Spatiotemporal system reconstruction using Fourier spectral operators and structure selection techniques

Abstract

A technique based on trigonometric spectral methods and structure selection is proposed for the reconstruction, from observed time series, of spatiotemporal systems governed by nonlinear partial differential equations of polynomial type with terms of arbitrary derivative order and nonlinearity degree. The system identification using Fourier spectral differentiation operators in conjunction with a structure selection procedure leads to a parsimonious model of the original system by detecting and eliminating the redundant parameters using orthogonal decomposition of the state data. Implementation of the technique is exemplified for a highly stiff reaction-diffusion system governed by the Kuramoto-Sivashinsky equation. Numerical experiments demonstrate the superior performance of the proposed technique in terms of accuracy as well as robustness, even with smaller sets of sampling data.

Robust and optimal sparse regression for nonlinear PDE models

“…Such reliability and high accuracy in the presence of noisy or sparse data is indispensable for analysis of experimental data. Notably, even in the absence of noise, our results compare very favorably with those of previous studies 11,14 because the discretization error of the algorithm can be made extremely small: for the Kuramoto-Sivashinsky equation, the relative error in all parameters can easily be reduced to 10 −10 . It is also important to mention that the computational cost of our algorithm is comparable to that of existing sparse regression methods.…”

“…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.…”

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

“…2 Our attempt is to extend the algebraic spectral operators to the field of system identification and highlight their superiority when dealing with inverse problems for spatiotemporal systems on nonperiodic domains. We will also introduce a filtering algorithm that builds on the accuracy of the proposed spectral operators and leads to automatic SR.…”

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

“…In our case, this is not an option, since f satisfies neither condition. Furthermore, taking a derivative greatly amplifies the noise present in the data, whether this is done using finite differences 6,12 , polynomial interpolation 11 , or spectral methods 13,14 . Instead, we use a weak form of the model to address both noise sensitivity and the dependence on latent variables.…”