  As traditional machine learning tools are increasingly applied to science and
engineering applications, physics-informed methods have emerged as effective
tools for endowing inferences with properties essential for physical
realizability. While promising, these methods generally enforce physics weakly
via penalization. To enforce physics strongly, we turn to the exterior calculus
framework underpinning combinatorial Hodge theory and physics-compatible
discretization of partial differential equations (PDEs). Historically, these
two fields have remained largely distinct, as graphs are strictly topological
objects lacking the metric information fundamental to PDE discretization. We
present an approach where this missing metric information may be learned from
data, using graphs as coarse-grained mesh surrogates that inherit desirable
conservation and exact sequence structure from the combinatorial Hodge theory.
The resulting data-driven exterior calculus (DDEC) may be used to extract
structure-preserving surrogate models with mathematical guarantees of
well-posedness. The approach admits a PDE-constrained optimization training
strategy which guarantees machine-learned models enforce physics to machine
precision, even for poorly trained models or small data regimes. We provide
analysis of the method for a class of models designed to reproduce nonlinear
perturbations of elliptic problems and provide examples of learning
$H(div)/H(curl)$ systems representative of subsurface flows and
electromagnetics.
