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Diffusion-based Models for Unpaired Super-resolution in Fluid Dynamics

Wuzhe Xu, Yulong Lu, Lian Shen, Anqing Xuan, Ali Barzegari

TL;DR

This work tackles the challenge of unpaired super-resolution in fluid dynamics by introducing a two-step diffusion-based pipeline. First, Enhanced Diffusion Domain Interpolation Bridge (EDDIB) translates low-fidelity, low-resolution data $u^l$ to a high-fidelity, low-resolution form $\tilde{u}^h$, preserving large-scale structures while recovering fine-scale details; second, cascaded SR3 upscales $\tilde{u}^h$ to high-fidelity, high-resolution outputs $u^h$. The framework is further extended to trajectory evolution by integrating neural operators (e.g., FNO) to learn system dynamics, enabling stable, long-term predictions. The authors provide theoretical guarantees for the translation step (KL and $\mathcal{W}_2$ bounds) and demonstrate effectiveness across 2D Navier–Stokes, Euler with shocks, and nonlinear water waves, with additional results for trajectory data. Overall, the approach offers a data-efficient, unpaired downscaling solution that preserves global structures while faithfully reconstructing multiscale features, with practical implications for accelerated, accurate simulations of chaotic fluid systems.

Abstract

High-fidelity, high-resolution numerical simulations are crucial for studying complex multiscale phenomena in fluid dynamics, such as turbulent flows and ocean waves. However, direct numerical simulations with high-resolution solvers are computationally prohibitive. As an alternative, super-resolution techniques enable the enhancement of low-fidelity, low-resolution simulations. However, traditional super-resolution approaches rely on paired low-fidelity, low-resolution and high-fidelity, high-resolution datasets for training, which are often impossible to acquire in complex flow systems. To address this challenge, we propose a novel two-step approach that eliminates the need for paired datasets. First, we perform unpaired domain translation at the low-resolution level using an Enhanced Denoising Diffusion Implicit Bridge. This process transforms low-fidelity, low-resolution inputs into high-fidelity, low-resolution outputs, and we provide a theoretical analysis to highlight the advantages of this enhanced diffusion-based approach. Second, we employ the cascaded Super-Resolution via Repeated Refinement model to upscale the high-fidelity, low-resolution prediction to the high-resolution result. We demonstrate the effectiveness of our approach across three fluid dynamics problems. Moreover, by incorporating a neural operator to learn system dynamics, our method can be extended to improve evolutionary simulations of low-fidelity, low-resolution data.

Diffusion-based Models for Unpaired Super-resolution in Fluid Dynamics

TL;DR

This work tackles the challenge of unpaired super-resolution in fluid dynamics by introducing a two-step diffusion-based pipeline. First, Enhanced Diffusion Domain Interpolation Bridge (EDDIB) translates low-fidelity, low-resolution data to a high-fidelity, low-resolution form , preserving large-scale structures while recovering fine-scale details; second, cascaded SR3 upscales to high-fidelity, high-resolution outputs . The framework is further extended to trajectory evolution by integrating neural operators (e.g., FNO) to learn system dynamics, enabling stable, long-term predictions. The authors provide theoretical guarantees for the translation step (KL and bounds) and demonstrate effectiveness across 2D Navier–Stokes, Euler with shocks, and nonlinear water waves, with additional results for trajectory data. Overall, the approach offers a data-efficient, unpaired downscaling solution that preserves global structures while faithfully reconstructing multiscale features, with practical implications for accelerated, accurate simulations of chaotic fluid systems.

Abstract

High-fidelity, high-resolution numerical simulations are crucial for studying complex multiscale phenomena in fluid dynamics, such as turbulent flows and ocean waves. However, direct numerical simulations with high-resolution solvers are computationally prohibitive. As an alternative, super-resolution techniques enable the enhancement of low-fidelity, low-resolution simulations. However, traditional super-resolution approaches rely on paired low-fidelity, low-resolution and high-fidelity, high-resolution datasets for training, which are often impossible to acquire in complex flow systems. To address this challenge, we propose a novel two-step approach that eliminates the need for paired datasets. First, we perform unpaired domain translation at the low-resolution level using an Enhanced Denoising Diffusion Implicit Bridge. This process transforms low-fidelity, low-resolution inputs into high-fidelity, low-resolution outputs, and we provide a theoretical analysis to highlight the advantages of this enhanced diffusion-based approach. Second, we employ the cascaded Super-Resolution via Repeated Refinement model to upscale the high-fidelity, low-resolution prediction to the high-resolution result. We demonstrate the effectiveness of our approach across three fluid dynamics problems. Moreover, by incorporating a neural operator to learn system dynamics, our method can be extended to improve evolutionary simulations of low-fidelity, low-resolution data.

Paper Structure

This paper contains 23 sections, 6 theorems, 47 equations, 8 figures, 5 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Problem description and methodology overview
  3. Related works
  4. Methodology
  5. Background on diffusion models
  6. Dual diffusion implicit bridges
  7. Enhanced DDIB
  8. Super-Resolution via SR3
  9. Neural Operator for dynamics
  10. Diffusion-based Unpaired SR
  11. Time-snapshot data
  12. Trajectory data
  13. Numerical Results
  14. Experimental settings
  15. 2D Navier-Stokes equation
  16. ...and 8 more sections

Key Result

Proposition 2.1

\newlabelprop:10 For any $t_1, t_2 \in (0, 1]$, the KL divergence between the translated LFLR distribution $p(\hat{u}^l(t_1, t_2))$ and the target HFLR distribution $p(\tilde{u}^h)$ equals the KL divergence between two perturbed distribution $p(\underline{u}^l(t_1))$ and $p(\underline{\tilde{u}}^h

Figures (8)

  • Figure 1: Relationship among HFHR (left), HFLR (middle), and LFLR (right) data in time-snapshot super-resolution.
  • Figure 1: Comparison of the translated LFLR obtained from various methods. The first and the second rows presents two instances.
  • Figure 2: Comparison of snapshot-wise refinement (left), HFLR dynamics learning (middle), and HFHR dynamics learning (right) for enhancing the trajectory data.
  • Figure 2: 2D Navier–Stokes equation: SR results from five baseline methods compared to the reference. The top two rows show the predictions for two different LFLR data. The bottom-left panel displays the boxplots comparing four distance metrics between each prediction and the reference. The bottom-right plot displays the log ratio of the predicted energy spectrum relative to the reference, illustrating that EDDIB most effectively preserves both large-scale structures and fine-scale details.
  • Figure 3: 2D Euler equation: SR results from five baseline methods compared to the reference. The top two rows show the predictions for two different LFLR data. The bottom-left panel displays the boxplots comparing four distance metrics between each prediction and the reference. The bottom-right plot displays the log ratio of the predicted energy spectrum relative to the reference.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Proposition 2.1
  • Proposition 2.2
  • Lemma E.1
  • Proof 1
  • Lemma E.2
  • Proof 2
  • Proposition E.3
  • Proof 3
  • Proposition E.4
  • Proof 4