex activation function eix→ is introduced to be the activation function of the neural network, PWNN significantly improves the computational speed and accuracy as compared to traditional activation based neural networks (TANN) and the finite difference method (FDM), for relatively large wave number problems. We establish a new upper bound for error estimates for homogeneous Helmholtz solutions in two dimensions by plane waves, which is different from the previous estimates by generalized harmonic polynomials. Based on the new error estimates, the theoretical guidance is given for choosing the number of neural network's neurons and the initial value to accelerate network training. The analyses of computational complexity related to the wave number k are given for PWNN with two layers, TANN, the plane wave partition of unity method (PWPUM) and FDM. Numerical experiments in 2D and 3D are performed to demonstrate the efficiency and accuracy of PWNN. Especially for large wave number problems, like k=500, PWNN can get the solution with relative error less than 10−4 in less than 20 seconds, which is more efficient than other methods.