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

TL;DR

The paper reveals a fundamental link between EDMD-DL with a state-space projection and neural ODEs, showing that projecting Koopman-based predictions back to the state space creates a nonlinear flow that mirrors neural network representations of the dynamics. By deriving both discrete-time and continuous-time formulations, the authors propose several hybrid architectures that blend EDMD-DL and neural ODE principles. Through experiments on the Lorenz system and a Moehlis nine-mode model of turbulence, they demonstrate that projection-based EDMD approaches achieve predictive performance on par with neural ODEs and significantly outperform standard linear EDMD, including for extreme-event statistics. The findings provide a unified perspective on data-driven time evolution and suggest practical pathways to leverage Koopman theory alongside neural ODE techniques for robust, scalable forecasting in chaotic systems.

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.

Figures (10)

• Figure 1: Schematic illustrating how EDMD-DL with a projection to the state space directly leads to a neural network representation of the nonlinear dynamics on the state space. Here, $\mathbf{x}_i=\mathbf{x}(t_i)$, $\mathbf{x}_{i+1}=\mathbf{x}(t_{i+1})=\mathbf{x}(t_i+\tau)$, and $\mathbf{F}_\tau$ is the flow map for time interval, $\tau$.
• Figure 2: Schematic illustrating how the continuous-time EDMD-DL formulation with projection to the state space leads to a neural ODE representation of the state space dynamics.
• Figure 3: Prediction results for the data-driven models on the Lorenz system for sampling interval $\tau=0.02$. (a) Normalized ensemble-averaged error over time. (b) Representative trajectory timeseries of the states with predictions from each model.
• Figure 4: Long time predictions of each model from a common initial condition for $100$ time units on the Lorenz system. Each of these models maintains the shape of the attractor for long times. These trajectories are generated by models trained with timestep $\tau=0.02$.
• Figure 5: Normalized ensemble-averaged error of each model evaluated at $t=2$ for varying training data sampling time, $\tau$.