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. 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.
We first train a neural operator to learn 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.
The first row represents the vorticity field in a Kolmogorov flow simulated using a pseudo-spectral solver. The second row shows the states inferred by the neural operator (FNO). The third row illustrates the prediction of the score-based diffusion model conditioned on the neural operator. The corresponding energy spectra are compared in the last row.
The sampling process of the score-based diffusion model conditioned on the Fourier neural operator's prediction.
Isosurfaces corresponding to Q-criterion = 1.5 of a turbulent jet is visualized. The color of the isosurfaces corresponds to the magnitude of the velocity. The first three rows represent solutions of the reference LES, neural operator (MATCHO), and diffusion model conditioned on the neural operator. The last row compares the energy spectra at different axial cross-sections corresponding to \(\Delta t = 4\tau, 8\tau\). (where \(\tau\) is the accoustic time units).
a) Schematic of knife-based single mirror Schlieren apparatus (Settles et al.) used to visualize turbulent jet of helium in the air. b) The density gradients captured using the schlieren velocimetry apparatus (row 1), estimated by neural operator, MATCHO (row 2) and estimated using diffusion model conditioned on neural operator in (row 3). In the last row, we compare the energy spectra of the respective snapshots captured/estimated by the three approaches.