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

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.

Figures (17)

• Figure 1: A schematic representation of a DeepONet, which is a neural operator trained to learn the mapping from the input function $u(x)$ to the output function $\mathcal{G}(u)(y)$, evaluated at $y \in \mathbb{R}^q$. DeepONet consists of two networks: a branch network, which processes the input function evaluations $[ u(x_1), u(x_2), \dots, u(x_m)]^\top$, and a trunk network, which takes in the coordinates $y$.
• Figure 2: Illustration of the two-stage process for training a combination of DeepONet models for different field components to impose the PEC boundary condition $\bm{\hat{n}} \times \bm{E} = 0$ on the electric field.
• Figure 3: Computational domain used for electromagnetic simulations. Subfigure (a) shows the simplified outer boundary with a metallic sphere (in green color) at the center. Subfigure (b) shows crinkled view of the tetrahedral meshes used to discretize the computational domain show in (a).
• Figure 4: Left subfigure shows training loss history for each electromagnetic field using complex-valued DeepONets (solid lines) and real-valued DeepONets (dashed lines), starting from epoch 500 on the logarithmic scale. Each field component is distinguished by color. The figure shows the superior convergence of the complex-valued DeepONet. The right subfigure shows testing performance of DeepONets (the complex-valued DeepONet and the reparametrized complex-valued DeepONet). The plot shows a superior performance of complex-valued DeepONets over real-valued DeepONet.
• Figure 5: Scatter plots showing the correlation between predicted and true values for the real ($\Re (E)$) and imaginary ($\Im(E)$) components of the electric field, and for the real ($\Re(H)$) and imaginary ($\Im(H)$) components of the magnetic field. Strong diagonal alignment demonstrates high prediction accuracy for both fields.