标题： 无界域声散射问题的交替优化SNN方法 显示英文标题

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

摘要： 在本文中，我们提出了一种基于机器学习的新方法来解决无限域中的声学散射问题。我们首先采用狄利克雷-诺伊曼（DtN）算子将物理上无限的区域截断为可计算的有界区域。这种转换将原始的无限域中的散射问题转化为有界区域内的边界值问题。为了解决这个边界值问题，我们设计了一个具有子空间层的神经网络，其中该层中的每个神经元代表一个基函数。因此，近似解可以表示为这些基函数的线性组合。此外，我们引入了一种创新的交替优化技术，通过训练和最小二乘法分别交替更新基函数及其线性组合系数。在我们的方法中，我们将基函数的系数设为1，并在每次训练子空间时使用一个新的损失函数。这些创新确保由这些基函数形成的子空间真正得到优化。我们将这种方法称为基于神经网络的交替优化子空间方法（AO-SNN）。大量的数值实验表明，我们的新方法可以显著地将相对$l^2$误差降低到$10^{-7}$或更低，据我们所知，这优于现有的基于机器学习的方法。 显示英文摘要

摘要： 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.