Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modeling

TL;DR

This work addresses the spectral bias of neural operators in turbulence surrogate modeling by conditioning a score-based diffusion model on neural-operator outputs. The diffusion prior recovers high-frequency content and reconciles the energy spectrum $E(k)$ with ground truth across multiple turbulence cases, including Kolmogorov flow, buoyancy-driven transport, turbulent airfoil wakes, a 3D jet, and Schlieren measurements. Across several neural-operator architectures (e.g., FNO, UNet, TC-UNet), the NO+Diffusion model yields improved spectral fidelity, stabilizes long-horizon autoregressive rollouts (diffusion-corrected AR), and enhances POD-mode alignment, demonstrating a general, extensible framework for turbulence surrogates. This integration offers a new paradigm for leveraging generative models with neural operators to accelerate DNS/LES and can extend to other sciences with microstructure and high-frequency content, with potential for further improvements as neural operators and diffusion techniques evolve.

Abstract

We integrate neural operators with diffusion models to address the spectral limitations of neural operators in surrogate modeling of turbulent flows. While neural operators offer computational efficiency, they exhibit deficiencies in capturing high-frequency flow dynamics, resulting in overly smooth approximations. To overcome this, we condition diffusion models on neural operators to enhance the resolution of turbulent structures. Our approach is validated for different neural operators on diverse datasets, including a high Reynolds number jet flow simulation and experimental Schlieren velocimetry. The proposed method significantly improves the alignment of predicted energy spectra with true distributions compared to neural operators alone. This enables the diffusion models to stabilize longer forecasts through diffusion-corrected autoregressive rollouts, as we demonstrate in this work. Additionally, proper orthogonal decomposition analysis demonstrates enhanced spectral fidelity in space-time. This work establishes a new paradigm for combining generative models with neural operators to advance surrogate modeling of turbulent systems, and it can be used in other scientific applications that involve microstructure and high-frequency content. See our project page: vivekoommen.github.io/NO_DM

Figures (14)

• Figure 1: Methodology. First, we train a neural operator to learn the mapping between the function spaces by minimizing, typically, the $L^2$ norm with respect to available ground truth. Next, we train a diffusion model conditioned on a neural operator's output to approximate the ground truth data distribution using denoising score matching.
• Figure 2: Kolmogorov Flow. The first row shows the vorticity field in a kolmogorov flow simulated using a pseudo-spectral solver. The following rows show the predictions of different types of neural operators - FNO, UNet & TC-UNet and that of a score-based diffusion model conditioned on the respective neural operator as a prior. The corresponding energy spectra at each timestep are also displayed.
• Figure 3: Sampling Process. a) An illustration of the sampling process in a score-based diffusion model conditioned on the Fourier neural operator's prediction (NO + Diffusion), and b) the comparison of the corresponding energy spectra with the ground truth.
• Figure 4: Buoyancy-driven Transport. The first row illustrates the true transport of the concentration field $d$ simulated using $\Phi$Flow. The second row is the prediction of the (UNet) neural operator. The third row visualizes the prediction of the diffusion model conditioned on the neural operator's output. The last row compares the square roots of the respective energy spectra to highlight the differences.
• Figure 5: Turbulent wake of NACA0012. Distribution of the streamwise velocity $u$ at the mid-span slice at $Re=23,000$. The first row is the ground truth generated from LES, second row is the prediction of neural operator (TC-UNet) and the third row is the prediction of a diffusion model conditioned on the neural operator's output. The last row plots the energy spectrum of the ground truth and predictions.