Koopman operator theory provides a principled framework for reduced-order modeling of nonlinear spatiotemporal dynamics through linear evolution in an observable space. In practice, however, learned finite-dimensional representations are typically approximately invariant, causing unresolved dynamics and accumulated closure errors. Recent approaches, inspired by the Mori-Zwanzig formalism, address these effects through deterministic memory or closure modeling, yet robust extrapolation beyond the training horizon remains challenging. Here, we propose KoopFlow, an efficient prediction-correction framework based on the key insight that finite-dimensional Koopman error accumulation can be reinterpreted as a progressive distributional degradation process. Accordingly, KoopFlow treats Koopman predictions as noisy intermediate estimates, which are then transported toward the target latent distribution through a generative model, thereby mitigating accumulated errors while retaining computational efficiency and temporal flexibility. Moreover, we reveal a non-monotonic relationship between Koopman spectral coherence and predictive accuracy. Experiments across diverse nonlinear dynamical systems demonstrate improved long-horizon prediction accuracy and stability over representative baselines, particularly beyond the training horizon.

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