Integrating neural operators with diffusion models improves spectral representation in turbulence modelling

We integrate neural operators with diffusion models to address the spectral limitations of neural operators in surrogate modelling 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 (AR) rollouts, as we demonstrate in this work. In addition, proper orthogonal decomposition (POD) analysis demonstrates enhanced spectral fidelity in space–time. This work establishes a new paradigm for combining generative models with neural operators to advance surrogate modelling 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 .