Physics-informed Spline Learning for Nonlinear Dynamics Discovery
This addresses a fundamental challenge in fields like physics and engineering for extracting analytical models from limited data, though it appears incremental as it builds on existing sparse representation and physics-informed approaches.
The authors tackled the problem of discovering governing equations for nonlinear dynamics from sparse, noisy data by proposing a Physics-informed Spline Learning (PiSL) framework, which demonstrated efficacy and superiority over state-of-the-art methods in multiple well-known systems.
Dynamical systems are typically governed by a set of linear/nonlinear differential equations. Distilling the analytical form of these equations from very limited data remains intractable in many disciplines such as physics, biology, climate science, engineering and social science. To address this fundamental challenge, we propose a novel Physics-informed Spline Learning (PiSL) framework to discover parsimonious governing equations for nonlinear dynamics, based on sparsely sampled noisy data. The key concept is to (1) leverage splines to interpolate locally the dynamics, perform analytical differentiation and build the library of candidate terms, (2) employ sparse representation of the governing equations, and (3) use the physics residual in turn to inform the spline learning. The synergy between splines and discovered underlying physics leads to the robust capacity of dealing with high-level data scarcity and noise. A hybrid sparsity-promoting alternating direction optimization strategy is developed for systematically pruning the sparse coefficients that form the structure and explicit expression of the governing equations. The efficacy and superiority of the proposed method have been demonstrated by multiple well-known nonlinear dynamical systems, in comparison with two state-of-the-art methods.