Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems (1708.06850v2)

Abstract: The Koopman operator has recently garnered much attention for its value in dynamical systems analysis and data-driven model discovery. However, its application has been hindered by the computational complexity of extended dynamic mode decomposition; this requires a combinatorially large basis set to adequately describe many nonlinear systems of interest, e.g. cyber-physical infrastructure systems, biological networks, social systems, and fluid dynamics. Often the dictionaries generated for these problems are manually curated, requiring domain-specific knowledge and painstaking tuning. In this paper we introduce a deep learning framework for learning Koopman operators of nonlinear dynamical systems. We show that this novel method automatically selects efficient deep dictionaries, outperforming state-of-the-art methods. We benchmark this method on partially observed nonlinear systems, including the glycolytic oscillator and show it is able to predict quantitatively 100 steps into the future, using only a single timepoint, and qualitative oscillatory behavior 400 steps into the future.

• The paper proposes a deep neural network approach that automatically learns Koopman operators, eliminating the need for manual dictionary tuning.
• It demonstrates significant improvements over traditional EDMD by achieving less than 1% one-step errors and reliable predictions up to 400 steps ahead.
• The framework paves the way for practical applications in diverse fields such as cyber-physical infrastructures and biological network analysis.

Deep Learning Framework for Koopman Operators in Nonlinear Systems

The research paper "Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems" presents an innovative approach to addressing the computational challenges associated with the analysis of nonlinear dynamical systems using the Koopman operator. Recognizing the limitations of traditional methods, such as the extended dynamic mode decomposition (EDMD), this work introduces the application of deep learning techniques to enhance the Koopman operator framework for handling complex, nonlinear dynamic systems.

The Koopman operator theory provides a means to linearize the progression of observables on nonlinear dynamical systems, offering the potential to apply robust linear analysis methodologies in nonlinear contexts. However, its practical application has been severely constrained by the need for a combinatorially large basis set in EDMD, making the process computationally intensive and heavily reliant on expert domain knowledge for dictionary tuning.

In this paper, the authors propose a novel methodology that leverages deep neural networks to learn efficient representations of the Koopman operator automatically. The proposed deep dynamic mode decomposition framework facilitates the automatic selection and training of Koopman dictionaries, which are traditionally curated manually. The paper details how this deep learning model surpasses existing approaches to predict future states of nonlinear systems more effectively, demonstrated through applications such as the glycolytic oscillator.

Key Findings and Numerical Results

The paper details significant performance enhancements of the deep Koopman methodology compared to EDMD:

• For partially observed nonlinear systems like the glycolytic oscillator, the deep learning-powered Koopman model predicts future states with high quantitative accuracy (100 steps) and qualitative behavior (400 steps) into the future from a single initial input point.
• Training and test prediction errors in several benchmark nonlinear systems were reduced significantly, with the deep Koopman operator achieving less than 1% one-step training and test errors, a marked improvement over the EDMD method.

Implications and Future Directions

The introduction of deep learning mechanisms into the Koopman operator framework marks a significant advancement in simplifying and enhancing the process of capturing dynamics in complex systems. It shifts the paradigm from static, manually curated dictionaries to dynamic, adaptable models capable of learning system-specific transformations autonomously. This advancement has vast implications:

• Practical: This methodology can be applied to a broad range of fields, including cyber-physical infrastructures, biological and social networks, and large-scale power systems. By facilitating multi-step forecasting, it can significantly improve system stability analyses and contingency planning in real-time applications.
• Theoretical: This work provides a foundation for further exploration into neural network capabilities in dynamical systems theory, particularly in terms of model generalization and learning efficiency.

The potential for further improvement and exploration is immense. Adapting the deep Koopman framework to specific types of dynamical systems or optimizing neural network architectures for various nonlinear dynamics remains an open area for research. Future studies might delve into how different neural network models or regularization frameworks impact the fidelity and interpretability of the learned Koopman operators across diverse application domains.

In summation, this paper addresses one of the key bottlenecks in Koopman operator application for nonlinear systems and opens pathways for more adaptive and scalable solutions, enhancing the robustness and applicability of data-driven model discovery methods.