Complex Valued Deep Operator Network (DeepONet) $[\mathcal{G}]$ for Three Dimensional Maxwell's Equations: $\mathcal{G} \in \mathbb{C}^{m \times n}$

TL;DR

This paper introduces a complex-valued DeepONet architecture tailored for solving three-dimensional Maxwell's equations, enabling efficient high-frequency electromagnetic field predictions with complex data handling.

Contribution

The work develops a complex-valued DeepONet framework, including a specialized forward pass and a multi-network approach for boundary conditions, advancing neural operator methods for electromagnetic PDEs.

Findings

Enhanced efficiency in predicting high-frequency scattered fields.

Effective handling of complex-valued solutions in Maxwell's equations.

Successful application to 3D metallic objects with singularities.

Abstract

Maxwell's equations, a system of linear partial differential equations (PDEs), describe the behavior of electric and magnetic fields in time and space and are essential for many important electromagnetic applications. Although numerical methods have been applied successfully in the past, the primary challenge in solving these equations arises from the frequency of electromagnetic fields, which depends on the shape and size of the objects to be resolved. Since the domain of influence for these equations is compactly supported, even a small perturbation in frequency necessitates a new discretization of Maxwell's equations, resulting in substantial computational costs. In this work, we investigate the potential of neural operators, particularly the Deep Operator Network (DeepONet) and its variants, as a surrogate model for Maxwell's equations. Existing DeepONet implementations are restricted to real-valued data in $R^n$, but since the time-harmonic Maxwell's equations yield solutions in the complex domain $C^n$, a specialized architecture is required to handle complex algebra. We propose a formulation of DeepONet for complex data, define the forward pass in the complex domain, and adopt a reparametrized version of DeepONet for more efficient training. We also propose a unified framework to combine a plurality of DeepONets, trained for multiple electromagnetic field components, to incorporate the boundary condition. We conduct computational experiments on a 3D metallic sphere without singularities and on a metallic almond-shaped target to demonstrate the effectiveness of the proposed method for problems involving singularity-prone solutions. As shown by computational experiments, our method significantly enhances the efficiency of predicting scattered fields from a spherical object at arbitrary high frequencies.