Using noisy or incomplete data to discover models of spatiotemporal dynamics

Abstract

Sparse regression has recently emerged as an attractive approach for discovering models of spatiotemporally complex dynamics directly from data. In many instances, such models are in the form of nonlinear partial differential equations (PDEs); hence sparse regression typically requires evaluation of various partial derivatives. However, accurate evaluation of derivatives, especially of high order, is infeasible when the data are noisy, which has a dramatic negative effect on the result of regression. We present a novel and rather general approach that addresses this difficulty by using a weak formulation of the problem. For instance, it allows accurate reconstruction of PDEs involving high-order derivatives, such as the Kuramoto-Sivashinsky equation, from data with a considerable amount of noise. The flexibility of our approach also allows reconstruction of PDE models that involve latent variables which cannot be measured directly with acceptable accuracy. This is illustrated by reconstructing a model for a weakly turbulent flow in a thin fluid layer, where neither the forcing nor the pressure field is known.

SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics

“…where Θ(X,Ẋ|θ j (X,Ẋ)) is the library Θ(X,Ẋ) with the θ j column removed. equation (3.1) is no longer in implicit form, and the sparse coefficient matrix corresponding to the remaining terms may be solved for using previously developed SINDy techniques [28][29][30]36,46,49,50,[59][60][61]. In particular, we solve for a sparse coefficient vector ξ j that minimizes the following loss function:…”

PySINDy: A Python package for the sparse identification of nonlinear dynamical systems from data

“…The SINDy method has been widely applied for model identification in applications such as chemical reaction dynamics (Hoffmann, Fröhner, & Noé, 2019), nonlinear optics (Sorokina, Sygletos, & Turitsyn, 2016), thermal fluids (Loiseau, 2019), plasma convection (Dam, Brøns, Juul Rasmussen, Naulin, & Hesthaven, 2017), numerical algorithms (Thaler, Paehler, & Adams, 2019), and structural modeling (Lai & Nagarajaiah, 2019). It has also been extended to handle more complex modeling scenarios such as partial differential equations (Rudy, Brunton, Proctor, & Kutz, 2017;Schaeffer, 2017), systems with inputs or control (Kaiser, Kutz, & Brunton, 2018), corrupt or limited data (Schaeffer, Tran, & Ward, 2018;Tran & Ward, 2017), integral formulations (Reinbold, Gurevich, & Grigoriev, 2020;Schaeffer & McCalla, 2017), physical constraints (Loiseau & Brunton, 2018), tensor representations (Gelß, Klus, Eisert, & Schütte, 2019), and stochastic systems (Boninsegna, Nüske, & Clementi, 2018). However, there is not a definitive standard implementation or package for applying SINDy.…”

Robust learning from noisy, incomplete, high-dimensional experimental data via physically constrained symbolic regression

“…This approach was originally introduced in the context of ordinary differential equations 15,16 . In the context of PDE models, it was shown to be as general as prior approaches based on the strong form 6,7 and superior in terms of both its flexibility and robustness 17,18 .…”

“…Here we adopt the computationally efficient iterative procedure introduced in ref. 18 , which is an adaptation of the latter algorithm. At each iteration, Eq.…”

“…(2) does not change η for a flow that is incompressible, but does change the model 11 . The robustness of the functional form of the model and the accuracy with which the coefficients c n are determined can both be quantified by performing symbolic regression for an ensemble of different samplings of the data (or even different data sets) 18 . Here, each ensemble includes different distributions of integration domains in the temporal direction.…”