# Data-driven model reduction and transfer operator approximation

## TL;DR

This review compares data-driven dimension reduction techniques for dynamical systems based on transfer operator theory, highlighting their similarities, differences, and cross-disciplinary applications to improve understanding and methodology.

## Contribution

It synthesizes methods like TICA and DMD across fields, clarifying their relationships and enabling transfer of best practices among them.

## Key findings

- Identifies commonalities between TICA and DMD methods.
- Highlights extensions and best practices applicable across methods.
- Provides a unified perspective on transfer operator-based dimension reduction.

## 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.