Koopman Autoencoders with Continuous-Time Latent Dynamics for Fluid Dynamics Forecasting

arXiv:2602.02832v3 Announce Type: replace Abstract: Forecasting physical systems over long horizons from irregularly sampled observations demands models that are stable, computationally efficient, and free of fixed-timestep assumptions. We address this with a continuous-time Koopman autoencoder whose latent dynamics obey dz/dt = mathbf{K}{cont} z, yielding closed-form inference via z(au) = exp(mathbf{K}{cont} au) z(0) at any horizon au in a single step. This decouples forecast cost from forecast length at inference time and supports data assimilation as gradient-based optimization with cost independent of the assimilation window. However, scaling continuous-time Koopman dynamics to high-dimensional chaotic systems causes severe latent instability, including spectral collapse and trajectory divergence over long horizons. In contrast, discrete Koopman methods train an operator mathbf{A} such that z_{t+Delta t} = mathbf{A} z_t; recovering the continuous generator could be theoretically done through matrix logarithm but requires conditions not guaranteed by training, and approximation errors grow with the Delta t imposed by the training data. These methods also require fixed, regular timesteps. We identify an empirically effective set of structural constraints -- rollout training, forward-backward consistency, latent regularization, and physics-conditioned LoRA -- sufficient for stable long-horizon latent dynamics. On challenging fluid benchmarks, our method outperforms strong diffusion and operator-learning baselines on long-horizon forecasting while achieving a 110imes inference speedup.

Source: arXiv cs.LG | 2026-05-11