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.
| Subjects: | Machine Learning (cs.LG); Fluid Dynamics (physics.flu-dyn) |
| Cite as: | arXiv:2602.02832 [cs.LG] |
| (or arXiv:2602.02832v3 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2602.02832 arXiv-issued DOI via DataCite |
From: Rares Grozavescu [view email]
[v1]
Mon, 2 Feb 2026 21:33:07 UTC (1,608 KB)
[v2]
Thu, 19 Mar 2026 10:03:18 UTC (2,340 KB)
[v3]
Fri, 8 May 2026 09:53:18 UTC (2,832 KB)
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