Score-Based Diffusion Models with Autocorrelation Regularization for Fluid Flow Prediction

The simulation of turbulent flows is an essential field of study with widespread applications. While traditional solvers struggle to provide rapid estimations of solutions due to the intrinsic chaotic nature of turbulence, recent breakthroughs in generative modeling, particularly diffusion models, have demonstrated promising performance for such complex problems. Our method, based on score-matching diffusion models, aims to sample a probability density function representing snapshots of a fluid simulation by learning a score function through reverse-time diffusion processes. This score function is conditioned on previous fluid flow states, following an autoregressive framework. By leveraging denoising score-matching (DSM) techniques, we further regularize the traditional loss function to improve prediction quality (total error on the autocorrelation matrices of the velocity field fluctuations). The conditioning is implemented using a denoising U-Net architecture with input concatenation (known for effectively learning conditional distributions). The model is trained on two-dimensional turbulent compressible flow data (Karman vortex street), and inference is conducted on an extrapolation task for velocities within the transonic regime (frequent shock waves apparition). Early results indicate that incorporating an energy loss function is essential for preserving temporal stability and coherence in intermediate predictions. However, achieving accurate predictions over extended time horizons remains difficult and resource-intensive.