Multiscale Neural PDE Surrogates for Prediction and Downscaling: Application to Ocean Currents

TL;DR

This work tackles the need for high-resolution ocean current fields by introducing a resolution-agnostic neural-operator framework that can downscale coarse observations and simultaneously serve as a PDE surrogate capable of producing solutions at arbitrary resolutions. It extends neural operators with temporal capabilities and develops multiple variants (including DUNO, SpecDFNO, SpecDFNODiff, MetaGradDFNO, and MultiGradDFNO) that incorporate gradient information, residual learning, and diffusion-based upsampling, all with a soft physics constraint layer. The approach is validated on Navier–Stokes simulations and real Copernicus ocean current data, showing strong gains over CNN baselines for downscaling at twofold and fourfold resolutions and robust multi-resolution PDE prediction without external solvers, though performance degrades at eightfold downscaling due to missing subgrid details. These results demonstrate a practical, scalable path to high-resolution ocean current maps and pave the way for uncertainty quantification and physics-informed extensions in geophysical surrogates.

Abstract

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.

Figures (5)

• Figure 1: The figure (inspired by yang2023fourierneuraloperatorsarbitrary) shows the overall structure of our Temporal/static downscaling model. The low-resolution input a goes through an optional preprocessing (gradient transformation for gradient based methods) then a neural network and an upsampling block. Then an embedding function $e(·)$ is returned. Finally, a neural operator takes in $e(·)$ and outputs a function which gets into a reconstruction block and an optional constraint layer.
• Figure 2: (Temp_DFNO) 5 steps low resolution inputs, and predictions of the model on both $16\times16$ and $32\times32$ resolutions, as well as the residuals in the last 2 rows.
• Figure 3: (Temp_specDFNO) 5 steps low resolution inputs, and predictions of the model on both $16\times16$ and $32\times32$ resolutions, as well as the residuals in the last 2 rows.
• Figure 4: Ground truth vs. predictions of SpecDFNO. Rows correspond to different output resolutions: $16\times16$, $32\times32$, $64\times64$, and $128\times128$. The first column shows the model predictions, the second shows the ground truth, and the third displays the difference between them.
• Figure 5: Ground truth vs. predictions of SpecDFNODiff. Rows correspond to different output resolutions: $16\times16$, $32\times32$, $64\times64$, and $128\times128$. The first column shows the model predictions, the second shows the ground truth, and the third displays the difference between them.