Physics-informed neural networks for modelling anisotropic and bi-anisotropic electromagnetic constitutive laws through indirect data

This paper investigates the application of PhysicsInformed Neural Networks (PINNs) in modelling constitutive laws for transverse electromagnetic polarized waves in all three space dimensions governed by Maxwell Faraday equation and Ampere's circuital law. The constitutive relationships for electrical permittivity and conductivity are modelled for the 1D case. In 2D, constitutive laws for electric displacement and polarization fields are modelled for anisotropic materials without using the data for either of the two fields. Constitutive laws for bi-anisotropic media are modelled in 3D, where the electromagnetic fields are coupled. It is observed that using PDEs as a guiding constraint regularizes the learning procedure efficiently for training the neural network when no direct measurements of the functions of interest are available. Furthermore, the presented method outperforms the state-of-the-art Gaussian process regression and data-driven deep neural network in the presence of sparse indirect noisy measurement data. For all the numerical experiments, the presented method solves the forward problem simultaneously modelling the constitutive law.