# Physics-Based Machine Learning Closures and Wall Models for Hypersonic Transition-Continuum Boundary Layer Predictions

arXiv: https://arxiv.org/abs/2507.08986 · HTML: https://mlsanity.com/paper/2507.08986/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 62 references that are in the map.
- First posted on arXiv 11 Jul 2025.

## Builds on (most cited first, 9 of 9)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [DPM: A deep learning PDE augmentation method (with application to large-eddy simulation)](https://mlsanity.com/paper/1911.09145/) (2019)
- [Embedded training of neural-network sub-grid-scale turbulence models](https://mlsanity.com/paper/2105.01030/) (2021)
- [PDE-constrained Models with Neural Network Terms: Optimization and Global Convergence](https://mlsanity.com/paper/2105.08633/) (2021)
- [Deep Learning Closure Models for Large-Eddy Simulation of Flows around Bluff Bodies](https://mlsanity.com/paper/2208.03498/) (2022)
- [Adaptive physics-informed neural operator for coarse-grained non-equilibrium flows](https://mlsanity.com/paper/2210.15799/) (2022)

## Abstract (arXiv)

Modeling rarefied hypersonic flows remains a fundamental challenge due to the breakdown of classical continuum assumptions in the transition-continuum regime, where the Knudsen number ranges from approximately 0.1 to 10. Conventional Navier-Stokes-Fourier (NSF) models with empirical slip-wall boundary conditions fail to accurately predict nonequilibrium effects such as velocity slip, temperature jump, and shock structure deviations. We develop a physics-constrained machine learning framework that augments transport models and boundary conditions to extend the applicability of continuum solvers in nonequilibrium hypersonic regimes. We employ deep learning PDE models (DPMs) for the viscous stress and heat flux embedded in the governing PDEs and trained via adjoint-based optimization. We evaluate these for two-dimensional supersonic flat-plate flows across a range of Mach and Knudsen numbers. Additionally, we introduce a wall model based on a mixture of skewed Gaussian approximations of the particle velocity distribution function. This wall model replaces empirical slip conditions with physically informed, data-driven boundary conditions for the streamwise velocity and wall temperature. Our results show that a trace-free anisotropic viscosity model, paired with the skewed-Gaussian distribution function wall model, achieves significantly improved accuracy, particularly at high-Mach and high-Knudsen number regimes. Strategies such as parallel training across multiple Knudsen numbers and inclusion of high-Mach data during training are shown to enhance model generalization. Increasing model complexity yields diminishing returns for out-of-sample cases, underscoring the need to balance degrees of freedom and overfitting. This work establishes data-driven, physics-consistent strategies for improving hypersonic flow modeling for regimes in which conventional continuum approaches are invalid.