On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions

TL;DR

This paper investigates the connection between Koopman operator methods and neural ODEs for predicting nonlinear dynamical systems, demonstrating their equivalence and comparing their performance on complex chaotic systems.

Contribution

It reveals the equivalence between EDMD-DL with state space projection and neural ODE representations, and explores their combined structures and training methods.

Findings

EDMD-DL with projection is equivalent to neural ODEs.

Methods perform similarly on chaotic Lorenz and turbulent flow models.

All methods effectively predict trajectories, statistics, and rare events.

Abstract

This work explores the relationship between state space methods and Koopman operator-based methods for predicting the time-evolution of nonlinear dynamical systems. We demonstrate that extended dynamic mode decomposition with dictionary learning (EDMD-DL), when combined with a state space projection, is equivalent to a neural network representation of the nonlinear discrete-time flow map on the state space. We highlight how this projection step introduces nonlinearity into the evolution equations, enabling significantly improved EDMD-DL predictions. With this projection, EDMD-DL leads to a nonlinear dynamical system on the state space, which can be represented in either discrete or continuous time. This system has a natural structure for neural networks, where the state is first expanded into a high dimensional feature space followed by a linear mapping which represents the discrete-time map or the vector field as a linear combination of these features. Inspired by these observations, we implement several variations of neural ordinary differential equations (ODEs) and EDMD-DL, developed by combining different aspects of their respective model structures and training procedures. We evaluate these methods using numerical experiments on chaotic dynamics in the Lorenz system and a nine-mode model of turbulent shear flow, showing comparable performance across methods in terms of short-time trajectory prediction, reconstruction of long-time statistics, and prediction of rare events. These results highlight the equivalence of the EDMD-DL implementation with a state space projection to a neural ODE representation of the dynamics. We also show that these methods provide comparable performance to a non-Markovian approach in terms of prediction of extreme events.