On the relationship between Koopman operator approximations and neural ordinary differential equations for...

Area of Science:

• Dynamical Systems Theory
• Machine Learning for Scientific Discovery
• Nonlinear Dynamics

Background:

• Predicting the time evolution of nonlinear dynamical systems is crucial in many scientific fields.
• Koopman operator-based methods and state space methods offer distinct approaches to this challenge.
• Bridging these methods can lead to more powerful predictive models.

Purpose of the Study:

• To explore the relationship between state space and Koopman operator-based methods.
• To demonstrate the equivalence between extended dynamic mode decomposition with dictionary learning (EDMD-DL) and neural network representations.
• To develop and evaluate novel hybrid models for nonlinear system prediction.

Main Methods:

• Utilized extended dynamic mode decomposition with dictionary learning (EDMD-DL) combined with state space projection.
• Implemented variations of neural ordinary differential equations (ODEs) and EDMD-DL.
• Employed numerical experiments on chaotic systems (Lorenz system, nine-mode turbulent flow).

Main Results:

• EDMD-DL with state space projection is equivalent to a neural network representation of the nonlinear discrete-time flow map.
• The projection step introduces nonlinearity, significantly improving EDMD-DL predictions.
• Hybrid models showed comparable performance to neural ODEs and non-Markovian approaches for various prediction tasks.

Conclusions:

• The equivalence between EDMD-DL with state space projection and neural ODEs is established.
• These methods provide robust predictions for chaotic dynamics, long-time statistics, and extreme events.
• The findings offer a unified perspective on data-driven modeling of nonlinear dynamical systems.