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

TL;DR

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 hybrid physical-machine learning climate models.

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 hybrid physical-machine learning climate models.

Figures (5)

• 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, with gradual attenuation at high frequencies indicating controlled dissipation rather than instability.
• 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.
• 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 \ref{['section:results']}.