A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

Spectral methods for nonlinear dynamics, notably Koopman analysis, have gained prominence for their ability to linearize complex systems in an infinite-dimensional space. Early methods like Dynamic Mode Decomposition (DMD) provided practical tools for extracting dominant modes but were limited to eigenvalues and modes, lacking eigenfunctions. Ulam's method and Galerkin approaches offered theoretical frameworks but faced computational challenges. Recent advances integrate machine learning and data-driven techniques, aiming to overcome these limitations. The need for scalable, accurate spectral analysis methods that handle high-dimensional, noisy, and stochastic data remains pressing, motivating the development of EDMD as a flexible, convergent algorithm.

Existing spectral methods struggle with capturing the full spectral structure, especially eigenfunctions, in complex systems. DMD's approximation is limited to eigenvalues and modes, insufficient for detailed long-term analysis. Ulam and Galerkin methods, while theoretically sound, are computationally intensive and sensitive to basis choice. Handling stochastic systems adds further complexity. The core challenge is developing a scalable, robust method that can accurately approximate the entire Koopman spectrum, including eigenfunctions, from limited data, and extend applicability to stochastic dynamics with theoretical convergence guarantees.

The key innovation is the formulation of EDMD, which generalizes DMD by incorporating a flexible dictionary of observables, enabling the approximation of eigenfunctions. The method leverages Galerkin theory, providing convergence guarantees as data volume increases. It supports multiple basis functions, including polynomials and Fourier modes, and extends naturally to stochastic systems by approximating eigenfunctions of the Kolmogorov backward operator. The approach balances computational efficiency with theoretical rigor, offering a unified framework for deterministic and stochastic spectral analysis, filling a critical gap in the field.

• �� Collect snapshot pairs \(\{(x_m, y_m)\}\), where \(y_m = F(x_m)\). • Choose a dictionary \(D = \{\psi_1, \psi_2, ..., \psi_K\}\) of observables. • Construct the data matrices \(X = [x_1, ..., x_M]\) and \(Y = [y_1, ..., y_M]\). • Compute the matrices \(G\) and \(A\) via least squares: \(G = rac{1}{M} \sum \Psi(x_m)^* \Psi(x_m)\), \(A = rac{1}{M} \sum \Psi(x_m)^* \Psi(y_m)\). • Solve \(K = G^+ A\) for the Koopman matrix. • Perform eigen-decomposition of \(K\) to obtain eigenvalues \(\mu_j\) and eigenvectors \(\xi_j\). • Approximate eigenfunctions as \(\varphi_j = \Psi \xi_j\). • Derive Koopman modes from the eigenvectors of \(K\) and the data matrices, enabling reconstruction of system states.

The experiments involved applying EDMD to Lorenz and double pendulum systems for deterministic validation, and stochastic differential equations and Markov processes for probabilistic validation. Data were generated via numerical simulations with varying sample sizes and dictionary choices. Performance metrics included eigenvalue errors, eigenfunction approximation accuracy, and mode reconstruction fidelity. Comparisons with standard DMD and Ulam methods demonstrated superior accuracy, especially in capturing eigenfunctions. Sensitivity analyses assessed the impact of dictionary selection and data volume, confirming robustness. The experiments validated theoretical convergence and practical effectiveness across diverse systems.

EDMD achieved eigenvalue errors below 10^-4 in deterministic examples, accurately capturing long-term dynamics. Eigenfunctions of stochastic systems, such as the Kolmogorov backward operator, were well approximated, confirming the method's extension to probabilistic systems. The influence of dictionary choice was significant; richer dictionaries improved accuracy. Limited data experiments showed that even with sparse samples, EDMD could identify key spectral features. These results demonstrate EDMD's potential for high-fidelity system analysis, model reduction, and control in both deterministic and stochastic contexts.

The method is directly applicable to fluid flow analysis, structural health monitoring, climate modeling, and control systems where explicit models are unavailable. It enables extraction of dominant modes and eigenfunctions for model reduction, system identification, and predictive control. Its flexibility allows adaptation to various data types, including experimental and simulation data, supporting real-time monitoring and fault detection. The ability to handle stochastic systems broadens its utility in probabilistic forecasting and risk assessment, promising significant impact across engineering and scientific disciplines.

The choice of dictionary functions remains heuristic; automatic learning methods are needed for optimal basis selection. High-dimensional systems pose computational challenges due to matrix sizes and sample requirements. The approach assumes stationarity, limiting applicability to non-stationary or rapidly evolving systems. Noise and measurement errors can affect eigenfunction accuracy, requiring robust regularization. Future work should address these issues to enhance scalability, adaptability, and robustness.