  Maxwell's equations govern light propagation and its interaction with matter.
Therefore, the solution of Maxwell's equations using computational
electromagnetic simulations plays a critical role in understanding light-matter
interaction and designing optical elements. Such simulations are often
time-consuming and recent activities have been described to replace or
supplement them with trained deep neural networks (DNNs). Such DNNs typically
require extensive, computationally demanding simulations using conventional
electromagnetic solvers to compose the training dataset. In this paper, we
present a novel scheme to train a DNN that solves Maxwell's equations speedily
and accurately without relying on other computational electromagnetic solvers.
Our approach is to train a DNN using the residual of Maxwell's equations as the
physics-driven loss function for a network that finds the electric field given
the spatial distribution of the material property. We demonstrate it by
training a single network that simultaneously finds multiple solutions of
various aspheric micro-lenses. Furthermore, we exploit the speed of this
network in a novel inverse design scheme to design a micro-lens that maximizes
a desired merit function. We believe that our approach opens up a novel way for
light simulation and optical design of photonics devices.
