Towards Efficient and Stable Ocean State Forecasting: A Continuous-Time Koopman Approach

Rares Grozavescu; Pengyu Zhang; Mark Girolami; Etienne Meunier

cs.LGMar 5, 2026

Towards Efficient and Stable Ocean State Forecasting: A Continuous-Time Koopman Approach

Authors: Rares Grozavescu, Pengyu Zhang, Mark Girolami, Etienne Meunier

Organizations: Department of Engineering, University of Cambridge, Cambridge, UK · Alan Turing Institute, London, UK · INRIA, Paris, France

Abstract

We investigate the Continuous-Time Koopman Autoencoder (CT-KAE) as a lightweight surrogate model for long-horizon ocean state forecasting in a two-layer quasi-geostrophic (QG) system. By projecting nonlinear dynamics into a latent space governed by a linear ordinary differential equation, the model enforces structured and interpretable temporal evolution while enabling temporally resolution-invariant forecasting via a matrix exponential formulation. Across 2083-day rollouts, CT-KAE exhibits bounded error growth and stable large-scale statistics, in contrast to autoregressive Transformer baselines which exhibit gradual error amplification and energy drift over long rollouts. While fine-scale turbulent structures are partially dissipated, bulk energy spectra, enstrophy evolution, and autocorrelation structure remain consistent over long horizons. The model achieves orders-of-magnitude faster inference compared to the numerical solver, suggesting that continuous-time Koopman surrogates offer a promising backbone for efficient and stable physical-machine learning climate models.

Figures & tables

Figure 1: CT-KAE architecture. High-dimensional QG states $x_{t}$ are encoded into a latent representation $z_{t}$ using a dual-stream encoder that incorporates both present and historical information. The latent state evolves according to a linear continuous-time ODE $\dot{z}=\mathbf{K}z$ , enabling resolution-invariant temporal querying via the matrix exponential. The decoder reconstructs physical fields from latent space.

Figure 2: Long-horizon qualitative comparison (2083 days). Top row: Ground truth QG potential vorticity fields. Middle row: CT-KAE predictions rolled out for 9999 steps ( $\Delta t=5$ h) without reinitialization. Bottom row: Kinetic Energy (KE) spectra averaged over the rollout for ground truth (blue) and CT-KAE (orange). The KE spectrum demonstrates preservation of bulk energy at low wavenumbers, while high-wavenumber modes are gradually attenuated. This controlled dissipation contributes to long-horizon stability, though it also indicates partial loss of fine-scale turbulent structures. Improving the capture of high-frequency dynamics while maintaining stability is a promising direction for future work.

Panel of Figure 3 — Figure 3: Stability over long horizons (20 trajectories). (a) CT-KAE preserves large-scale persistence patterns. (b) MSE remains bounded over 10,000 rollout steps. (c) Enstrophy evolves without runaway amplification.

Metric	CT-KAE	ViT (AR)
RMSE $\downarrow$	0.938 $\pm$ 0.031	0.993 $\pm$ 0.033
ACC $\uparrow$	0.033	0.004
KE Drift (%) $\to 0$	-49.4	23.3
Enstrophy Drift (%) $\to 0$	-34.6	37.0
Error Growth Rate $\downarrow$	-0.0267	0.0217

Table 1: Long-horizon performance at 2083-day horizon (10,000 steps). Averaged across trajectories.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Value
Latent Dimension ( $d$ )	128
Batch Size	32
Optimizer	AdamW
Learning Rate	$2\times 10^{-4}$
Weight Decay	$1\times 10^{-5}$
ODE Integration	Runge-Kutta 4 (Train), Matrix Exp (Test)

Appendix

Table 2: Hyperparameters used for the CT-KAE experiments.

Figure 4: Temporal resolution invariance. CT-KAE trained at $\Delta t=5$ h is evaluated at different integration resolutions ( $1h$ and $10h$ ) using the continuous-time formulation. Each row corresponds to a different query time-step. The qualitative similarity across resolutions demonstrates that latent dynamics generalize across temporal discretizations without retraining, validating the matrix exponential formulation.

Figure 5: Eigenvalue spectrum of the learned Koopman operator $\mathbf{K}$ . Eigenvalues are plotted in the complex plane. The majority lie in the left half-plane (negative real part), indicating dissipative modes, while a subset lies close to the imaginary axis, corresponding to oscillatory dynamics. The learned operator spectrum exhibits no eigenvalues with positive real part, which is consistent with the empirically observed bounded latent trajectories. While this does not constitute a formal stability guarantee of the decoded dynamics, it provides structural evidence supporting long-horizon stability as from Section 5 .

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KFTD: Koopman-Fourier Time-Differentiable Network for Continuous Ocean Spatiotemporal Forecasting

Qinghui Chen, Zekai Zhang, Hailong Liu, Jinglin Zhang+1

Shandong University School of Control Science and Engineering Jinan, China · Laoshan Laboratory Qingdao, China · Chinese Academy of sciences Institute of Atmospheric Physics Beijing, China+1

Koopman Operator Learning Physics-Informed ML Spatiotemporal Forecasting

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Accurate oceanic forecasting is critical for climate monitoring and disaster early warning. However, ocean spatiotemporal forecasting encounters the double challenges of modeling complex dynamical systems and ensuring computational efficiency. We present Koopman Fourier Time-Differentiable (KFTD) Network, a time continuous twostage paradigm that decouples interpolation from prediction to achieve efficient and scalable spatiotemporal modeling. We map complex nonlinear dynamics into the Koopman linear space and exploit Fourier analysis to enable continuous time interpolation at arbitrary sub-steps. A lightweight residual network consumes the high fidelity intermediate states to yield the final forecast. Unlike diffusion models, KFTD eliminates multi step noise sampling and directly evolves the system in continuous time, yielding a 4 computational speedup. We further introduce a DPP Loss that supports arbitrary PDE constraints in an endtoend manner, breaking the physical consistency bottleneck of pure data-driven approaches. Empirical results on four ocean datasets confirm that our continuous time framework reduces MSE by an average of 5.6% (up to 12.7% for SST) and improves efficiency over MCVD by 76.25%.

Abstract

Spatiotemporal Forecasting

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