Sparse identification of nonlinear dynamics for rapid model recovery

Abstract

Big data have become a critically enabling component of emerging mathematical methods aimed at the automated discovery of dynamical systems, where first principles modeling may be intractable. However, in many engineering systems, abrupt changes must be rapidly characterized based on limited, incomplete, and noisy data. Many leading automated learning techniques rely on unrealistically large data sets, and it is unclear how to leverage prior knowledge effectively to re-identify a model after an abrupt change. In this work, we propose a conceptual framework to recover parsimonious models of a system in response to abrupt changes in the low-data limit. First, the abrupt change is detected by comparing the estimated Lyapunov time of the data with the model prediction. Next, we apply the sparse identification of nonlinear dynamics (SINDy) regression to update a previously identified model with the fewest changes, either by addition, deletion, or modification of existing model terms. We demonstrate this sparse model recovery on several examples for abrupt system change detection in periodic and chaotic dynamical systems. Our examples show that sparse updates to a previously identified model perform better with less data, have lower runtime complexity, and are less sensitive to noise than identifying an entirely new model. The proposed abrupt-SINDy architecture provides a new paradigm for the rapid and efficient recovery of a system model after abrupt changes.

Discovery of Algebraic Reynolds-Stress Models Using Sparse Symbolic Regression

“…In [25,28,29] this was done using ordinary least-squares regression for problems in the domains of dynamical systems and biological networks. As mentioned above, the ability of the CFD solver, in which the models will be implemented, to produce a converged solution is sensitive to large coefficients, which has been reported in [11,12,22].…”

Variational system identification of the partial differential equations governing the physics of pattern-formation: Inference under varying fidelity and noise

“…In this work we treat them as distinct bases, thus allowing us to separately identify the Neumann boundary term. We also call attention to the fact that for the unsteady diffusion problem, equating (12) to the sum of (15) and (16) amounts to the Backward Euler time integration algorithm, which is first-order accurate. The mid-point rule, on the other hand is second-order in time.…”

“…We note that our treatment is finite-dimensional by being discretization-based, similar to previous work that has used the finite-difference method for representing operators [13,14,15,16,17,18]. A different, neural network-based approach [12] enforced PDE constraints at random collocation points in one or two spatial dimensions and in time, thus effectively introducing a finite-dimensional representation with the global basis induced by activation functions.…”

“…A different, neural network-based approach [12] enforced PDE constraints at random collocation points in one or two spatial dimensions and in time, thus effectively introducing a finite-dimensional representation with the global basis induced by activation functions. We also note that the neural network approach, while illustrated for learning coefficients in a predefined PDE, remains to be tested for discovery of PDE structures by downselection from a large set of operators, which is central to the treatment in this work and other recent approaches [13,14,15,16,17,18].…”

Hybrid constitutive modeling: data-driven learning of corrections to plasticity models

“…These approaches can be embedded in an even more general context. Several works have been devoted to unveil governing equations from data [16][17][18]. These may include laws in the form of partial differential equations, for instance [19][20][21].…”