SINDy-BVP: Sparse identification of nonlinear dynamics for boundary value problems

Abstract

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 heterogeneous material systems. The technique models forced systems governed by linear or nonlinear operators of the form L[u(x)]=f(x) on a prescribed domain x[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 the governing operator and spatially-dependent parameters that describe the heterogenous, physical quantities of the system. Our SINDy-BVP framework enables the characterization of a broad range of systems, including for instance, the discovery of anisotropic materials with heterogeneous variability.