Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds

2026-03-16

AI Summary

This work addresses the closure errors and spurious spectral artifacts commonly arising in data-driven Koopman methods due to the lack of invariance in the learned feature space, which undermines prediction reliability. To overcome this, the authors propose a unified posterior diagnostic framework that quantifies invariance via principal angles and introduces Principal Angle Decomposition (PAD) of observables as a principled alternative to conventional SVD truncation. By integrating multi-step pointwise error bounds in a reproducing kernel Hilbert space (RKHS) with Gaussian process-based error surrogates, the framework enables certifiable dictionary learning and spectral analysis. Experiments on chaotic systems, high-dimensional benchmarks, and real-world datasets—including cavity flow and the Pluto–Charon system—demonstrate substantial improvements in modal accuracy and predictive reliability.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning
• Data Mining & Knowledge Management: Data Compression

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Koopman operator theory provides a global linear representation of nonlinear dynamics and underpins many data-driven methods. In practice, however, finite-dimensional feature spaces induced by a user-chosen dictionary are rarely invariant, so closure failures and projection errors lead to spurious eigenvalues, misleading Koopman modes, and overconfident forecasts. This paper addresses a central validation problem in data-driven Koopman methods: how to quantify invariance and projection errors for an arbitrary feature space using only snapshot data, and how to use these diagnostics to produce actionable guarantees and guide dictionary refinement? A unified a posteriori methodology is developed for certifying when a Koopman approximation is trustworthy and improving it when it is not. Koopman invariance is quantified using principal angles between a subspace and its Koopman image, yielding principal observables and a principal angle decomposition (PAD), a dynamics-informed alternative to SVD truncation with significantly improved performance. Multi-step error bounds are derived for Koopman and Perron--Frobenius mode decompositions, including RKHS-based pointwise guarantees, and are complemented by Gaussian process expected error surrogates. The resulting toolbox enables validated spectral analysis, certified forecasting, and principled dictionary and kernel learning, demonstrated on chaotic and high-dimensional benchmarks and real-world datasets, including cavity flow and the Pluto--Charon system.

Problem

• Koopman operator
• invariance
• projection error
• data-driven dynamics
• error bounds

Innovation

• Koopman operator
• invariance diagnostics
• principal angle decomposition
• error bounds
• dictionary learning