  Physics-Informed Neural Networks (PINNs) are a class of deep learning neural
networks that learn the response of a physical system without any simulation
data, and only by incorporating the governing partial differential equations
(PDEs) in their loss function. While PINNs are successfully used for solving
forward and inverse problems, their accuracy decreases significantly for
parameterized systems. PINNs also have a soft implementation of boundary
conditions resulting in boundary conditions not being exactly imposed
everywhere on the boundary. With these challenges at hand, we present
first-order physics-informed neural networks (FO-PINNs). These are PINNs that
are trained using a first-order formulation of the PDE loss function. We show
that, compared to standard PINNs, FO-PINNs offer significantly higher accuracy
in solving parameterized systems, and reduce time-per-iteration by removing the
extra backpropagations needed to compute the second or higher-order
derivatives. Additionally, FO-PINNs can enable exact imposition of boundary
conditions using approximate distance functions, which pose challenges when
applied on high-order PDEs. Through three examples, we demonstrate the
advantages of FO-PINNs over standard PINNs in terms of accuracy and training
speedup.
