Data-Driven Model Reduction and Transfer Operator Approximation

Abstract

In this review paper, we will present different data-driven dimension reduction techniques for dynamical systems that are based on transfer operator theory as well as methods to approximate transfer operators and their eigenvalues, eigenfunctions, and eigenmodes. The goal is to point out similarities and differences between methods developed independently by the dynamical systems, fluid dynamics, and molecular dynamics communities such as time-lagged independent component analysis (TICA), dynamic mode decomposition (DMD), and their respective generalizations. As a result, extensions and best practices developed for one particular method can be carried over to other related methods.

On the application of the dynamic mode decomposition in problems of computational fluid dynamics

“…Эта линеаризация основана на том, что оператор Купмана (линейный) является сопряженным оператором Перрона-Фробениуса [15][16][17][18][19].…”

“…Оператор Купмана тесно связан (сопряжен) с оператором Перрона-Фробениуса (пропагатором обобщенного уравнения Лиувилля), описывающим линейную динамику плотности вероятности [14][15][16][17][18][19].…”

Dynamic Distribution Decomposition for Single-Cell Snapshot Time Series Identifies Subpopulations and Trajectories during iPSC Reprogramming

“…for initial condition (t = 0, x) = 0 (x). The P term is the continuous-time Perron-Frobenius operator 226 [19,20]. This operator is known by many names depending on the underlying dynamical system for the state 227 evolution x(t) ∈ M for t ∈ (t 1 , t R ).…”

“…This situation results in a combinatorial explosion 18 of model variants that cannot be exhaustively evaluated [4,5]. Alternative approaches are agnostic with 19 regards to parametric form and model structure and use a probabilistically-motivated rule to map between 20 distributions, e.g., one optimal transport method maps neighbours at one time point to the nearest neighbour 21 at the next time point [6]. However, such generic approach are rather extreme in their agnosticism and 22 abandon reasonable assumptions on the dynamics of cellular systems, e.g., that cells can be modelled as an 23 autonomous dynamical systems in continuous time, such as a Markov chain, where the cell's current state 24 infers its likely future state independent of the current time within the experiment.…”

“…Eigenfunctions of linear operators show the fundamental 34 building blocks of possible behaviours available to a dynamical system, e.g., exponential growth/decay, 35 oscillations, a steady state. Data-driven approximations to the operators have been investigated in the past 36 years, originating in the computational fluid dynamics community [16][17][18][19][20]. Their focus is to approximate 37 finite-dimensional projections of the Koopman operator with a family of algorithms known as Dynamic 38 Mode Decomposition (DMD).…”