We propose neural network operator inference (NN-OpInf): a structure-preserving, composable, and minimally restrictive operator inference framework for the non-intrusive reduced-order modeling of dynamical systems. The approach learns latent dynamics from snapshot data, enforcing local operator structure such as skew-symmetry, (semi-)positive definiteness, and gradient preservation, while also reflecting complex dynamics by supporting additive compositions of heterogeneous operators. We present practical training strategies and analyze computational costs relative to linear and quadratic polynomial OpInf (P-OpInf). Numerical experiments across several nonlinear and parametric problems demonstrate improved accuracy, stability, and robustness over P-OpInf and prior NN-ROM formulations, particularly when the dynamics are not well represented by polynomial models. These results suggest that NN-OpInf can serve as an effective drop-in replacement for P-OpInf when the dynamics to be modeled contain non-polynomial nonlinearities, offering potential gains in accuracy and out-of-distribution performance at the expense of higher training computational costs and a more difficult, non-convex learning problem.

Hypersonic flow conditions pose exceptional challenges for Reynolds-Averaged Navier-Stokes (RANS) turbulence modeling. Critical phenomena include compressibility effects, shock/turbulent boundary layer interactions, turbulence-chemistry interaction in thermo-chemical non-equilibrium, and ablation-induced surface roughness and blowing effects. This comprehensive review synthesizes recent developments in adapting turbulence models to hypersonic applications, examining approaches ranging from empirical modifications to physics-based reformulations and novel data-driven methodologies. We provide a systematic evaluation of current RANS-based turbulence modeling capabilities, comparing eddy viscosity and Reynolds stress transport formulations in their ability to predict engineering quantities of interest such as separation characteristics and wall heat transfer. Our analysis encompasses the latest experimental and direct numerical simulation datasets for validation, specifically addressing two- and three-dimensional equilibrium turbulent boundary layers and shock/turbulent boundary layer interactions across both smooth and rough surfaces. Key multi-physics considerations including catalysis and ablation phenomena along with the integration of conjugate heat transfer into a RANS solver for efficient design of a thermal protection system are also discussed. We conclude by identifying the critical gaps in the available validation databases and limitations of the existing turbulence models and suggest potential areas for future research to improve the fidelity of turbulence modeling in the hypersonic regime.

This paper presents a novel hybrid approach for coupling subdomain-local non-intrusive Operator Inference (OpInf) reduced order models (ROMs) with each other and with subdomain-local high-fidelity full order models (FOMs) with using the overlapping Schwarz alternating method (O-SAM). The proposed methodology addresses significant challenges in multiscale modeling and simulation, particularly the long runtime and complex mesh generation requirements associated with traditional high-fidelity simulations. By leveraging the flexibility of O-SAM, we enable the seamless integration of disparate models, meshes, and time integration schemes, enhancing computational efficiency while maintaining high accuracy. Our approach is demonstrated through a series of numerical experiments on complex three-dimensional (3D) solid dynamics problems, showcasing speedups of up to 106x compared to conventional FOM-FOM couplings. This work paves the way for more efficient simulation workflows in engineering applications, with potential extensions to a wide range of partial differential equations.

This work investigates transmission conditions for the domain decomposition-based coupling of subdomain-local models using the non-overlapping Schwarz alternating method (NO-SAM). Building on prior efforts involving overlapping SAM (O-SAM), we formulate and assess two NO-SAM variants, based on alternating Dirichlet-Neumann and Robin-Robin transmission conditions. For the subdomain-local models, we consider a mix of full order models (FOMs) and non-intrusive reduced order models (ROMs) constructed via an emerging model reduction technique known as operator inference (OpInf). Of particular novelty is the first application of NO-SAM to couple non-intrusive OpInf ROMs with each other, and with FOMs. Numerical studies on a one-dimensional linear elastic wave propagation benchmark problem demonstrate that transmission condition choice and parameter tuning significantly impact convergence rate, accuracy, and stability. Robin-Robin coupling often yields faster convergence than alternating Dirichlet-Neumann, though improper parameter selection can induce spurious oscillations at subdomain interfaces. For FOM-OpInf and OpInf-OpInf couplings, sufficient modal content in the ROM basis improves accuracy and mitigates instability, in some cases outperforming the coupled FOM-FOM reference solutions in both accuracy and efficiency. These findings highlight NO-SAM's potential for enabling flexible, non-intrusive, and efficient multi-model coupling across independently meshed subdomains, while emphasizing the need for careful interface condition design in higher-dimensional and predictive settings.

We study the role of the wall-normal Reynolds stress in hypersonic boundary layers and consider implications for RANS modeling. In low-Mach-number, zero-pressure-gradient boundary layers, the contribution of the wall-normal Reynolds stress to the wall-normal momentum equation is negligible. As a result, the wall-normal momentum equation indicates that pressure is uniform throughout the boundary layer. However, in hypersonic boundary layers, analysis indicates that the wall-normal Reynolds stress becomes significant compared to mean pressure, and the wall-normal momentum equation indicates that pressure must balance the wall-normal Reynolds stress. We validate this hypothesis using direct numerical simulation data of supersonic and hypersonic boundary layers. The results show that the wall-normal Reynolds stress balances with pressure, resulting in non-negligible pressure deficits in the log layer. We consider implications for RANS modeling and show that errors in standard models for the wall-normal Reynolds stress have a noticeable impact (upward of 5% for a Mach 14 cold-wall flow) on wall quantities of interest, such as skin friction.

