# Dynamic mode decomposition for multiscale nonlinear physics

## TL;DR

This paper introduces a recursive dynamic mode decomposition method that separates multiscale nonlinear systems into their components, enabling better analysis and short-term prediction by capturing spatial and temporal coherencies.

## Contribution

It extends multi-resolution DMD to handle a broader class of multiscale systems with improved component separation and reconstruction capabilities.

## Key findings

- Successfully applied to two physical systems
- Achieves faithful reconstruction of multiscale components
- Enables short-term prediction of system dynamics

## 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.