🤖 AI Summary
This work addresses the susceptibility of numerical Koopman spectral analysis to spurious eigenvalues, which compromises spectral reliability. To mitigate this issue, the authors propose a novel neural dictionary learning approach that uniquely formulates Koopman residual minimization as the central objective of dictionary learning and jointly optimizes the condition number of the data matrix to enhance numerical stability. The method simultaneously preserves one-step prediction accuracy while significantly improving the fidelity and certifiability of the Koopman spectrum. Experimental evaluations across diverse dynamical systems and real-world sea surface temperature data demonstrate that the proposed framework effectively suppresses spectral pollution, enhances pseudospectral inclusiveness, and yields superior overall approximation quality.
📝 Abstract
Koopman theory promises linear structure in nonlinear dynamics, but numerical Koopman spectra are easy to compute and hard to trust. A finite EDMD matrix always has eigenvalues; the problem is that many of them may have nothing to do with the infinite-dimensional operator. In this paper we make spectral reliability the objective of dictionary learning. We train neural-network dictionaries not merely to predict the next snapshot, but to minimize Residual Dynamic Mode Decomposition residuals: operator-level a posteriori errors that test whether computed eigenvalues and modes are genuine Koopman spectral objects. To keep the learned observables from collapsing into an unstable coordinate system, the loss also penalizes the condition number of the lifted data matrix. Thus the method couples two requirements that should not be separated: small Koopman residuals and a well-conditioned representation. The result is a learned dictionary that is expressive, numerically stable, and spectrally disciplined. Across conservative and dissipative benchmark systems, the method sharply reduces spectral pollution, improves residual pseudospectral inclusion, and lowers forecast error relative to standard fixed dictionaries. On sea-surface temperature data, it gives cleaner Koopman diagnostics and substantially better one-step forecasts from noisy observations with no governing equations. The message is simple: neural Koopman learning should be judged not by prediction alone, but by whether its spectral claims can be certified. Residuals provide the certificate; conditioning makes it computable.