[Paper Review] Linear identification of nonlinear systems: A lifting technique based on the Koopman operator

Alexandre Mauroy, Jorge Gonçalves|arXiv (Cornell University)|May 14, 2016

Model Reduction and Neural Networks13 references4 citations

TL;DR

This paper proposes a novel indirect method for identifying nonlinear dynamical systems by lifting the problem to the infinite-dimensional space of observables via the Koopman operator, enabling linear system identification techniques to recover polynomial vector fields. The method achieves robust, noise-tolerant identification of unstable, chaotic, and open systems using low-sampling-rate data, with demonstrated accuracy in reconstructing vector fields and network topologies.

ABSTRACT

We exploit the key idea that nonlinear system identification is equivalent to linear identification of the socalled Koopman operator. Instead of considering nonlinear system identification in the state space, we obtain a novel linear identification technique by recasting the problem in the infinite-dimensional space of observables. This technique can be described in two main steps. In the first step, similar to the socalled Extended Dynamic Mode Decomposition algorithm, the data are lifted to the infinite-dimensional space and used for linear identification of the Koopman operator. In the second step, the obtained Koopman operator is "projected back" to the finite-dimensional state space, and identified to the nonlinear vector field through a linear least squares problem. The proposed technique is efficient to recover (polynomial) vector fields of different classes of systems, including unstable, chaotic, and open systems. In addition, it is robust to noise, well-suited to model low sampling rate datasets, and able to infer network topology and dynamics.

Motivation & Objective

• To address the challenge of identifying nonlinear dynamical systems from limited, noisy, or low-sampling-rate data.
• To bridge the gap between Koopman operator theory and data-driven system identification by recasting nonlinear identification as linear identification in the space of observables.
• To develop a method that avoids direct estimation of time derivatives, which is infeasible at low sampling rates.
• To enable accurate recovery of vector fields, including unstable and chaotic systems, and to infer network topology from trajectory data.
• To provide a robust, linear-method-based framework for system identification that is scalable to small datasets and complex dynamics.

Proposed method

• The method lifts the nonlinear system to an infinite-dimensional space of observables using the Koopman operator, where the system's dynamics become linear.
• It performs linear identification of the Koopman operator in this space using snapshot data, analogous to a component of the Extended Dynamic Mode Decomposition (EDMD) algorithm.
• The identified Koopman operator is projected back to the finite-dimensional state space via a linear least squares problem to recover the original vector field coefficients.
• The approach relies on a finite basis of polynomial functions to approximate the Koopman operator, enabling numerical computation.
• The infinitesimal generator of the Koopman operator is analytically connected to the vector field through a derivation similar to that in [4], enabling reconstruction of the nonlinear dynamics.
• The method is extended to handle open systems with process noise and inputs by incorporating noise models into the data generation and identification process.

Experimental results

Research questions

• RQ1 Can nonlinear system identification be effectively reformulated as linear identification in the space of observables using the Koopman operator?
• RQ2 How well can this lifting technique recover polynomial vector fields from low-sampling-rate, noisy trajectory data?
• RQ3 To what extent is the method robust to measurement and process noise in unstable or chaotic systems?
• RQ4 Can the method accurately reconstruct network topology and dynamics from limited trajectory data?
• RQ5 How does the performance of the method compare to direct methods that require time derivative estimation?

Key findings

• The method achieves a root mean square error (RMSE) of 0.025 in recovering the vector field coefficients of a 2D nonlinear system with forcing terms, averaged over 10 simulations.
• With strong process noise (σproc = 1), the method maintains robustness, achieving an RMSE of 0.078, demonstrating resilience to both measurement and process noise.
• For a 12-dimensional system with complex nonlinear interactions, the method achieves an RMSE of 0.021 with 1500 data points, and remains effective with only 300 trajectories (RMSE = 0.151).
• Network reconstruction using a threshold of 0.1 yields a true positive rate of 0.875 and a false positive rate of 0.023, indicating high accuracy in identifying true links.
• The method successfully infers not only the presence of links in a network but also their type (e.g., quadratic, cubic) and weight (coefficient values).
• The approach is well-suited for low-sampling-rate data and avoids the need for time derivative estimation, making it applicable to real-world biological and engineering systems with sparse measurements.