Neural Operators for Rarefied Gas Dynamics: BGK relaxation problem, Polyatomic Shock waves, and Hypersonic Cylinder Flow

Abstract

This work introduces a suite of targeted methodologies utilizing neural operators—including physicsinformed, constrained, and data-driven approaches—for creating computationally efficient and robust surrogate models of rarefied gas dynamics, a domain where high-fidelity kinetic solvers are often prohibitively expensive. We present three key contributions demonstrating stability, physics-discovery capabilities, and powerful generalization across distinct challenges. First, for the BGK kinetic relaxation problem, we introduce a novel perturbation ansatz that ensures numerical stability. We leverage this stabilized physics-informed neural network (PINN) framework to solve a challenging combined forward and inverse problem, demonstrating its capability as a physics-discovery tool by successfully inferring the unknown, velocity-dependent relaxation time using only the initial conditions and governing equations. Second, for one-dimensional shock waves in polyatomic gases, we develop a physics-constrained Deep Operator Network (DeepONet) that accurately captures complex non-equilibrium structures for unseen viscosity ratios by embedding monotonicity constraints directly into the learning process, eliminating non-physical oscillations. Finally, we demonstrate significant data efficiency and powerful generalization for two-dimensional hypersonic flow over a cylinder. A data-driven DeepONet ensemble, trained on a dataset spanning a wide hypersonic regime from Mach 5 to 15, accurately predicts the flow field for multiple unseen conditions. The resulting surrogate excels at interpolation (e.g., predicting Mach 7, 9, 12, and 14), and the extrapolation case (predicting Mach 15 while trained on Machs 5-14) with quantitative validation confirming high accuracy across the parametric range. This work establishes a powerful, flexible methodology for building neural operator surrogates that ensure physical consistency, perform physics-discovery, and generalize robustly, significantly lowering the computational barrier for design and analysis in high-speed aerodynamics