KoopFM: Base-Anchored Flow Matching for Long-Term Emulation of Forced Dynamical Systems

Long-term rollouts of chaotic dynamical systems are critical across computational science, from turbulence to climate projection. Under time-varying external forcing the problem shifts from an initial-value to a boundary condition one, in which long-run statistics are set by the forcing trajectory. Neural operators are increasingly being used in this setting, but deterministic models are spectrally biased and may drift, whereas generative ones fix the spectra issue but transport mass from isotropic Gaussian noise. Thus, the vector field must rebuild global structure, forcing response and fine-scale turbulence at once, at many network evaluations per step. Motivated by scale separation and the Mori–Zwanzig formalism, we propose KoopFM, which couples a Koopman-autoencoder base model with base-anchored flow matching, where we replace the standard Gaussian source of the probability path with a source centered on the base prediction, so the residual travels a much shorter path. We identify the source variance σ as the central mechanism, as it prevents degeneracy and determines that path length, as well as the correction range of the flow. Across three chaotic dynamical systems, we show that the anchoring construction has two main advantages. First, whenever the one-step conditional is narrow, inference needs as few as two NFEs without any distillation. Second, the damped, forcing-aware Koopman anchor improves the out-of-distribution forcing response on the chaotic Lorenz–96 system, and on forced Rayleigh–Benard convection the anchored model outperforms vanilla flow matching beyond the training range of the forcing, up to a metric-dependent limit.