Dynamic Deep Learning Based Super-Resolution For The Shallow Water Equations

TL;DR

The work presents a hybrid dynamic super-resolution framework that couples the ICON-O shallow-water ocean model with a local-patch U-Net to correct coarse-grid velocity fields during runtime. By training the network to map low-resolution outputs to high-resolution ground truth from Galewsky-type tests, the authors demonstrate that a 20 km ICON-O run with 12 h ML corrections can achieve $L_2$-accuracy comparable to a 10 km reference after about $8$ days, while conserving mass. Energy and enstrophy analyses reveal that the ML corrections mainly impact small scales, injecting some energy at high wavenumbers and introducing artifacts that slightly perturb the energy pathways. The approach shows potential for reducing wallclock time in dynamical core simulations, though it requires physics-informed constraints and uncertainty handling to stabilize long-term integrations and curb spurious energy generation.

Abstract

Using the nonlinear shallow water equations as benchmark, we demonstrate that a simulation with the ICON-O ocean model with a 20km resolution that is frequently corrected by a U-net-type neural network can achieve discretization errors of a simulation with 10km resolution. The network, originally developed for image-based super-resolution in post-processing, is trained to compute the difference between solutions on both meshes and is used to correct the coarse mesh every 12h. Our setup is the Galewsky test case, modeling transition of a barotropic instability into turbulent flow. We show that the ML-corrected coarse resolution run correctly maintains a balance flow and captures the transition to turbulence in line with the higher resolution simulation. After 8 day of simulation, the $L_2$-error of the corrected run is similar to a simulation run on the finer mesh. While mass is conserved in the corrected runs, we observe some spurious generation of kinetic energy.

Figures (12)

• Figure 1: Hybrid approach combining numerical simulation using the ICON-O model with machine-learning-based super-resolution. The data flow during runtime is indicated by the continuous arrows (top panel), whereas the dashed blue lines indicate data movement during training. $\Delta t$ is the time step used in the numerical simulation (ICON-O). At $t=0$ and $\tau \gg \Delta t$ we use a distance weighted interpolation method to map the high resolution simulation ground truth onto the low resolution grid for the input and the ground truth of ML-model $\Theta$ training.
• Figure 2: ICON-O's horizontal discretization design. The scalar height field is located in the blue diamond shape, while the vector velocity field is decomposed in the midpoint edges of each triangle.
• Figure 3: Our super-resolution network consists of a U-Net core model and two modules handling the transition between the ICON-O grid and the regular grid. Nearest neighbor interpolation is used to map the local patch $\mathbf{u}_{n,m}$ to a regular grid. The corrected patch is obtained by sampling data points from the output of the U-Net according to their coordinates.
• Figure 4: U-Net core of the ML model. In the encoder and decoder branch, we use a kernel size of $7\times7$ and $3\times3$, respectively. The dotted arrows indicate concatenation of tensors.
• Figure 5: ML model learning progress as a function of mini-batch iteration. Validation was done at a frequency of 200 iterations, averaged over 10iterations. To compare both losses, we applied a moving average window to the training loss. The learning rate as a function of the training iteration is shown in the bottom graph.