[Paper Review] A Convex Optimization Approach to Learning Koopman Operators

Mario Sznaier|arXiv (Cornell University)|Feb 7, 2021

Model Reduction and Neural Networks14 references4 citations

TL;DR

This paper proposes a convex optimization framework for learning Koopman operators from data by formulating the problem as rank-constrained semi-definite programs (SDPs), which are relaxed to convex SDPs. The method jointly learns the Koopman operator, a nonlinear embedding of the state space, and the associated Hilbert space of observables without requiring prior specification of the embedding dimension or dynamics order, enabling provably optimal and low-order models for nonlinear dynamical systems.

ABSTRACT

Koopman operators provide tractable means of learning linear approximations of non-linear dynamics. Many approaches have been proposed to find these operators, typically based upon approximations using an a-priori fixed class of models. However, choosing appropriate models and bounding the approximation error is far from trivial. Motivated by these difficulties, in this paper we propose an optimization based approach to learning Koopman operators from data. Our results show that the Koopman operator, the associated Hilbert space of observables and a suitable dictionary can be obtained by solving two rank-constrained semi-definite programs (SDP). While in principle these problems are NP-hard, the use of standard relaxations of rank leads to convex SDPs. Further, these SDPs exhibit chordal sparsity leading to algorithms that scale linearly with the number of data points.

Motivation & Objective

• To address the challenge of selecting appropriate dictionaries and bounding approximation errors in Koopman operator learning, which are critical in nonlinear system identification.
• To develop a data-driven method that jointly learns the Koopman operator, the Hilbert space of observables, and a nonlinear embedding of the state space from time-series data.
• To eliminate the need for a priori specification of the embedding dimension or the order of the dynamics, which are typically heuristic choices in existing methods.
• To provide a certification mechanism for the solution, ensuring that the obtained Koopman operator and embedding are optimal under the convex relaxation.
• To enable the discovery of simpler, lower-order models compared to traditional methods like Hankel matrix factorization or fixed-dictionary EDMD.

Proposed method

• The method formulates Koopman operator learning as a rank-constrained semi-definite program (SDP), where the rank constraint encodes the low-dimensional dynamics of the system.
• The rank-constrained SDP is relaxed into a convex SDP using standard nuclear norm relaxation, enabling tractable solution via standard SDP solvers.
• The approach uses delay coordinates and kernel-based nonlinear embeddings to map the original state space into a higher-dimensional space where the Koopman operator acts linearly.
• Two sequential SDPs are solved: the first learns the embedding and Koopman operator via a Loewner matrix formulation, and the second learns the inverse mapping from the latent space back to the original state space.
• The pipeline leverages chordal sparsity in the SDP structure, enabling scalable solution via ADMM-based solvers with linear complexity in the number of data points.
• After learning the Koopman operator and embedding, the dynamics are predicted using a linear model in the latent space, requiring only O(r) operations per step, where r is the delay horizon.

Experimental results

Research questions

• RQ1 Can Koopman operator learning be formulated as a convex optimization problem that jointly identifies the operator, the Hilbert space of observables, and the nonlinear embedding without prior assumptions on dimensionality?
• RQ2 How can the rank-constrained SDP formulation be relaxed to a convex problem while preserving optimality and enabling certification of the solution?
• RQ3 Can the proposed method discover lower-order models than traditional EDMD or Hankel-based factorization methods, especially for chaotic systems?
• RQ4 To what extent can the method generalize to unseen initial conditions and long-horizon predictions in nonlinear and chaotic systems?
• RQ5 Does the use of kernel-based embeddings and chordal sparsity enable scalable, linear-time solution of the SDP for large datasets?

Key findings

• The proposed method successfully learns Koopman operators for chaotic systems such as the Duffing oscillator and a 4D nonlinear system, achieving accurate one-step-ahead predictions on both training and unseen data.
• For the Duffing oscillator, the method achieved a 3D embedding and used a second-order model to predict trajectories over a 100-step horizon with high accuracy.
• In the predator-prey model, an 8th-order model was estimated from a 120-point trajectory, and the pipeline predicted 80 steps ahead with minimal error, even when starting from an initial condition not in the training set.
• For a 4D chaotic system with extreme sensitivity to initial conditions, the method produced a 3D embedding and predicted 100 steps ahead with reasonable accuracy, demonstrating robustness to chaos.
• The method discovered lower-order dynamics than traditional Hankel matrix factorization, indicating improved model simplicity and potential for better generalization.
• The SDP structure exhibits chordal sparsity, enabling scalable solution via ADMM-based solvers with linear complexity in the number of data points, making the approach feasible for larger datasets.