  We present a method that employs physics-informed deep learning techniques
for parametrically solving partial differential equations. The focus is on the
steady-state heat equations within heterogeneous solids exhibiting significant
phase contrast. Similar equations manifest in diverse applications like
chemical diffusion, electrostatics, and Darcy flow. The neural network aims to
establish the link between the complex thermal conductivity profiles and
temperature distributions, as well as heat flux components within the
microstructure, under fixed boundary conditions. A distinctive aspect is our
independence from classical solvers like finite element methods for data. A
noteworthy contribution lies in our novel approach to defining the loss
function, based on the discretized weak form of the governing equation. This
not only reduces the required order of derivatives but also eliminates the need
for automatic differentiation in the construction of loss terms, accepting
potential numerical errors from the chosen discretization method. As a result,
the loss function in this work is an algebraic equation that significantly
enhances training efficiency. We benchmark our methodology against the standard
finite element method, demonstrating accurate yet faster predictions using the
trained neural network for temperature and flux profiles. We also show higher
accuracy by using the proposed method compared to purely data-driven approaches
for unforeseen scenarios.
