Machine Learning for Auxiliary Sources

TL;DR

This paper transforms the Method of Auxiliary Sources, a computational electromagnetics technique, into a neural network framework, enabling optimization and singularity detection through training algorithms like Adam.

Contribution

It introduces a neural network formulation of MAS, allowing for optimized source coefficients, positions, and singularity detection in PDE solutions.

Findings

MAS can be trained as a neural network using standard optimization algorithms.

The neural MAS can accurately detect the position of singularities in unknown functions.

The approach bridges classical PDE methods with modern machine learning techniques.

Abstract

We rewrite the numerical ansatz of the Method of Auxiliary Sources (MAS), typically used in computational electromagnetics, as a neural network, i.e. as a composed function of linear and activation layers. MAS is a numerical method for Partial Differential Equations (PDEs) that employs point sources, which are also exact solutions of the considered PDE, as radial basis functions to match a given boundary condition. In the framework of neural networks we rely on optimization algorithms such as Adam to train MAS and find both its optimal coefficients and positions of the central singularities of the sources. In this work we also show that the MAS ansatz trained as a neural network can be used, in the case of an unknown function with a central singularity, to detect the position of such singularity.