Alternately-optimized SNN method for acoustic scattering problem in unbounded domain

Abstract

In this paper, we propose a novel machine learning-based method to solve the acoustic scattering problem in unbounded domain. We first employ the Dirichlet-to-Neumann (DtN) operator to truncate the physically unbounded domain into a computable bounded domain. This transformation reduces the original scattering problem in the unbounded domain to a boundary value problem within the bounded domain. To solve this boundary value problem, we design a neural network with a subspace layer, where each neuron in this layer represents a basis function. Consequently, the approximate solution can be expressed by a linear combination of these basis functions. Furthermore, we introduce an innovative alternating optimization technique which alternately updates the basis functions and their linear combination coefficients respectively by training and least squares methods. In our method, we set the coefficients of basis functions to 1 and use a new loss function each time train the subspace. These innovations ensure that the subspace formed by these basis functions is truly optimized. We refer to this method as the alternately-optimized subspace method based on neural networks (AO-SNN). Extensive numerical experiments demonstrate that our new method can significantly reduce the relative $l^2$ error to $10^{-7}$ or lower, outperforming existing machine learning-based methods to the best of our knowledge.