Probabilistic Super-Resolution for Urban Micrometeorology via a Schrödinger Bridge

This study employs a neural network that represents the solution to a Schrödinger bridge problem to perform super-resolution of 2-m temperature in an urban area. Schrödinger bridges generally describe transformations between two data distributions based on diffusion processes. We use a specific Schrödinger-bridge model (SM) that directly transforms low-resolution data into high-resolution data, unlike denoising diffusion probabilistic models (simply, diffusion models; DMs) that generate high-resolution data from Gaussian noise. Low-resolution and high-resolution data were obtained from separate numerical simulations with a physics-based model under common initial and boundary conditions. Compared with a DM, the SM attains comparable accuracy at one-fifth the computational cost, requiring 50 neural-network evaluations per datum for the DM and only 10 for the SM. Furthermore, high-resolution samples generated by the SM exhibit larger variance, implying superior uncertainty quantification relative to the DM. Owing to the reduced computational cost of the SM, our results suggest the feasibility of real-time ensemble micrometeorological prediction using SM-based super-resolution.

💡 Research Summary

This paper introduces a probabilistic super‑resolution (SR) framework for 2‑meter air temperature in an urban environment, based on a Schrödinger Bridge (SB) model (referred to as the SM). The authors compare the SB approach with a conventional denoising diffusion probabilistic model (DM) that has become popular for SR in meteorology.

The data consist of separate high‑resolution (HR, 5 m grid, 320 × 320) and low‑resolution (LR, 20 m grid, 80 × 80) simulations from the Multi‑Scale Simulator for the Geo‑environment (MSSG). LR and HR fields are generated by independent runs with identical initial and boundary conditions, making the SR task more challenging than simple down‑sampling. Auxiliary inputs include low‑level wind and temperature fields, as well as static high‑resolution building height and land‑use maps. The dataset spans 2013‑2020, with 2,387 training, 493 validation, and 540 test samples.

Both the DM and SM use the same U‑Net architecture (four down‑sampling and four up‑sampling blocks, multi‑head self‑attention, sinusoidal diffusion‑time embeddings). The DM follows the Palette model (Saharia et al. 2022) and is trained to predict the residual x_HR − x_LR; during inference it adds the LR field back to the generated residual. The SM directly learns the drift of a single‑direction SDE that transports the Dirac mass at x_LR to the conditional HR distribution p_HR(x | x_LR, ξ). Consequently, the SM generates HR samples by integrating this SDE from the LR state, whereas the DM must start from a standard Gaussian and integrate a reverse‑time SDE for many steps.

Training uses AdamW (lr = 1e‑4, batch = 32, 1,000 epochs). The SM loss is the mean‑squared error between the true drift and the U‑Net approximation; the DM loss is the denoising score‑matching MSE. Noise schedules differ: SM’s γ_t decays from 0.2 to 0, while DM’s λ_t decays linearly from 10 to 0.001, yielding a maximum noise amplitude of ≈3.16 for the DM.

Evaluation metrics are RMSE and SSIM loss for pointwise accuracy, and ensemble diagnostics (spread‑skill ratio and rank histogram) for uncertainty quantification, using 64‑member ensembles per test case. The SM achieves comparable RMSE (~0.42 K) and SSIM loss (~0.018) with only N_T = 10 diffusion‑time steps, whereas the DM requires N_T ≈ 50 to reach similar performance; below this threshold the DM error rises sharply. The SM’s ensemble spread is larger, and its spread‑skill ratio is close to 1, indicating well‑calibrated uncertainty, while the DM shows under‑dispersion. Rank histograms confirm the SM’s superior dispersion.

Computationally, because both models share the same U‑Net, the SM’s 10 forward passes per sample translate to roughly one‑fifth the GPU time of the DM’s 50 passes. This efficiency makes real‑time or large‑scale ensemble SR feasible for urban micrometeorology.

In summary, the study demonstrates that a Schrödinger Bridge‑based SR model can transform low‑resolution urban temperature fields into high‑resolution counterparts with accuracy on par with diffusion models, while offering markedly lower computational cost and more reliable probabilistic forecasts. The results open the door to operational, real‑time ensemble micrometeorological prediction in complex urban settings.