A tensor-based dynamic mode decomposition based on the $\star_{\boldsymbol{M}}$-product

TL;DR

This paper introduces a tensor-based dynamic mode decomposition method using the $oldsymbol{igstar}_{oldsymbol{M}}$-product, offering improved compression and computational efficiency for tensor-structured data in dynamical systems.

Contribution

It develops a novel tensor-based DMD framework with the $igstar_{oldsymbol{M}}$-product, connecting it to traditional DMD and providing efficient algorithms including a randomized streaming version.

Findings

Achieves comparable or better accuracy than standard DMD.

Offers significant compression advantages over matrix-based methods.

Demonstrates computational efficiency and suitability for streaming data.

Abstract

Dynamic mode decomposition (DMD) is a data-driven method for estimating the dynamics of a discrete dynamical system. This paper proposes a tensor-based approach to DMD for applications in which the states can be viewed as tensors. Specifically, we use the $\star_{\boldsymbol{M}}$-product framework for tensor decompositions which we demonstrate offers excellent compression compared to matrix-based methods and can be implemented in a computationally efficient manner. We show how the proposed approach is connected to the traditional DMD and physics-informed DMD frameworks. We give a computational framework for computing the tensor-based DMD and detail the computational costs. We also give a randomized algorithm that enables efficient $\star_{\boldsymbol{M}}$-DMD computations in the streaming setting. The numerical results show that the proposed method achieves equal or better accuracy for the same storage compared to the standard DMD on these examples and is more efficient to compute.