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

TL;DR

This paper introduces a continuous-time Koopman autoencoder model for fluid dynamics forecasting that achieves stable, long-horizon predictions with efficient inference, addressing limitations of discrete methods.

Contribution

The authors develop a continuous-time Koopman autoencoder with structural constraints that enable stable long-term forecasting and faster inference compared to existing methods.

Findings

Outperforms diffusion and operator-learning baselines on fluid benchmarks.

Achieves 110× inference speedup over previous methods.

Maintains stability and accuracy over long prediction horizons.

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 $dz/dt = \mathbf{K}_{\mathrm{cont}} z$, yielding closed-form inference via $z(\tau) = \exp(\mathbf{K}_{\mathrm{cont}} \tau) z(0)$ at any horizon $\tau$ 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 110$\times$ inference speedup.