Exploring denoising diffusion models for compressible fluid field prediction

Building upon our prior success with Pix2Pix generative adversarial networks (GANs), this work explores the potential of denoising diffusion probabilistic models (DDPMs) for supersonic flow prediction. DDPMs, renowned for their stable training and superior mode coverage, are constructed to predict key flow field quantities for compressible flows over generic aerodynamic geometries. We employ fully-conditioned DDPMs to generate high-resolution predictions of density, temperature, and Mach number fields for supersonic flows over ramps. For flows around supersonic airfoils, DDPMs are used to generate high-resolution synthetic Schlieren images, enabling detailed analysis of complex shock wave phenomena in analogy to classical experimental approaches. For ramp flows, where the training dataset is relatively small, we address residual noise in the DDPM outputs by training a U-Net to remove the noise. This approach significantly improves the accuracy of the predicted flow fields. Comparative analysis against Pix2Pix GANs reveals that DDPMs achieve superior performance, particularly in capturing shock-waves and secondary-shock details around airfoils. Furthermore, we explore the generative capabilities of DDPMs by introducing degrees of freedom into the flow problems. This is achieved, for instance, by removing ramp geometry constraints, allowing the model to generate new flow field configurations not present in the training data. To address the challenge of evaluating semi-conditioned models in scenarios lacking ground truth data, we introduce a novel proxy evaluator method. This method leverages the superior quality of fully-conditioned DDPMs to assess the outputs of semi-conditioned models. We validate this approach by comparing generated outputs to a limited set of actual ground truth samples obtained from high-fidelity numerical simulations. This work highlights the significant potential of DDPMs not only as surrogate models for predicting flow field data but also for rapidly generating synthetic data and augmenting datasets, paving the way for advancements in supersonic flow analysis and design.