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Generative Modelling of Stochastic Rotating Shallow Water Noise

Dan Crisan, Oana Lang, Alexander Lobbe

TL;DR

This work tackles uncertainty quantification in coarse-resolution fluid models by reframing stochastic subgrid-scale noise from Gaussian PCA increments to a data-driven, non-Gaussian generative approach. It employs a Diffusion Schrödinger Bridge (DSB) score-based diffusion model to learn the distribution of unresolved-scale increments $N_n$ from fine-scale RS shallow water data, enabling realistic ensemble generation. The method is validated on a non-dimensional rotating shallow water system, showing that the learned noise is non-Gaussian and produces competitive RMSE and CRPS scores, particularly under low initial uncertainty. The results suggest that DSB-based noise calibration can improve uncertainty representation and forecast skill, with future work aimed at broader model classes and comparisons to alternative decompositions like Karhunen–Loève.

Abstract

In recent work, the authors have developed a generic methodology for calibrating the noise in fluid dynamics stochastic partial differential equations where the stochasticity was introduced to parametrize subgrid-scale processes. The stochastic parameterization of sub-grid scale processes is required in the estimation of uncertainty in weather and climate predictions, to represent systematic model errors arising from subgrid-scale fluctuations. The previous methodology used a principal component analysis (PCA) technique based on the ansatz that the increments of the stochastic parametrization are normally distributed. In this paper, the PCA technique is replaced by a generative model technique. This enables us to avoid imposing additional constraints on the increments. The methodology is tested on a stochastic rotating shallow water model with the elevation variable of the model used as input data. The numerical simulations show that the noise is indeed non-Gaussian. The generative modelling technology gives good RMSE, CRPS score and forecast rank histogram results.

Generative Modelling of Stochastic Rotating Shallow Water Noise

TL;DR

This work tackles uncertainty quantification in coarse-resolution fluid models by reframing stochastic subgrid-scale noise from Gaussian PCA increments to a data-driven, non-Gaussian generative approach. It employs a Diffusion Schrödinger Bridge (DSB) score-based diffusion model to learn the distribution of unresolved-scale increments from fine-scale RS shallow water data, enabling realistic ensemble generation. The method is validated on a non-dimensional rotating shallow water system, showing that the learned noise is non-Gaussian and produces competitive RMSE and CRPS scores, particularly under low initial uncertainty. The results suggest that DSB-based noise calibration can improve uncertainty representation and forecast skill, with future work aimed at broader model classes and comparisons to alternative decompositions like Karhunen–Loève.

Abstract

In recent work, the authors have developed a generic methodology for calibrating the noise in fluid dynamics stochastic partial differential equations where the stochasticity was introduced to parametrize subgrid-scale processes. The stochastic parameterization of sub-grid scale processes is required in the estimation of uncertainty in weather and climate predictions, to represent systematic model errors arising from subgrid-scale fluctuations. The previous methodology used a principal component analysis (PCA) technique based on the ansatz that the increments of the stochastic parametrization are normally distributed. In this paper, the PCA technique is replaced by a generative model technique. This enables us to avoid imposing additional constraints on the increments. The methodology is tested on a stochastic rotating shallow water model with the elevation variable of the model used as input data. The numerical simulations show that the noise is indeed non-Gaussian. The generative modelling technology gives good RMSE, CRPS score and forecast rank histogram results.
Paper Structure (11 sections, 2 theorems, 46 equations, 9 figures)

This paper contains 11 sections, 2 theorems, 46 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Outline of the Paper
  3. Rotating Shallow Water Model
  4. Score-Based Generative Models: Diffusion Schrödinger Bridge
  5. Learning Diffusion Schrödinger Bridge
  6. Numerical Results
  7. Fine vs Coarse Scale
  8. Training Data
  9. Generative Model Output
  10. Forecast Studies
  11. Conclusions and Future Work

Key Result

Proposition 1

Assume that $\mathrm{KL}\left(p_{\text{data }} \otimes p_{\text{prior }} \mid p_{0, N}\right)<+\infty$. Then for any $n \in \mathbb{N}, \pi^{2 n}$ and $\pi^{2 n+1}$ admit positive densities w.r.t. the Lebesgue measure denoted as $p^n$ resp. $q^n$ and for any $x_{0: N} \in \mathcal{X}$, we have $p^0\

Figures (9)

  • Figure 1: Initial Condition for the non-dimensional height variable on the fine (128x128) and coarse (32x32) grids
  • Figure 2: Snapshot of non-dimensional height variable at fine and coarse run after 4150 fine scale timesteps. We can observe the development of fine scale features which are absent (unresolved) in the coarse field.
  • Figure 3: Training Samples and Samples from the generative model. Samples of the training data after transformation. The fields are outputs of the calibration equation thought of a stream functions for the velocity perturbations in the SPDE. The data have been transformed by a $arcsinh$ transformation and normalized to the interval $[0,1]$. Samples from the generative model.
  • Figure 4: (a) Distribution of pixelvalues throughout the whole training data set. The data transformation has been choosen s.th. the pixel values achieve a good coverage of the data range $[0,1]$. This way the model can better distinguish between variations in shade.(b) Distribution of pixelvalues in central grid locations in the generated dataset, i.e. the output from the generative model.
  • Figure 5: Distribution of the generated noise values compared to Gaussians at every pixel in a central region. A Kolmogorov-Smirnov one-sample test has been performed to check if the data come from a normal distribution. The hypothesis was rejected for all locations indicating that the generated noise does not come from a simple normal distribution.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Proposition 1: Proposition 2 in de2021diffusion
  • Proposition 2: Proposition 3 in de2021diffusion