Koopman Autoencoders with Continuous-Time Latent Dynamics for Fluid Dynamics Forecasting
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917470554554368 |
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| author | Grozavescu, Rares Zhang, Pengyu Meunier, Etienne Girolami, Mark |
| author_facet | Grozavescu, Rares Zhang, Pengyu Meunier, Etienne Girolami, Mark |
| contents | 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 $z(τ) = \exp(\mathbf{K}_{\mathrm{cont}} τ) z(0)$ at any horizon $τ$ in a single step. This decouples forecast cost from forecast length at inference time 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 $z_{t+Δt} = \mathbf{A} z_t$; recovering the continuous generator could be theoretically done through matrix logarithm but requires conditions not guaranteed by training, and approximation errors grow with the $Δt$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02832 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Koopman Autoencoders with Continuous-Time Latent Dynamics for Fluid Dynamics Forecasting Grozavescu, Rares Zhang, Pengyu Meunier, Etienne Girolami, Mark Machine Learning Fluid Dynamics 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 $z(τ) = \exp(\mathbf{K}_{\mathrm{cont}} τ) z(0)$ at any horizon $τ$ in a single step. This decouples forecast cost from forecast length at inference time 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 $z_{t+Δt} = \mathbf{A} z_t$; recovering the continuous generator could be theoretically done through matrix logarithm but requires conditions not guaranteed by training, and approximation errors grow with the $Δt$ 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. |
| title | Koopman Autoencoders with Continuous-Time Latent Dynamics for Fluid Dynamics Forecasting |
| topic | Machine Learning Fluid Dynamics |
| url | https://arxiv.org/abs/2602.02832 |