Data-Driven Learning for the Mori--Zwanzig Formalism: A Generalization of the Koopman Learning Framework

Lin, Yen Ting; Tian, Yifeng; Anghel, Marian; Livescu, Daniel

doi:10.1137/21m1401759

Cited by 22 publications

(52 citation statements)

References 39 publications

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“…To address the aforementioned difficulties, we propose to adopt general regression as a projection operator to bridge the gap between the linear Mori's projection operator and the fully nonlinear Zwanzig projection operator. We will show that when the regression is linear, the regression-based projection operator converges to our recent result of data-driven methods for Mori's projection operator [9]. Thus, the method we propose here is a superset of the previously published algorithms [9].…”

Section: Introductionmentioning

confidence: 53%

“…We will show that when the regression is linear, the regression-based projection operator converges to our recent result of data-driven methods for Mori's projection operator [9]. Thus, the method we propose here is a superset of the previously published algorithms [9]. The regression-based projection has the capability to incorporate flexible nonlinear functions that are adaptable to the data.…”

Section: Introductionmentioning

confidence: 61%

“…(2) finite-rank projection operator, which is essentially Mori's projection operator with a proper ortho-normalization, and (3) Zwanzig's fully nonlinear projection [2,3]. Several recent studies established that it is possible to adopt a data-driven approach to learn the Mori-Zwanzig operators using simulation data, if the projection operator is Mori's linear projection operator [4,5,6,7,8,9,10]. In addition, we also showed that our previously proposed algorithms [9] provide higher-order and memory-dependent corrections to existing data-driven learning of the approximate Koopman operators [11,12,13,14].…”

Section: Introductionmentioning

confidence: 99%

“…To the authors' best knowledge, there is no principled way to select the set of functions for linearly spanning the optimal subspace for general dynamical systems. With a non-optimal choice of the set of functions, the difference between the accurately predicted projected dynamics and the true dynamics can be very large [9]. The same problem also manifests itself in the approximate Koopman learning framework [14].…”

Section: Introductionmentioning

confidence: 99%

“…neural networks (NNs), for identifying nonlinear relationship between the independent variables (what we know at the present) and the dependent ones (what we want to predict in the future). With Mori's projection operator, our data-driven learning framework [9] cannot be generalized to capture these built-in and flexible nonlinearities that are informed and constrained by data in these ML-based approaches.…”

Section: Introductionmentioning

confidence: 99%

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Regression-based projection for learning Mori--Zwanzig operators

Tian¹,

Pérez²,

Livescu³

2022

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We propose to adopt statistical regression as the projection operator to enable datadriven learning of the operators in the Mori-Zwanzig formalism. We present a principled algorithm to extract the Markov and memory operators for any regression models. We show that the choice of linear regression results in a recently proposed data-driven learning algorithm based on Mori's projection operator, which can be considered as a higher-order approximate Koopman learning method. We show that more expressive, potentially nonlinear regression models naturally fill in the gap between the highly idealized and computationally efficient Mori's projection operator and the most optimal yet computationally infeasible Zwanzig projection operator. We performed numerical experiments and extracted the operators for an array of regression-based projections, including linear, polynomial, spline, and neural-network-based regression, showing a progressive improvement as the complexity of the regression model increased. Our proposition provides a general framework to extract memory-dependent corrections and can be readily applied to an array of data-driven learning methods for stationary dynamical systems in the literature.

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Section: Introductionmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 61%