Spatiotemporal system identification on nonperiodic domains using Chebyshev spectral operators and system reduction algorithms

Abstract

A system identification methodology based on Chebyshev spectral operators and an orthogonal system reduction algorithm is proposed, leading to a new approach for data-driven modeling of nonlinear spatiotemporal systems on nonperiodic domains. A continuous model structure is devised allowing for terms of arbitrary derivative order and nonlinearity degree. Chebyshev spectral operators are introduced to realm of inverse problems to discretize that continuous structure and arrive with spectral accuracy at a discrete form. Finally, least squares combined with an orthogonal system reduction algorithm are employed to solve for the parameters and eliminate the redundancies to achieve a parsimonious model. A numerical case study of identifying the Allen-Cahn metastable equation demonstrates the superior accuracy of the proposed Chebyshev spectral identification over its finite difference counterpart.

Robust and optimal sparse regression for nonlinear PDE models

“…Sparse regression aims to convert the PDE (1) to a tractable (and ideally, robust) linear algebra problem. Conventionally this is done by evaluating all of the terms in the PDE at a random collection of points (x k , t k ) using finite differences 17,18 , spectral methods 11,12 , or polynomial approximation 14,15 . All of these approaches are extremely sensitive to noise, especially when high-order derivatives are present.…”

“…The genetic algorithms used in these earlier studies are however computationally expensive, preventing application of this approach to high-dimensional systems. Thus, the recent emergence of a sparse regression approach for model discovery [11][12][13][14][15] has made a significant impact. Applied to spatially extended systems, this approach allows data-driven discovery of governing equations in the form of PDEs by evaluating a library of candidate terms containing partial derivatives at a large number of points and using a regularized regression procedure to compute the coefficients of each term and select a parsimonious model.…”

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

“…so that w j , ∇p i = − ∇ • w j , p i = 0, (14) eliminating the dependence on pressure. All of the above constraints can be satisfied by choosing the scalar fields φ j in the form φ j (x, y, t) = P λ (x )P µ (y )P ν (t )E α (x )E β (y )E γ (t ), (15) where P m (•) is a Legendre polynomial,…”

Machine Discovery of Partial Differential Equations from Spatiotemporal Data

“…The first kind of such methods builds on the ordinary least squares regression. Examples include orthogonal reduction (Xu and Khanmohamadi, 2008;Khanmohamadi and Xu, 2009), the backward elimination scheme (Bär et al, 1999;Guo and Billings, 2006), least squares estimator (Müller and Timmer, 2004) and the alternating conditional expectation algorithm (Breiman and Friedman, 1985;Voss et al, 1999Voss et al, , 1998. These approaches do have an good fitting accuracy as described by (Tibshirani et al, 2015) which, however, have two major issues: prediction accuracy and interpretability in modeling a wider range of data.…”