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This research introduces a continuous-time Koopman autoencoder that decouples forecast cost from time horizon, achieving significant speedups over diffusion models for forward simulation while excelling at inverse data assimilation tasks. The study demonstrates that high performance in forward forecasting does not automatically translate to effective state recovery from sparse observations, highlighting the distinct challenges of solving inverse problems in physical systems. By enabling closed-form inference and gradient-based optimization independent of the assimilation window, this approach offers a robust surrogate model for complex fluid dynamics simulations.
Invertible continuous latent dynamic for long-term data assimilation in complex physical systems
arXiv:2602.02832v4 Announce Type: replace Abstract: Forward forecasting and data assimilation are the two important aspects in physical simulation: one propagates the state forward, the other recovers unknown states from sparse observations. Learned surrogates are normally built and benchmarked for forward forecasting, however, whether a surrogate could attain good performance in data assimilation tasks is valuable as well, as inverse problems are of paramount importance in the scientific domain. In this paper, we propose a continuous-time Koopman autoencoder whose latent dynamics obey $\frac{dz}{dt} = \mathbf{K}_{\mathrm{cont}} z$, 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, showing long-term stability and high efficiency in forward simulation, and also supports data assimilation as gradient-based optimization with cost independent of the assimilation window. Experiments are performed on the Kuramoto--Sivashinsky equation and a transient flow, and we compare our method against a range of baselines on the forward problem, including diffusion models and operator-learning models, and obtain a 110x inference speedup over strong diffusion baselines. We further test these baselines on an initial-state inference data assimilation task, and find that a strong forecaster does not guarantee a strong assimilator, while the continuous-time Koopman autoencoder achieves both higher accuracy and efficiency than surrogates of comparable forward performance.
Abstract:Forward forecasting and data assimilation are the two important aspects in physical simulation: one propagates the state forward, the other recovers unknown states from sparse observations. Learned surrogates are normally built and benchmarked for forward forecasting, however, whether a surrogate could attain good performance in data assimilation tasks is valuable as well, as inverse problems are of paramount importance in the scientific domain. In this paper, we propose a continuous-time Koopman autoencoder whose latent dynamics obey $\frac{dz}{dt} = \mathbf{K}_{\mathrm{cont}} z$, 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, showing long-term stability and high efficiency in forward simulation, and also supports data assimilation as gradient-based optimization with cost independent of the assimilation window. Experiments are performed on the Kuramoto--Sivashinsky equation and a transient flow, and we compare our method against a range of baselines on the forward problem, including diffusion models and operator-learning models, and obtain a 110x inference speedup over strong diffusion baselines. We further test these baselines on an initial-state inference data assimilation task, and find that a strong forecaster does not guarantee a strong assimilator, while the continuous-time Koopman autoencoder achieves both higher accuracy and efficiency than surrogates of comparable forward performance.
Machine Learning (cs.LG); Fluid Dynamics (physics.flu-dyn)
arXiv:2602.02832 [cs.LG]
(or arXiv:2602.02832v4 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2602.02832 Focus to learn more arXiv-issued DOI via DataCite
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