  We develop a data-driven model discovery and system identification technique
for spatially-dependent boundary value problems (BVPs). Specifically, we
leverage the sparse identification of nonlinear dynamics (SINDy) algorithm and
group sparse regression techniques with a set of forcing functions and
corresponding state variable measurements to yield a parsimonious model of the
system. The approach models forced systems governed by linear or nonlinear
operators of the form $L[u(x)] = f(x)$ on a prescribed domain $x \in [a, b]$.
We demonstrate the approach on a range of example systems, including
Sturm-Liouville operators, beam theory (elasticity), and a class of nonlinear
BVPs. The generated data-driven model is used to infer both the operator and/or
spatially-dependent parameters that describe the heterogenous, physical
quantities of the system. Our SINDy-BVP framework will enables the
characterization of a broad range of systems, including for instance, the
discovery of anisotropic materials with heterogeneous variability.
