New Robust Streaming DMD with Forecasting

TL;DR

This paper introduces an improved, robust streaming Dynamic Mode Decomposition method that enhances efficiency, numerical stability, and forecasting accuracy for analyzing high-dimensional dynamical systems in real-time.

Contribution

It proposes a new streaming DMD algorithm with residual bounds, Exact DMD vectors, and better numerical robustness, building on prior foundational work.

Findings

Enhanced forecasting accuracy over previous methods

Reduced computational complexity and memory usage

Improved numerical stability and robustness

Abstract

The Dynamic Mode Decomposition (DMD) and the more general Extended DMD (EDMD) are powerful tools for computational analysis of dynamical systems in data-driven scenarios. They are built on the theoretical foundation of the Koopman composition operator and can be considered as numerical methods for data snapshot-based extraction of spectral information of the composition operator associated with the dynamics, spectral analysis of the structure of the dynamics, and for forecasting. In high fidelity numerical simulations, the state space is high dimensional and efficient numerical methods leverage the fact that the actual dynamics evolves on manifolds of much smaller dimension. This motivates computing low rank approximations in a streaming fashion and the DMD matrix is adaptively updated with newly received data. In this way, large number of high dimensional snapshots can be processed very efficiently. Low dimensional representation also requires fast updating for online applications. This paper revisits the pioneering works of Hemati, Williams and Rowley (Physics of Fluids, 2014), and Zhang, Rowley, Deem and Cattafesta (SIAM Journal on Applied Dynamical Systems, 2019) on the streaming DMD and proposes improvements in functionality (using residual bounds, Exact DMD vectors), computational efficiency (more efficient algorithm with smaller memory footprint) and numerical robustness (smaller condition numbers and better forecasting skill).