SSTODE: Ocean-Atmosphere Physics-Informed Neural ODEs for Sea Surface Temperature Prediction

AI summary

Overview

Research area: Machine learning for ocean and climate science — specifically physics-informed deep learning applied to sea surface temperature (SST) forecasting.

Technical level: Advanced. The paper builds on partial differential equations (PDEs), Neural Ordinary Differential Equations, PDE-constrained inverse problems, and variational optimization. The summary below explains the ideas in plain language, but the underlying method assumes familiarity with continuous-time dynamical systems and fluid transport equations.

Scope: The paper introduces SSTODE, a Neural ODE framework that embeds advection–diffusion physics and surface heat-flux forcing to forecast global and regional SST, and reports state-of-the-art results on the OceanVP and ERA5 benchmarks with visualizations of the learned physical components.

What This Paper Is About

SST governs how the ocean and atmosphere exchange heat, and it shapes phenomena such as ENSO that affect weather worldwide. Existing deep learning models predict SST reasonably well but act as black boxes: they ignore the physical processes that move and mix heat in seawater, and they typically operate only at fixed time intervals. SSTODE addresses this by writing the evolution of SST as a continuous-time differential equation built from fluid transport principles, then adding a module that accounts for energy exchanged between the ocean and atmosphere at the surface.

Key Contributions

1. A continuous-time Neural ODE framework (SSTODE) that explicitly models coupled advection and diffusion for spatiotemporal SST prediction, rather than assuming idealized pure advection as prior Neural ODE work does.
2. An Energy Exchanges Integrator (EEI), derived from the ocean heat budget equation, that incorporates four surface heat fluxes from ERA5 reanalysis — shortwave radiation, longwave radiation, latent heat flux, and sensible heat flux — to capture external forcing on SST over the forecast horizon.
3. State-of-the-art performance on global and regional SST forecasting benchmarks (OceanVP and ERA5), outperforming recurrent-based, recurrent-free, and physics-informed baselines across multiple forecast horizons.
4. Visual evidence of three physical mechanisms the model learns: coherent advection flows, boundary-aware diffusion patterns, and external forcing of diurnal variations.

Main Findings

• Global forecasting gains: On OceanVP, SSTODE achieved the lowest MSE among all compared models at every tested horizon — 0.0527 at q=5, 0.0638 at q=7, and 0.0954 at q=12 — with ACC of 0.9990, 0.9987, and 0.9981 respectively. On ERA5, SSTODE reached MSE of 0.0180, 0.0232, and 0.0349 at q=5, 7, and 12, with ACC of 1.0000 in all three cases. The closest physics-informed baseline, ClimODE, recorded OceanVP MSE of 0.0622, 0.0705, and 0.1004 at the same horizons.

• Regional forecasting gains: Evaluated on the Equatorial Pacific, Northwest Atlantic, and Southern Ocean at q=5 and q=12, SSTODE produced the lowest MSE in every column of Table 2 (values scaled ×1000). Examples include the Equatorial Pacific at q=12 (0.7473 versus ClimODE's 0.8466), the Northwest Atlantic at q=12 (1.6445 versus ClimODE's 1.7046), and the Southern Ocean at q=12 (0.5597 versus ClimODE's 0.5716).

• Diffusion term matters: Removing the diffusion term degraded performance to MSE 0.0596, MAE 0.1185, ACC 0.9989 at q=5. A learnable global scalar diffusivity performed best (MSE 0.0527), a fixed κ=1 gave MSE 0.0562, and a learnable spatially-varying 2D diffusivity map gave MSE 0.0578. The authors report that the 2D map often fails through over-parameterization, since with only SST observed, the diffusion term becomes partly interchangeable with the advection term in explaining SST tendencies.

• Every surface heat flux helps, and shortwave contributes most: Removing the source term entirely raised MSE to 0.0595, MAE to 0.1198, and lowered ACC to 0.9988. Adding individual fluxes produced MSE of 0.0546 (shortwave), 0.0560 (longwave), 0.0559 (latent heat flux), and 0.0554 (sensible heat flux), while the full SSTODE reached 0.0527 with MAE 0.1107 and ACC 0.9990.

• Diffusion acts as a physical regularizer near coastlines: The authors report that regions with strong SST gradients, particularly near coastlines and straits in the Northwest Pacific, are highly sensitive to diffusion modeling, and that excluding diffusion produced numerical artifacts and larger errors in those areas.

• Stable initial velocity estimation: Varying the optimization epochs for the initial velocity field from 50 to 400 (50, 100, 200, 300, 400) on OceanVP at q=5 produced MSE values of 0.0556, 0.0548, 0.0527, 0.0539, and 0.0538, with a mean of 0.0542 ± 0.0010. Convergence typically occurred within 200 epochs.

• Parameter efficiency: At 2.8125° resolution, SSTODE uses 4.62M parameters versus ClimODE's 3.67M, and the authors report an approximately 10% improvement in forecasting accuracy for that marginal increase.

• Longer horizons and finer resolution: The paper reports experiments over a 7-day horizon at standard 6-hourly (q=28) and sparser 12-hourly (q=14) intervals on OceanVP (Appendix A.6), and at 2.8125° spatial resolution (Appendix A.7). The specific numerical results of those experiments are not included in the provided content.

Methodology in Plain English

The model treats SST as a field that changes continuously in time according to the advection–diffusion equation. Advection is the heat carried horizontally by ocean currents; diffusion is the spreading of heat by unresolved small-scale processes such as turbulence and mesoscale eddies. The authors reformulate this PDE as a system of first-order ODEs using the Method of Lines, so a Neural ODE solver can integrate it forward from the last observed SST field to any future time, at any resolution.

