Beyond Invariant Dictionary: Data-Driven Koopman Spectral Recovery with Filtered Extended Dynamic Mode Decomposition

2026-08-01

AI Summary

This work addresses spectral pollution in extended dynamic mode decomposition (EDMD) caused by non-invariant dictionaries, which distorts the eigenvalues of the Koopman operator. To resolve this issue, the authors propose a projected Koopman approximation framework that constructs compatible subspaces via forward-intersection chains, enabling accurate preservation of Koopman eigenpairs associated with nonzero eigenvalues without requiring dictionary invariance. The approach integrates coordinate-wise and $L^2(\mu)$-orthogonal projection geometries, leveraging singular value decomposition to formulate a filtered EDMD variant. Numerical experiments on the Kronecker flow, polynomial systems, and the Van der Pol oscillator demonstrate significant suppression of spectral pollution: the coordinate projector recovers the local equilibrium spectrum independently of the sampling measure, while the $L^2(\mu)$ projector accurately approximates the limit-cycle spectrum.

Abstract

The Koopman operator provides a linear framework for analyzing nonlinear dynamical systems through spectral properties. Extended Dynamic Mode Decomposition (EDMD) approximates this operator from data, but non-invariant dictionaries can introduce spurious eigenvalues. We introduce the Projected Koopman Operator Approximation framework for constructing Filtered EDMD operators. The framework projects the Koopman action onto admissible dictionary subspaces that need not be invariant, while exactly preserving every represented Koopman eigenpair with nonzero eigenvalue. A forward--intersection chain provides a canonical hierarchy of compatible subspaces, connecting the full dictionary to its maximal invariant core while retaining useful intermediate models. We analyze two projection geometries: a coordinate-orthogonal projector, which requires no function-space Gram-matrix estimate but is basis-dependent, and a function-space orthogonal projector, which recovers population EDMD at the unfiltered level. We characterize their relationship to EDMD and existing subspace-selection methods. We also develop SVD-based algorithms for constructing sampled forward--intersection spaces and implementing the coordinate projector. Under independent noiseless sampling and exact-rank identifiability, the resulting empirical operators converge almost surely to their population counterparts. Experiments on a Kronecker flow, a polynomial system, and the Van~der~Pol oscillator demonstrate reduced spectral pollution. For Van~der~Pol, the coordinate projector recovers the local equilibrium spectrum independently of the sampling measure, whereas the $L^2(μ)$ projector approximates the limit-cycle spectrum on the same certified subspace.

Problem

• Koopman operator
• Extended Dynamic Mode Decomposition
• spectral pollution
• non-invariant dictionary
• eigenvalue spuriousness

Innovation

• Filtered EDMD
• Projected Koopman Operator
• Forward–Intersection Chain
• Non-invariant Dictionary
• Spectral Pollution Reduction