Temporally-consistent koopman autoencoders for forecasting dynamical systems.

Area of Science:

• Dynamical Systems and Control Theory
• Machine Learning and Artificial Intelligence
• Scientific Computing

Background:

• Data-driven modeling of high-dimensional spatio-temporal systems faces challenges due to insufficient data quality.
• Koopman Autoencoders (KAEs) combine deep neural networks, autoencoders, and Koopman operator theory for reduced-order modeling but struggle with limited/noisy data.
• Poor generalizability of existing KAEs hinders their application in real-world scenarios.

Purpose of the Study:

• To introduce a novel Temporally-Consistent Koopman Autoencoder (tcKAE) for enhanced long-term predictions.
• To improve the robustness and generalizability of KAEs using limited and noisy datasets.
• To provide analytical and empirical validation of the tcKAE approach.

Main Methods:

• Developed the Temporally-Consistent Koopman Autoencoder (tcKAE) by incorporating a consistency regularization term.
• The regularization enforces prediction coherence across time steps, enhancing model stability.
• Analytical justification derived from Koopman spectral theory.

Main Results:

• tcKAE demonstrates superior performance compared to state-of-the-art KAE models.
• The model achieves accurate long-term predictions even with limited and noisy training data.
• Empirical validation across diverse test cases including pendulum oscillations, kinetic plasma, and fluid flow.

Conclusions:

• The tcKAE effectively addresses the limitations of traditional KAEs in data-scarce environments.
• The proposed temporal consistency regularization significantly enhances model robustness and predictive accuracy.
• tcKAE offers a promising approach for modeling complex spatio-temporal dynamical systems.