Galerkin and Petrov-Galerkin projection-based reduced-order models (ROMs) of transient partial differential equations are typically obtained by performing a dimension reduction and projection process that is defined at either the spatially continuous or spatially discrete level. In both cases, it is common to add stabilization to the resulting ROM to increase the stability and accuracy of the method; the addition of stabilization is particularly common for advection-dominated systems when the ROM is under-resolved. While these two approaches can be equivalent in certain settings, differing techniques have emerged in both contexts. This work outlines these two approaches within the setting of finite element method (FEM) discretizations (in which case a duality exists between the continuous and discrete levels) of the convection-diffusion-reaction equation, and compares residual-based stabilization techniques that have been developed in both contexts. In the spatially continuous case, we examine the Galerkin, streamline upwind Petrov-Galerkin (SUPG), Galerkin/least-squares (GLS), and adjoint (ADJ) stabilization methods. For the GLS and ADJ methods, we examine formulations constructed from both the "discretize-then-stabilize" technique and the space-time technique. In the spatially discrete case, we examine the Galerkin, least-squares Petrov-Galerkin (LSPG), and adjoint Petrov-Galerkin (APG) methods. We summarize existing analyses for these methods, and provide numerical experiments, which demonstrate that residual-based stabilized methods developed via continuous and discrete processes yield substantial improvements over standard Galerkin methods when the underlying FEM model is under-resolved.

This work investigates the use of the reduced-basis method for multifidelity uncertainty quantification (UQ) with application to the Reynolds-Averaged Navier-Stokes equations. We examine the use of a greedy algorithm that leverages easily computable error estimates for constructing a reduced basis. This reduced basis is used to define a Galerkin reduced-order model that leverages entropy-variable transforms to maintain robustness. We embed our reduced-order model within sampling multifidelity UQ methods and assess its utility for more-efficiently propagating parametric uncertainties with the Spalart-Allmaras turbulence model. Results are presented on several hypersonic turbulent flow configurations.

The Spalart-Allmaras turbulence model is re-examined in light of recent direct numerical simulations (DNS) of hypersonic turbulent boundary layers. Inspirations from the DNS data, in combination with other published modifications, are used to propose a model form for compressible boundary layers. The new form is comprised of: 1) adoption of the Catris and Aupoix (2000) form of the diffusion terms that satisfy the van Driest log layer solution; 2) adoption of the Spalart and Garburuk (2020) low-Reynolds number modification; 3) a new near-wall damping function based on a newly identified eddy viscosity transformation for the "far"' buffer layer; and 4) a modification of the eddy viscosity relation to provide a more accurate wall-normal turbulent stress for high-Mach number boundary layers. The new form of the model is tested on various zero-pressure-gradient flat plate and cone flows, as well as on the more complex HIFiRE-1 vehicle case. It provides improved prediction of wall shear stress and wall heat flux for sufficiently high Reynolds numbers. A more general modification is needed for low-Reynolds number compressible boundary layers. To further improve the model, a machine learning approach is take to infer a correction to the near-wall destruction term. The machine-learned correction improves predictions at low Reynolds numbers, but does not retain the accuracy of the modified model at higher Reynolds numbers.

We develop a neural-network-based variable turbulent Prandtl number model for the k-& varepsilon; turbulence model for improved wall heating predictions in hypersonic shock-boundary-layer interactions (SBLIs). The model is developed by performing a finite-dimensional field inference for a spatially varying turbulent Prandtl number on six canonical SBLIs: three compression ramps at Mach 8 and three impinging shocks at Mach 5. The inference results identify a turbulent Prandtl number that reduces wall heating by systematically directing heat transfer away from the wall. An ensemble of Lipschitz-continuous neural networks is then trained on the inferred turbulent Prandtl number fields to develop a predictive model. We evaluate the resulting variable turbulent Prandtl number model on a suite of test cases, including the hollow cylinder flare and HIFiRE ground test experiments. The machine-learning-augmented model systematically increases Prt near the wall to reduce negative turbulent heat flux while decreasing Prt away from the wall to enhance positive turbulent heat flux, collectively reducing overall heat transfer to the surface. Results show that the learned model consistently improves peak heating predictions by 40-70% compared to the baseline k-& varepsilon; model, a k-& varepsilon; model augmented with various high-speed corrections, and the shear stress transport model across a range of conditions.

We present a machine-learning strategy for finite element analysis of solid mechanics wherein we replace complex portions of a computational domain with a data-driven surrogate. In the proposed strategy, we decompose a computational domain into an "outer" coarse-scale domain that we resolve using a finite element method (FEM) and an "inner" fine-scale domain. We then develop a machine-learned (ML) model for the impact of the inner domain on the outer domain. In essence, for solid mechanics, our machine-learned surrogate performs static condensation of the inner domain degrees of freedom. This is achieved by learning the map from (virtual) displacements on the inner-outer domain interface boundary to forces contributed by the inner domain to the outer domain on the same interface boundary. We consider two such mappings, one that directly maps from displacements to forces without constraints, and one that maps from displacements to forces by virtue of learning a symmetric positive semi-definite (SPSD) stiffness matrix. We demonstrate, in a simplified setting, that learning an SPSD stiffness matrix results in a coarse-scale problem that is well-posed with a unique solution. We present numerical experiments on several exemplars, ranging from finite deformations of a cube to finite deformations with contact of a fastener-bushing geometry. We demonstrate that enforcing an SPSD stiffness matrix is critical for accurate FEM-ML coupled simulations, and that the resulting methods can accurately characterize out-of-sample loading configurations with significant speedups over the standard FEM simulations.