Because satellite data do not provide the ocean surface velocity field needed to drive the advection term, the framework first estimates it. It approximates the local temporal derivative of SST from past observations using cubic spline interpolation, then optimizes a learnable latent velocity field to satisfy the advection–diffusion residual, with ℓ2 regularization and an RBF-kernel Gaussian prior for spatial smoothness. The diffusion coefficient is a shared learnable scalar kept positive through a softplus activation, with α set to 1e-7, 200 epochs of optimization, and Adam.

Once initialized, the SST-ODE module integrates two coupled equations: one for SST evolution and one for latent velocity evolution. The velocity dynamics are not derived from Navier-Stokes; instead a neural network f_v predicts the velocity time derivative from the current velocity, SST, SST gradients, and a spatiotemporal embedding. That network uses a hybrid architecture combining ResNet blocks and attention modules.

The Energy Exchange Integrator then adds a source term for air–sea energy exchange. Following the ocean mixed-layer heat budget, the net energy flux is the sum of longwave radiation, shortwave radiation, latent heat flux, and sensible heat flux, divided by seawater density times specific heat capacity times mixed layer depth. Since future flux values are unavailable at inference, the four flux variables from the initial time are extended across the forecast horizon and concatenated with the predicted SST fields and spatiotemporal embeddings, then passed to a time-dependent source network that estimates future source terms. These estimates are added as a correction to the ODE forecast on the assumption that this captures both the forcing of SST and the feedback from SST to surface energy exchange.

Spatial positions are encoded with trigonometric functions and interactions of latitude and longitude, plus a static land-sea mask and orography. Temporal embeddings encode daily and seasonal periodicity via sine and cosine terms at periods of one day and 365 days.

Why This Matters

Impact on research. SSTODE shows that embedding ocean-specific physics — diffusion and surface energy exchange, not just advection — into a Neural ODE improves both accuracy and interpretability relative to generic deep learning and to prior physics-informed models like ClimODE. The paper positions this as a promising component for foundation models supporting regional refinement and physically consistent downscaling.

Real-world applications.

• Short-range SST forecasting to support ENSO monitoring, since the Equatorial Pacific is one of the three evaluated regions and ENSO is described as the dominant climate fluctuation of ocean–atmosphere coupling.
• Coastal and boundary-region forecasting where the diffusion term demonstrably reduces error and numerical artifacts, relevant to fisheries, upwelling-driven ecosystems, and coastal hazard planning.
• Operational marine and weather services that need forecasts at multiple lead times (30h, 42h, 72h, and up to 7 days evaluated here) rather than a single fixed interval.
• Climate monitoring that benefits from models whose learned terms can be inspected — velocity fields, diffusion patterns, and diurnal source terms — rather than only consulted for output numbers.

Industry relevance. The paper reports superior parameter efficiency relative to recurrent-based and recurrent-free baselines, and characterizes the ~10% accuracy gain over ClimODE (4.62M vs 3.67M parameters) as a trade-off acceptable for practical deployment. Accurate SST forecasts feed into shipping, offshore energy, aquaculture, and tourism planning.

Future Directions

• Extending the framework to ocean subsurface thermohaline dynamics, which the authors name explicitly as future work.
• Extending the framework to multiscale dynamics, also named by the authors.
• Investigating the 2D spatially-varying diffusivity parameterization further, since the paper reports it failed through over-parameterization and became partly interchangeable with the advection term.
• Testing whether the framework can serve as a component of foundation models for regional refinement and physically consistent downscaling, as the conclusion suggests.

Target Audience

This paper suits machine learning researchers working on physics-informed neural networks and Neural ODEs, oceanographers and climate scientists interested in interpretable SST prediction, and practitioners in marine forecasting who need continuous-time predictions over flexible horizons. Readers without a background in PDEs or continuous-time dynamical systems will find the method sections demanding, though the ablation and visualization sections are accessible on their own.

Authors’ abstract

Sea Surface Temperature (SST) is crucial for understanding upper-ocean thermal dynamics and ocean-atmosphere interactions, which have profound economic and social impacts. While data-driven models show promise in SST prediction, their black-box nature often limits interpretability and overlooks key physical processes. Recently, physics-informed neural networks have been gaining momentum but struggle with complex ocean-atmosphere dynamics due to 1) inadequate characterization of seawater movement (e.g., coastal upwelling) and 2) insufficient integration of external SST drivers (e.g., turbulent heat fluxes). To address these challenges, we propose SSTODE, a physics-informed Neural Ordinary Differential Equations (Neural ODEs) framework for SST prediction. First, we derive ODEs from fluid transport principles, incorporating both advection and diffusion to model ocean spatiotemporal dynamics. Through variational optimization, we recover a latent velocity field that explicitly governs the temporal dynamics of SST. Building upon ODE, we introduce an Energy Exchanges Integrator (EEI)-inspired by ocean heat budget equations-to account for external forcing factors. Thus, the variations in the components of these factors provide deeper insights into SST dynamics. Extensive experiments demonstrate that SSTODE achieves state-of-the-art performances in global and regional SST forecasting benchmarks. Furthermore, SSTODE visually reveals the impact of advection dynamics, thermal diffusion patterns, and diurnal heating-cooling cycles on SST evolution. These findings demonstrate the model's interpretability and physical consistency.