## Physics-based Deep Learning for electromagnetics

Numerical simulations of electromagnetic wave propagation
problems primarily rely on spatially and temporally
discretization of the system of time-domain Maxwell equations
using finite difference or finite element type methods. For
complex and realistic three-dimensional situations, such a
process can be computationally prohibitive, especially in view
of many-query analyses (e.g., optimization design and
uncertainty quantification). Therefore, developing
cost-effective surrogate models is of great practical
significance. Among the different possible approaches for
building a surrogate model of a given PDE system in a
non-intrusive way (i.e., with minimal modifications to an
existing discretization-based simulation methodology),
approaches based on neural networks and Deep Learning (DL) has
recently shown new promises due to their capability of
handling nonlinear or/and high dimensional problems. In the
present study, we propose to focus on the particular case of
Physics-Informed Neural Networks (PINNs). PINNs are neural
networks trained to solve supervised learning tasks while
respecting any given laws of physics described by a general
(possibly nonlinear) PDE system. They seamlessly integrate the
information from both the measurements and partial
differential equations (PDEs) by embedding the PDEs into the
loss function of a neural network using automatic
differentiation.

Scattering of a plane wave by a dielectric disk: comparison
between exact and predicted solution. In the proposed approach,
we train two neural networks (one for each medium) so as to
include the transmission condition between the two media in the
loss definitiones, The two networks share an identical MLP with
three hidden layers and 50 neurons in each layer. We use sin as
the activation function, and a combination of Adam then L-BFGS
optimizations (500 iterations for Adam followed by 15000
iterations for L-BFGS).