A Novel Highly Parallelizable Machine-Learning Based Method for the Fast Solution of Integral Equations for Electromagnetic Scattering Problems

TL;DR

The paper addresses the computational bottlenecks of frequency-domain integral equation solvers for electromagnetic scattering, focusing on load-balancing in MLFMA and low-frequency breakdown.It introduces a group-by-group interaction framework that maps subdomains to a fixed set of uniformly oriented Hertzian dipoles, uses a direct Green's function approach–style translation augmented with machine-learning predictions to handle far-field interactions, and employs a two-stage offline training strategy for complex-valued neural networks.This approach yields identical per-box workloads, strong parallel scalability, broadband accuracy, and reusability across scatterers and IEs, with numerical results showing close agreement to analytical or conventional solutions and resilience to LFB.The work has practical impact for scalable electromagnetic scattering simulations of large or multi-scale objects and paves the way for hierarchical extensions that could further improve load-balancing and efficiency.

Abstract

We propose a novel method for the efficient and accurate iterative solution of frequency domain integral equations (IEs) that are used for large/multi-scale electromagnetic scattering problems. The proposed method uses a novel group-by-group interaction strategy to accurately evaluate far-zone interactions within the framework of the one-box-buffer scheme during the matrix-vector multiplication at each iteration. Briefly, subdomain basis functions that are used to model the scatterer at each box are represented by a fixed number of uniformly distributed and arbitrarily oriented Hertzian dipoles (referred to as uniform basis functions), and then the dipole-to-dipole interactions are predicted in a group-wise manner by employing machine learning algorithms, thereby showcasing efficiency, strong scalability for parallelization and accuracy without the low-frequency breakdown (LFB) problem. Since the dipole representation is independent of the underlying material properties of the scatterer, the proposed method is valid for all types of IEs (surface or volume). Moreover, because the training is performed offline, the resulting networks can be used for any scatterer under any IE, without extra training, as long as the size of, and the distances among the boxes are preserved. The efficiency and accuracy of the proposed method are assessed by comparing our results with those obtained from the conventional multilevel fast multipole algorithm for various scattering problems. The proposed method's parallelization performance is showcased through scalability tests, and its resilience to LFB is demonstrated.

Figures (16)

• Figure 1: 2D projection of a sample scatterer inside a mother box, which is divided into child boxes. For the selected box $SB$, $NF$ boxes correspond to near interactions, and $FF$ boxes correspond to far interactions.
• Figure 2: Mapping from subdomain basis functions to uniform basis functions
• Figure 3: Illustration of the translation stage.
• Figure 4: Inverse mapping from uniform basis functions to subdomain functions
• Figure 5: The structure of the feed-forward and complex-valued fully connected neural network with one hidden layer.