Stabilizing Neural Emulation of Spatiotemporal Physical Dynamics through Denoising Koopman Latent Dynamics

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 Koopman representations are typically approximately invariant, causing unresolved dynamics and accumulated closure errors. Recent approaches, motivated by the Mori-Zwanzig formalism, address these unresolved 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 built on two key insights: First, we identify a non-monotonic relationship between Koopman spectral coherence and predictive accuracy, revealing a trade-off between long-term information preservation and transient dynamics fidelity. Second, we reinterpret finite-dimensional Koopman error accumulation as a progressive distributional degradation process. Accordingly, KoopFlow treats Koopman predictions as noisy intermediate estimates and leverages a pretrained flow matching model as a probabilistic transport bridge toward the target latent distribution. By coupling structured linear Koopman prediction with generative correction, KoopFlow mitigates accumulated errors while retaining computational efficiency and temporal flexibility. Experiments across diverse nonlinear dynamical systems demonstrate improved long-horizon prediction accuracy and stability over representative baselines, particularly beyond the training horizon.