  The Space Domain Awareness (SDA) community routinely tracks satellites in
orbit by fitting an orbital state to observations made by the Space
Surveillance Network (SSN). In order to fit such orbits, an accurate model of
the forces that are acting on the satellite is required. Over the past several
decades, high-quality, physics-based models have been developed for satellite
state estimation and propagation. These models are exceedingly good at
estimating and propagating orbital states for non-maneuvering satellites;
however, there are several classes of anomalous accelerations that a satellite
might experience which are not well-modeled, such as satellites that use
low-thrust electric propulsion to modify their orbit. Physics-Informed Neural
Networks (PINNs) are a valuable tool for these classes of satellites as they
combine physics models with Deep Neural Networks (DNNs), which are highly
expressive and versatile function approximators. By combining a physics model
with a DNN, the machine learning model need not learn astrodynamics, which
results in more efficient and effective utilization of machine learning
resources. This paper details the application of PINNs to estimate the orbital
state and a continuous, low-amplitude anomalous acceleration profile for
satellites. The PINN is trained to learn the unknown acceleration by minimizing
the mean square error of observations. We evaluate the performance of pure
physics models with PINNs in terms of their observation residuals and their
propagation accuracy beyond the fit span of the observations. For a two-day
simulation of a GEO satellite using an unmodeled acceleration profile on the
order of $10^{-8} \text{ km/s}^2$, the PINN outperformed the best-fit physics
model by orders of magnitude for both observation residuals (123 arcsec vs 1.00
arcsec) as well as propagation accuracy (3860 km vs 164 km after five days).
