A Domain-adaptive Physics-informed Neural Network for Inverse Problems
  of Maxwell's Equations in Heterogeneous Media
Maxwell's equations are a collection of coupled partial differential
equations (PDEs) that, together with the Lorentz force law, constitute the
basis of classical electromagnetism and electric circuits. Effectively solving
Maxwell's equations is crucial in various fields, like electromagnetic
scattering and antenna design optimization. Physics-informed neural networks
(PINNs) have shown powerful ability in solving PDEs. However, PINNs still
struggle to solve Maxwell's equations in heterogeneous media. To this end, we
propose a domain-adaptive PINN (da-PINN) to solve inverse problems of Maxwell's
equations in heterogeneous media. First, we propose a location parameter of
media interface to decompose the whole domain into several sub-domains.
Furthermore, the electromagnetic interface conditions are incorporated into a
loss function to improve the prediction performance near the interface. Then,
we propose a domain-adaptive training strategy for da-PINN. Finally, the
effectiveness of da-PINN is verified with two case studies.
Shiyuan Piao, Hong Gu, Aina Wang, Pan Qin
A Domain-adaptive Physics-informed Neural Network for Inverse Problems of Maxwell's Equations in Heterogeneous Media

###### Abstract

Maxwell's equations are a collection of coupled partial differential equations (PDEs) that, together with the Lorentz force law, constitute the basis of classical electromagnetism and electric circuits. Effectively solving Maxwell's equations is crucial in various fields, like electromagnetic scattering and antenna design optimization. Physics-informed neural networks (PINNs) have shown powerful ability in solving PDEs. However, PINNs still struggle to solve Maxwell's equations in heterogeneous media. To this end, we propose a domain-adaptive PINN (da-PINN) to solve inverse problems of Maxwell's equations in heterogeneous media. First, we propose a location parameter of media interface to decompose the whole domain into several sub-domains. Furthermore, the electromagnetic interface conditions are incorporated into a loss function to improve the prediction performance near the interface. Then, we propose a domain-adaptive training strategy for da-PINN. Finally, the effectiveness of da-PINN is verified with two case studies.

 Domain-adaptive physics-informed neural network, domain-adaptive training strategy, Maxwell's equations in heterogeneous media.

## I Introduction

Maxwell's equations are used to describe the fundamental laws of electromagnetic fields [1], which are widely used in geological exploration [2], medical imaging [3], and many other fields [4, 5]. The Maxwell's equations are classically solved by numerical methods [6], such as finite-difference time-domain method [7], finite element method [8], Born iterative method [9], and variational Born iteration method [10]. However, one of the challenges these numerical methods face is the high computational cost [11].

The deep learning methods with remarkable approximation capability have attracted increasing attention in solving partial differential equations (PDEs) [12, 13]. The physics-informed neural networks (PINNs) proposed in [14] are prevalent mesh-free methods for solving PDEs. In the framework of PINNs, the prior knowledge of physics and sparse measurements are incorporated into the loss function to train PINNs. Successful achievements in estimating the parameters of PDEs with homogeneous media, such as supersonic flows [15], beam systems [16], and nano-optics [17], have been reported [18, 19]. Meanwhile, the Maxwell's equations in heterogeneous media have wide applications, such as nondestructive testing of multilayered coating [20, 21]. However, it remains a challenge for PINNs to solve inverse problems in heterogeneous media composed of several homogeneous media [22]. In the heterogeneous media case, the parameters of PDEs are not fixed in the whole domain and will jump at media interface. Such parameter jumping can potentially lead to PINNs could not work well. In [23, 24], PINNs were used to solve inverse problems of the Maxwell's equations in heterogeneous media. However, these methods obtain inaccurate estimations near the interface due to the interface conditions not being considered, as indicated in our experiments.

To solve the aforementioned problems, this letter proposes a domain-adaptive PINN (da-PINN) for the Maxwell's equations in heterogeneous media. The goal of da-PINN is to estimate parameters of the Maxwell's equations and predict the electric and magnetic fields. We first propose a location parameter of media interface to decompose the whole domain into several sub-domains under the assumption of each sub-domain with fixed parameters of the Maxwell's equations. Then, sub-networks are constructed to approximate electric and magnetic fields in each sub-domain. The electromagnetic interface conditions are incorporated into a loss function. In addition, a domain-adaptive training strategy is proposed to optimize da-PINN. Finally, the performance of da-PINN is verified with one- and two- dimensional Maxwell's equations in heterogeneous media.

This letter is organized as follows. In Section II, we introduce PINNs for a derivative of vector-valued functions and da-PINN. In Section III, da-PINN is verified with one- and two- dimensional cases. Section IV concludes this letter.

## II Methodology

### _PINNs for Derivatives of Vector-valued Functions_

In this section, we briefly introduce PINNs for inverse problems in the derivative of vector-valued functions, the general form is as follows:

\[\mathbf{u}_{t}(t,\mathbf{x})+\mathbf{\mathcal{N}}\left[\mathbf{u}(t,\mathbf{x});\mathbf{\lambda} \right]=0,\ \mathbf{x}\in\Omega\subseteq\mathbb{R}^{D},\ t\in\left[0,T\right]. \tag{1}\]

Here, \(\mathbf{u}:\mathbb{R}\times\mathbb{R}^{p}\rightarrow\mathbb{R}^{q}\) denotes the solution with spatiotemporal dependence, \(\mathbf{u}_{t}\) is the partial derivative of \(\mathbf{u}\) with respect to \(t\), \(\mathbf{\mathcal{N}}\) is a vector differential operator parameterized by an unknown real-valued vector \(\mathbf{\lambda}\in\mathbb{R}^{m}\), \((t,\mathbf{x})\) is a pair-wise