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 $z(τ) = \exp(\mathbf{K}_{\mathrm{cont}} τ) z(0)$ at any horizon $τ$ 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.

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