Learning POD of Complex Dynamics Using Heavy-ball Neural ODEs

TL;DR

This paper introduces a novel approach using heavy-ball neural ODEs to learn reduced-order models from data, enhancing the accuracy and efficiency of modeling complex dynamical systems with POD.

Contribution

It applies heavy-ball neural ODEs to POD-based reduced-order modeling, providing theoretical guarantees and improved long-term dependency learning for complex dynamical systems.

Findings

HBNODE effectively learns long-term dependencies from sequential data.

HBNODE is computationally efficient in training and testing.

HBNODE outperforms other ROMs on complex systems.

Abstract

Proper orthogonal decomposition (POD) allows reduced-order modeling of complex dynamical systems at a substantial level, while maintaining a high degree of accuracy in modeling the underlying dynamical systems. Advances in machine learning algorithms enable learning POD-based dynamics from data and making accurate and fast predictions of dynamical systems. In this paper, we leverage the recently proposed heavy-ball neural ODEs (HBNODEs) [Xia et al. NeurIPS, 2021] for learning data-driven reduced-order models (ROMs) in the POD context, in particular, for learning dynamics of time-varying coefficients generated by the POD analysis on training snapshots generated from solving full order models. HBNODE enjoys several practical advantages for learning POD-based ROMs with theoretical guarantees, including 1) HBNODE can learn long-term dependencies effectively from sequential observations and 2) HBNODE is computationally efficient in both training and testing. We compare HBNODE with other popular ROMs on several complex dynamical systems, including the von K\'{a}rm\'{a}n Street flow, the Kurganov-Petrova-Popov equation, and the one-dimensional Euler equations for fluids modeling.