  Physics-informed neural network architectures have emerged as a powerful tool
for developing flexible PDE solvers which easily assimilate data, but face
challenges related to the PDE discretization underpinning them. By instead
adapting a least squares space-time control volume scheme, we circumvent issues
particularly related to imposition of boundary conditions and conservation
while reducing solution regularity requirements. Additionally, connections to
classical finite volume methods allows application of biases toward entropy
solutions and total variation diminishing properties. For inverse problems, we
may impose further thermodynamic biases, allowing us to fit shock hydrodynamics
models to molecular simulation of rarefied gases and metals. The resulting
data-driven equations of state may be incorporated into traditional shock
hydrodynamics codes.
