Principled Operator Learning in Ocean Dynamics: The Role of Temporal Structure

TL;DR

The paper addresses the challenge of achieving physically faithful, long-horizon ocean forecasts with neural operators. It introduces a temporally extended Fourier Neural Operator (FNOtD) that learns spatiotemporal kernels by incorporating temporal frequency, enabling internalization of the dispersion relation for shallow-water dynamics. In Baltic Sea experiments, FNOtD delivers more stable autoregressive predictions with improved spectral fidelity and competitive RMSE relative to a state-of-the-art numerical model, at a fraction of the computational cost. This principled operator-learning approach advances the goal of fast, reliable ocean digital twins and can be extended to three-dimensional, irregular-data settings.

Abstract

Neural operators are becoming the default tools to learn solutions to governing partial differential equations (PDEs) in weather and ocean forecasting applications. Despite early promising achievements, significant challenges remain, including long-term prediction stability and adherence to physical laws, particularly for high-frequency processes. In this paper, we take a step toward addressing these challenges in high-resolution ocean prediction by incorporating temporal Fourier modes, demonstrating how this modification enhances physical fidelity. This study compares the standard Fourier Neural Operator (FNO) with its variant, FNOtD, which has been modified to internalize the dispersion relation while learning the solution operator for ocean PDEs. The results demonstrate that entangling space and time in the training of integral kernels enables the model to capture multiscale wave propagation and effectively learn ocean dynamics. FNOtD substantially improves long-term prediction stability and consistency with underlying physical dynamics in challenging high-frequency settings compared to the standard FNO. It also provides competitive predictive skill relative to a state-of-the-art numerical ocean model, while requiring significantly lower computational cost.

Figures (4)

• Figure 1: Iterative predictions with different initial conditions. Time series of outputs at a representative location are shown for (a) the standard FNO and (c) FNOtD. The right panels illustrate the convergence of the predictions under identical forcing by displaying the difference between each prediction and a reference prediction initialized on September 10. Their time axis is relative to each prediction’s origin time, while differences are computed at corresponding calendar timestamps. Repeating the model training with different random initializations yields consistent results.
• Figure 2: Radially averaged power spectra of iterative hourly predictions at (a) 7 days and (b) 30 days after prediction initialization. (c) Relative RMSE of the predictions in the spectral domain (see Appendix \ref{['appendixC']}). All panels correspond to forecasts initialized on October 1.
• Figure 3: (a) Architecture of the Fourier Neural Operator FNOtD, with input and output dimensions of h×w×t×c, where h and w correspond to the latitude and longitude grid dimensions, t denotes time steps, and c is the number of physical variables. Eliminating the temporal dimension yields the standard FNO model architecture. The input is first projected into a higher-dimensional channel space by a point-wise encoder P. After passing through four Fourier layers, the feature representations are mapped to the output variables via a point-wise decoder Q. (b) Schematic of a single Fourier layer. $\mathcal{M}$ is a point-wise multilayer perceptron layer to enhance local nonlinear representation learning.
• Figure 4: Time series of iterative sea level (in meter) predictions by the standard FNO and FNOtD are shown, alongside Nemo outputs at the nearest grid point to the corresponding tide gauge observation. The data corresponds to the period of the out-of-sample test dataset.