Linearly-Recurrent Autoencoder Networks for Learning Dynamics (1712.01378v2)

Abstract: This paper describes a method for learning low-dimensional approximations of nonlinear dynamical systems, based on neural-network approximations of the underlying Koopman operator. Extended Dynamic Mode Decomposition (EDMD) provides a useful data-driven approximation of the Koopman operator for analyzing dynamical systems. This paper addresses a fundamental problem associated with EDMD: a trade-off between representational capacity of the dictionary and over-fitting due to insufficient data. A new neural network architecture combining an autoencoder with linear recurrent dynamics in the encoded state is used to learn a low-dimensional and highly informative Koopman-invariant subspace of observables. A method is also presented for balanced model reduction of over-specified EDMD systems in feature space. Nonlinear reconstruction using partially linear multi-kernel regression aims to improve reconstruction accuracy from the low-dimensional state when the data has complex but intrinsically low-dimensional structure. The techniques demonstrate the ability to identify Koopman eigenfunctions of the unforced Duffing equation, create accurate low-dimensional models of an unstable cylinder wake flow, and make short-time predictions of the chaotic Kuramoto-Sivashinsky equation.

• The paper presents an innovative LR autoencoder method that approximates nonlinear dynamics by leveraging a neural Koopman operator.
• It employs balanced model reduction techniques to mitigate overfitting in high-capacity EDMD systems and ensure robust performance on limited data.
• Nonlinear reconstruction via multi-kernel regression improves prediction accuracy for complex systems, demonstrated on examples like the Duffing equation and unstable wake flows.

An Expert Overview of "Linearly-Recurrent Autoencoder Networks for Learning Dynamics"

The paper under review introduces an innovative method for approximating nonlinear dynamical systems using linearly-recurrent autoencoder networks (LRANs). This research leverages the neural network-based approximation of the Koopman operator, promising significant advancements in modeling complex systems. The paper addresses key challenges associated with the Extended Dynamic Mode Decomposition (EDMD), focusing on the trade-off between the representational capacity of the chosen dictionary and overfitting due to insufficient data.

Key Contributions and Methodology

The paper makes several substantial contributions to the field:

1. Neural Network Architecture: The paper combines an autoencoder architecture with linear recurrence to form a compact and efficient neural network framework. This architecture is adept at discovering low-dimensional, Koopman-invariant subspaces of observables which can describe the dynamics of complex systems. The approach effectively marries neural network capabilities with the linear evolution dictated by the Koopman operator.
2. Balanced Model Reduction: The researchers introduce a novel method for balanced model reduction, catering to over-specified EDMD systems within feature space. This approach helps mitigate overfitting, a common issue when working with high-capacity models on limited data.
3. Nonlinear Reconstruction: To enhance reconstruction accuracy from low-dimensional representations, the paper employs partially linear multi-kernel regression techniques. This innovation is particularly useful when dealing with data that is complex in structure yet intrinsically low-dimensional.
4. Practical Applications: The paper demonstrates the efficacy of these techniques through several applications, including identifying Koopman eigenfunctions of the Duffing equation, modeling the dynamics of an unstable cylinder wake flow, and making short-term predictions of the chaotic Kuramoto-Sivashinsky equation.

Implications and Future Directions

The implications of this research are manifold for both theoretical understanding and practical application. The proposed LRAN framework enhances our ability to model nonlinear systems with greater precision and less computational complexity. The paper’s findings suggest that advanced machine learning architectures can uncover dynamics in high-dimensional spaces that classical methods might miss or address inefficiently.

Theoretical Implications: This methodology extends the applicability of the Koopman operator in dynamical systems theory, providing a robust framework for linearizing complex systems into simpler, analyzable components. The proposed approach can lead to new insights into the behavior of complex systems by efficiently capturing their dynamics in reduced-order models.

Practical Applications: In real-world applications, the ability to accurately predict system dynamics from sparse and noisy data has significant implications. For instance, this could improve weather forecasting models, control systems in engineering, and understanding fluid dynamics behaviors with potentially millions of degrees of freedom.

Future Developments: Future research could expand on the inclusion of uncertainty quantification within the LRAN framework, allowing for more robust models that can predict not just dynamics but also their uncertainty bounds. Additionally, incorporating these models into control frameworks could revolutionize how we develop intelligent control strategies for complex engineered systems.

In conclusion, this paper offers a compelling advancement in the modeling of dynamical systems. While challenges remain, particularly in optimizing and scaling these models for even more complex systems, the foundation laid here opens new pathways for both theoretical exploration and practical innovation in the field of data-driven dynamical systems modeling.