Loss functions and constraints improve sea surface height prediction

In order to understand currents, tides and other ocean dynamics, scientists need to accurately capture sea surface height, or a snapshot of the ocean’s surface, including peaks and valleys due to changes in wind, currents and temperature, at any given moment. In order to more accurately forecast ocean circulation and other processes, climate variability, air-sea interactions and extreme weather events, researchers need to be able to accurately predict sea surface height into the future.

Current physics-driven, statistical and data-driven sea surface height modeling systems are only capable of producing accurate predictions approximately 14–15 days into the future due to design limitations. In order to improve predictions further into the future, a group of oceanographers from Zhejiang University and the State Key Laboratory of Satellite Ocean Environment Dynamics characterized the limitations of existing sea surface height models and developed a new system designed to improve the reliability of medium- and long-range predictions beyond 14–15 days.

The team published the study on December 16, 2025 in the journal Ocean-Land-Atmosphere Research.

Many of today’s data-driven sea surface height prediction models rely on deep learning systems capable of automatically learning how ocean features, such as temperature, currents, heat level and carbon dioxide flux, change over space and time from large-scale historical datasets. Despite their power and accuracy in the short term, current deep-learning models require substantial computing power and long training times and suffer from error accumulation over longer timescales.

“Specifically, we [sought] to explore an approach within a deep learning framework that simultaneously enhances temporal sequence modeling capability and incorporates physical consistency constraints, so as to ensure that [sea surface height] predictions remain accurate, stable and physically reasonable over extended forecast horizons,” said Chengmin Si, graduate student at Ocean College in Zhejiang University and first author of the research paper.

The researchers explained that existing deep learning systems rely purely on data fitting, which eventually leads to error accumulation and gradually distorted prediction results. To combat this, the team integrated a physics-informed loss function in their deep learning model, which gently steers the model to follow well-known physical rules, helping keep its predictions realistic while still allowing the system to learn from data.

To address the excessive computing power and training times required for existing sea surface models, the team developed a geostrophic Temporal Attention Unit (TAU) U-Net (GTU-Net) model. Here, geostrophic refers to the balance between large-scale fluid motions, such as wind and ocean currents, which helps guide the model toward more physically realistic predictions, while the TAU mechanism allows the model to effectively focus on key spatiotemporal features, or ocean features that change over time.

 “[T]he proposed GTU-Net combines a TAU for enhanced long-term temporal dependency modeling with a two-step inference loss function incorporating geostrophic balance constraints, effectively improving long-term prediction stability and mitigating error accumulation. This study demonstrates that embedding physical prior knowledge into deep learning models is not only feasible but also a key approach for achieving reliable ocean dynamic predictions,” said Si.

One key to prediction stability was introducing temporal recurrence constraints to the model, which impose structure and order on data and operations involving time to ensure model consistency and reflect real-world phenomena. Geostrophic balance constraints additionally penalize model outputs that violate geostrophic equations, improving the predictive performance of the model based on pressure gradient force and the Coriolis force, or the deflection of moving objects caused by Earth’s rotation.

GTU-Net experiments performed in the western North Pacific outperformed existing mainstream models, such as ConvLSTM, PredRNN, SimVP, U-Net and the Persistence Forecast, across multiple metrics and demonstrated higher accuracy. Further, GTU-Net mitigated the error growth characteristic of existing sea surface height models and improved prediction accuracy to a span of up to 60 days.

The team plans to test the validity of GTU-Net predictions by assessing the physical plausibility of the model predictions, such as mesoscale eddy (a large rotating loop of ocean water) identification. Additionally, the current geostrophic constraints of the GTU-Net model primarily capture large-scale flow features. Future studies could explore multiscale physical information strategies, such as employing a dual-branch network architecture that integrates geostrophic balance constraints with momentum or shallow-water equations, to further enhance physical consistency across different spatial scales.

“Ultimately, our goal is to develop a robust and interpretable deep learning prediction framework that can serve as a practical complement to traditional numerical ocean models, supporting long-term ocean monitoring, climate studies and operational ocean forecasting,” said Si.

Jianhao Gao and Wenxia Zhang from the State Key Laboratory of Satellite Ocean Environment Dynamics in the Second Institute of Oceanography at the Ministry of Natural Resources in Hangzhou, China, and the Observation and Research Station of Yangtze River Delta Marine Ecosystems at the Ministry of Natural Resources in Zhoushan, China; and Feng Zhou from Ocean College at Zhejiang University in Zhoushan, China, the State Key Laboratory of Satellite Ocean Environment Dynamics in the Second Institute of Oceanography at the Ministry of Natural Resources in Hangzhou, China, and the Observation and Research Station of Yangtze River Delta Marine Ecosystems at the Ministry of Natural Resources in Zhoushan, China also contributed to this research.

This work was supported by the National Science Foundation of China (No. 42476216) and the United Nations Ocean Decade Project (No. 63.5. Coastal Oxygen and Hypoxia in Asian waters).