Marco Panesi

Ivan Zanardi, Simone Venturi, Marco Panesi

This work proposes a new machine learning (ML)-based paradigm aiming to enhance the computational efficiency of non-equilibrium reacting flow simulations while ensuring compliance with the underlying physics. The framework combines dimensionality reduction and neural operators through a hierarchical and adaptive deep learning strategy to learn the solution of multi-scale coarse-grained governing equations for chemical kinetics. The proposed surrogate's architecture is structured as a tree, with leaf nodes representing separate neural operator blocks where physics is embedded in the form of multiple soft and hard constraints. The hierarchical attribute has two advantages: i) It allows the simplification of the training phase via transfer learning, starting from the slowest temporal scales; ii) It accelerates the prediction step by enabling adaptivity as the surrogate's evaluation is limited to the necessary leaf nodes based on the local degree of non-equilibrium of the gas. The model is applied to the study of chemical kinetics relevant for application to hypersonic flight, and it is tested here on pure oxygen gas mixtures. In 0-D scenarios, the proposed ML framework can adaptively predict the dynamics of almost thirty species with a maximum relative error of 4.5% for a wide range of initial conditions. Furthermore, when employed in 1-D shock simulations, the approach shows accuracy ranging from 1% to 4.5% and a speedup of one order of magnitude compared to conventional implicit schemes employed in an operator-splitting integration framework. Given the results presented in the paper, this work lays the foundation for constructing an efficient ML-based surrogate coupled with reactive Navier-Stokes solvers for accurately characterizing non-equilibrium phenomena in multi-dimensional computational fluid dynamics simulations.

Audrey Gaymann, Juan M. Cardenas, Sung Min Jo et al.

This paper presents the development of an algorithm, termed the Global-Local Hybrid Surrogate (GLHS), designed to efficiently compute the probability of rare failure events in complex systems. The primary goal is to enhance the accuracy of reliability analysis while minimizing computational cost, particularly for high-dimensional problems where traditional methods, such as Monte Carlo simulations, become prohibitively expensive. The proposed GLHS builds upon the foundational work of Li et al., by integrating an adaptive strategy based on the General Domain Adaptive Strategy (Adcock et al.). The algorithm aims to approximate the failure domain of a given system, defined as the region in the input domain where the system transitions from safe to failure modes, described by a limit state surface. This failure domain is not explicitly known and must be learned iteratively during the analysis. The method employs a buffer zone, defined as the region surrounding the limit state surface. Within this buffer zone, Christoffel Adaptive Sampling is utilized to select new samples for constructing localized surrogate models, which are designed to refine the approximation in regions critical to failure probability estimation. The iterative process proceeds until convergence is reached. This results in a hybrid methodology that integrates a global surrogate to capture the overall trend with local surrogates that concentrate on critical regions near the limit state function. By adopting this strategy, the GLHS method balances computational efficiency with accuracy in estimating the failure probability.

Ivan Zanardi, Simone Venturi, Marco Panesi

We introduce MENO (''Matrix Exponential-based Neural Operator''), a hybrid surrogate modeling framework for efficiently solving stiff systems of ordinary differential equations (ODEs) that exhibit a sparse nonlinear structure. In such systems, only a few variables contribute nonlinearly to the dynamics, while the majority influence the equations linearly. MENO exploits this property by decomposing the system into two components: the low-dimensional nonlinear part is modeled using conventional neural operators, while the linear time-varying subsystem is integrated using a novel neural matrix exponential formulation. This approach combines the exact solution of linear time-invariant systems with learnable, time-dependent graph-based corrections applied to the linear operators. Unlike black-box or soft-constrained physics-informed (PI) models, MENO embeds the governing equations directly into its architecture, ensuring physical consistency (e.g., steady states), improved robustness, and more efficient training. We validate MENO on three complex thermochemical systems: the POLLU atmospheric chemistry model, an oxygen mixture in thermochemical nonequilibrium, and a collisional-radiative argon plasma in one- and two-dimensional shock-tube simulations. MENO achieves relative errors below 2% in trained zero-dimensional settings and maintains good accuracy in extrapolatory multidimensional regimes. It also delivers substantial computational speedups, achieving up to 4 800$\times$ on GPU and 185$\times$ on CPU compared to standard implicit ODE solvers. Although intrusive by design, MENO's physics-based architecture enables superior generalization and reliability, offering a scalable path for real-time simulation of stiff reactive systems.

Ashish S. Nair, Narendra Singh, Marco Panesi et al.

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.