Dynamic mode decomposition for multiscale nonlinear physics

Abstract

We present a data-driven method for separating complex, multiscale systems into their constituent time-scale components using a recursive implementation of dynamic mode decomposition (DMD). Local linear models are built from windowed subsets of the data, and dominant time scales are discovered using spectral clustering on their eigenvalues. This approach produces time series data for each identified component, which sum to a faithful reconstruction of the input signal. It differs from most other methods in the field of multiresolution analysis (MRA) in that it 1) accounts for spatial and temporal coherencies simultaneously, making it more robust to scale overlap between components, and 2) yields a closed-form expression for local dynamics at each scale, which can be used for short-term prediction of any or all components. Our technique is an extension of multi-resolution dynamic mode decomposition (mrDMD), generalized to treat a broader variety of multiscale systems and more faithfully reconstruct their isolated components. In this paper we present an overview of our algorithm and its results on two example physical systems, and briefly discuss some advantages and potential forecasting applications for the technique.

Adaptive symplectic model order reduction of parametric particle-based Vlasov-Poisson equation

“…In this situation, a different version of the DMD algorithm, proposed in [43], projects the data into a low-rank subspace instead of deriving A directly from the data, as described in Algorithm 1. Moreover, given the sensitivity of the DMD algorithm to the duration and sampling of the series Y and Y , [44] proposes a sliding-window approach where the measurement data are not taken in the whole temporal interval but only in the sampling window [t τ −T , t τ ] of length T ∈ N. The rationale is that if the system is time-varying and the incoming data is harvested in a streaming fashion, it may be beneficial to accuracy and memory storage to consider only the most recent data. The only computational overhead is the computation of the DMD modes and weights in the DMD approximation (4.4) as new data are collected.…”

“…We observe that the DMD window length T + 1 has no impact on the error when p * is large, and a small accuracy degradation is even registered for p * = 8 when T = 5 is chosen over T = 3. The optimal choice of T remains an open problem: as pointed out in [44], it should capture slow and fast scales of the local dynamics, but a rigorous optimization strategy would require a study of the multi-scale properties of the solution to the Vlasov-Poisson equation for each of the parameter realization considered. However, we stress that the results are relatively robust concerning this parameter.…”

Survey of Dynamic Mode Decomposition Methods

“…In this context, slow modes are those that are prevalent over the full sampling period and fast modes are those which are prevalent only in some slices of the sampling period. To the author's knowledge, little literature exists outside of [4] and [7] on physics applications of this algorithm. This algorithm is best suited for problems in which the fundamental assumptions of DMD do not hold over the full domain, but may approximately hold over sub-slices.…”

Sparse Identification of Slow Timescale Dynamics

“…There are potentially two unknowns that are required to find G: the fast period T > 0 and the mapping F . Here we use the method of sliding-window dynamic mode decomposition (DMD) [7] to extract the fast period and the sparse identification of nonlinear dynamics (SINDy) algorithm [3,5] to obtain the mapping F . We summarize the method visually in Figure 1 and explain the individual components in more detail in the following section.…”