Dans ce document, 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.

Cet article explore les excursions dans le temps et leurs implications.

Télécharger PDF: