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

TL;DR

Dynamic Mode Decomposition (DMD) provides a data-driven framework to decompose time-dependent PDE fields into spatial modes with associated temporal frequencies by analyzing the eigenstructure of the map between successive snapshots, with frequencies given by $ν_i=\frac{\arg(λ_i)}{2π Δt}$ and mode amplitudes inferred from projection coefficients. The paper details the compact-SVD based algorithm, discusses rank truncation and mode ordering, and interprets modes through physical structures. Through a damped vibrating membrane and a non-Newtonian natural convection example, it demonstrates accurate recovery of analytic eigenfrequencies for many modes, identification and filtering of spurious modes, and the ability to reveal regime-specific spatial patterns, highlighting DMD as a practical tool for complex dynamical systems and a Koopman-theoretic perspective. These results illustrate the method's utility for hydrodynamics and related fields, with implications for data-driven analysis and model reduction.

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.

Figures (7)

• Figure 1: Analytic eigenvectors for the oscillation of a square membrane, ordered by increasing frequency.
• Figure 2: left: The sum of height values provides a simple insight into the system's behaviour. The time evolution of thus derived observable is shown for different damping rates $\gamma$. right: A selection of snapshots used in the decomposition providing an insight into the system's behaviour during the observed time. A heat map display is used to visualise the scalar values with blues for negative, white for zero and reds for positive values.
• Figure 3: top: Eigenfrequencies calculated with DMD for different damping rates compared to analytic values. centre: The relative error between DMD eigenfrequencies and analytic values for the square membrane. bottom: Mode powers for the calculated DMD modes.
• Figure 4: DMD mode eigenvectors ordered by frequency for the $\gamma = 0$ case. The positions of modes are shifted by one relative to Figure \ref{['fig:rectangularFrequencyComparison']} because of the constant, zero frequency, background mode. The grayscale image displays absolute value with contours ranging from blue to red for the imaginary part.
• Figure 5: The DMD decomposition applied to a duck-shaped oscillating membrane. top left: Positioning of computational nodes with yellow for interior nodes where we solve the wave equation and purple for boundary nodes with enforced Dirichlet condition. top right: A subset of membrane state snapshots used for DMD. centre: The DMD spectrum showing the frequencies and powers of modes ordered by the former. bottom: Eigenvectors of DMD modes ordered by frequency.