Dynamic mode decomposition as an analysis tool for time-dependent partial differential equations

TL;DR

This paper introduces Dynamic Mode Decomposition as a powerful data-driven technique to analyze complex, time-dependent partial differential equations by extracting oscillating spatial structures and their frequencies, aiding in understanding temporal behaviors.

Contribution

The paper presents the algorithm for Dynamic Mode Decomposition and demonstrates its application to complex PDEs with physical interpretation of the results.

Findings

DMD effectively identifies oscillating modes in PDE solutions.

Application to complex examples shows clear physical insights.

Method enhances analysis of high-dimensional time-dependent data.

Abstract

The time-dependent fields obtained by solving partial differential equations in two and more dimensions quickly overwhelm the analytical capabilities of the human brain. A meaningful insight into the temporal behaviour can be obtained by using scalar reductions, which, however, come with a loss of spatial detail. Dynamic Mode Decomposition is a data-driven analysis method that solves this problem by identifying oscillating spatial structures and their corresponding frequencies. This paper presents the algorithm and provides a physical interpretation of the results by applying the decomposition method to a series of increasingly complex examples.