  Recent work in scientific machine learning (SciML) has focused on
incorporating partial differential equation (PDE) information into the learning
process. Much of this work has focused on relatively "easy" PDE operators
(e.g., elliptic and parabolic), with less emphasis on relatively "hard" PDE
operators (e.g., hyperbolic). Within numerical PDEs, the latter problem class
requires control of a type of volume element or conservation constraint, which
is known to be challenging. Delivering on the promise of SciML requires
seamlessly incorporating both types of problems into the learning process. To
address this issue, we propose ProbConserv, a framework for incorporating
conservation constraints into a generic SciML architecture. To do so,
ProbConserv combines the integral form of a conservation law with a Bayesian
update. We provide a detailed analysis of ProbConserv on learning with the
Generalized Porous Medium Equation (GPME), a widely-applicable parameterized
family of PDEs that illustrates the qualitative properties of both easier and
harder PDEs. ProbConserv is effective for easy GPME variants, performing well
with state-of-the-art competitors; and for harder GPME variants it outperforms
other approaches that do not guarantee volume conservation. ProbConserv
seamlessly enforces physical conservation constraints, maintains probabilistic
uncertainty quantification (UQ), and deals well with shocks and
heteroscedasticities. In each case, it achieves superior predictive performance
on downstream tasks.
