[Paper Review] Learning Parametric Koopman Decompositions for Prediction and Control

Yue Guo, Milan Korda+2 more|arXiv (Cornell University)|Oct 2, 2023

Model Reduction and Neural Networks4 citations

TL;DR

This paper introduces a learning-based method to construct parametric Koopman decompositions for dynamical systems with static or time-varying parameters by jointly learning a common invariant subspace and a parametric family of projected Koopman operators using neural networks. The approach enables accurate prediction and data-driven optimal control in high-dimensional, strongly non-linear systems, outperforming existing methods in both prediction accuracy and control performance.

ABSTRACT

We present an approach to construct approximate Koopman-type decompositions for dynamical systems depending on static or time-varying parameters. Our method simultaneously constructs an invariant subspace and a parametric family of projected Koopman operators acting on this subspace. We parametrize both the projected Koopman operator family and the dictionary that spans the invariant subspace by neural networks and jointly train them with trajectory data. We show theoretically the validity of our approach, and demonstrate via numerical experiments that it exhibits significant improvements over existing methods in solving prediction problems, especially those with large state or parameter dimensions, and those possessing strongly non-linear dynamics. Moreover, our method enables data-driven solution of optimal control problems involving non-linear dynamics, with interesting implications on controllability.

Motivation & Objective

• To address the challenge of modeling parametric dynamical systems with incomplete knowledge of dynamics, relying only on trajectory data.
• To extend Koopman operator theory to non-autonomous, parameter-dependent systems with strong non-linearities.
• To jointly learn an invariant subspace and a parametric family of Koopman operators from data, overcoming limitations of prior linear or bilinear approximations.
• To enable data-driven solution of optimal control problems involving complex, non-linear dynamics.

Proposed method

• Parametrize both the dictionary spanning the invariant subspace and the projected Koopman operator family using deep neural networks.
• Simultaneously train the neural networks using trajectory data to jointly optimize the subspace and operator approximation.
• Use a neural network architecture for the Koopman operator that ensures continuity and approximates the operator family over the entire parameter space.
• Leverage the function approximation power of neural networks to embed non-linear parameter dependence directly into the finite-dimensional Koopman operator.
• Apply theoretical guarantees from compactness and uniform continuity to ensure convergence of the approximation under mild conditions.
• Use a residual structure in the neural network for $ K(m{u}) $ to ensure that the matrix rank is bounded by the number of last-layer neurons, enabling low-rank approximation.

Figure 1 : The neural network architecture of PK-NN. The trainable parameters of $K(\bm{u};\bm{\theta}_{K})$ are integrated within the structure of $NN_{K}$ , and a distinct neural network $NN_{\bm{\Psi}}$ is used to parametrize the dictionaries $\bm{\Psi}(\bm{x};\bm{\theta}_{\psi})$ .

Experimental results

Research questions

• RQ1 Can a unified, data-driven method learn both an invariant subspace and a parametric family of Koopman operators for non-linear, parameter-dependent dynamical systems?
• RQ2 How can neural networks be used to jointly learn the Koopman operator and the dictionary in a way that preserves the linear structure of Koopman dynamics?
• RQ3 To what extent does the proposed method improve prediction accuracy over existing methods in high-dimensional or strongly non-linear systems?
• RQ4 Can the learned parametric Koopman model be effectively used for optimal control of non-linear systems?

Key findings

• The proposed method achieves significant improvements in prediction accuracy over existing methods, especially in systems with large state or parameter dimensions.
• The method successfully captures complex, non-linear relationships between high-dimensional states and parameters through joint neural network training.
• Numerical experiments show that the learned parametric Koopman operator enables accurate long-term prediction in non-autonomous, non-linear systems.
• The approach supports data-driven optimal control of non-linear systems, with demonstrated improvements in controllability analysis and trajectory optimization.
• Theoretical analysis confirms that the approximation error can be made arbitrarily small under mild regularity conditions on the Koopman operator family.
• The rank of the Koopman operator matrix is bounded by the number of neurons in the last hidden layer of the neural network, enabling low-rank, efficient representations.

Figure 2 : Comparison on different $(\nu_{1},\nu_{2})$ with $\nu_{1}\nu_{2}=10000$ . The topmost panel displays the prediction results for PK-NN and M1 on both training and testing data. The horizontal axis represents the values of $(\nu_{1},\nu_{2})$ . In the bottom panels, we provide a detailed er