Improved Sampling of Diffusion Models in Fluid Dynamics with Tweedie's Formula

Abstract

State-of-the-art Denoising Diffusion Probabilistic Models (DDPMs) rely on an expensive sampling process with a large Number of Function Evaluations (NFEs) to provide high-fidelity predictions. This computational bottleneck renders diffusion models less appealing as surrogates for the spatio-temporal prediction of physics-based problems with long rollout horizons. We propose Truncated Sampling Models, enabling single-step and few-step sampling with elevated fidelity by simple truncation of the diffusion process, reducing the gap between DDPMs and deterministic single-step approaches. We also introduce a novel approach, Iterative Refinement, to sample pre-trained DDPMs by reformulating the generative process as a refinement process with few sampling steps. Both proposed methods enable significant improvements in accuracy compared to DDPMs, DDIMs, and EDMs with NFEs $\leq$ 10 on a diverse set of experiments, including incompressible and compressible turbulent flow and airfoil flow uncertainty simulations. Our proposed methods provide stable predictions for long rollout horizons in time-dependent problems and are able to learn all modes of the data distribution in steady-state problems with high uncertainty.