Design of computer experiments for modeling hypersonic transition using deep operator networks

Using data from direct numerical simulations, we develop a machine-learned model to predict laminar-to-turbulence transition in high-speed boundary layers. Unlike conventional neural networks that learn functions, our model is a deep operator network (DeepONet) that learns operators, specifically the map from the upstream disturbances to the skin-friction profile. A primary challenge in the model development is the generation of training data, since transition is sensitive to various parameters, for example the upstream disturbance frequency and energy, Mach number, and Reynolds number. Our strategy for sampling the training space aims to minimize the uncertainty of the model predictions for a given number of training samples, or equivalently computational cost for data generation. Specifically, we adopt a Bayesian DeepONet that predicts both the mean skin-friction curve and its uncertainty. We then progressively identify points of the input space where the Bayesian DeepONet output shows large uncertainty in the skin friction, perform DNS at these points, and retrain the Bayesian DeepONet to improve the model accuracy. The method is demonstrated and assessed for transition prediction in high-speed, zero-pressure-gradient boundary layer over a flat plate.