Accurate modeling of physical systems governed by partial differential equations is a central challenge in scientific computing. In oceanography, high-resolution current data are critical for coastal management, environmental monitoring, and maritime safety. However, available satellite products, such as Copernicus data for sea water velocity at ~0.08 degrees spatial resolution and global ocean models, often lack the spatial granularity required for detailed local analyses. In this work, we (a) introduce a supervised deep learning framework based on neural operators for solving PDEs and providing arbitrary resolution solutions, and (b) propose downscaling models with an application to Copernicus ocean current data. Additionally, our method can model surrogate PDEs and predict solutions at arbitrary resolution, regardless of the input resolution. We evaluated our model on real-world Copernicus ocean current data and synthetic Navier-Stokes simulation datasets.

翻译：精确建模由偏微分方程控制的物理系统是科学计算领域的核心挑战。在海洋学中，高分辨率洋流数据对于海岸管理、环境监测和海上安全至关重要。然而，现有的卫星产品（如空间分辨率约0.08度的哥白尼海水流速数据）以及全球海洋模型，往往缺乏详细局部分析所需的空间粒度。在本研究中，我们（a）提出了一种基于神经算子的监督深度学习框架，用于求解偏微分方程并提供任意分辨率的解；（b）构建了降尺度模型并将其应用于哥白尼洋流数据。此外，我们的方法能够建模代理偏微分方程，并在任意输入分辨率下预测任意分辨率的解。我们在真实世界的哥白尼洋流数据以及合成Navier-Stokes模拟数据集上评估了所提出的模型。