Adaptive physics-informed neural operator for coarse-grained non-equilibrium flows.

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-[Formula: see text] 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-[Formula: see text] 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.

Efficient quasi-classical trajectory calculations by means of neural operator architectures.

An accurate description of non-equilibrium chemistry relies on rovibrational state-to-state (StS) kinetics data, which can be obtained through the quasi-classical trajectory (QCT) method for high-energy collisions. However, these calculations still represent one of the major computational bottlenecks in predictive simulations of non-equilibrium reacting gases. This work addresses this limitation by proposing SurQCT, a novel machine learning-based surrogate for efficiently and accurately predicting StS chemical reaction rate coefficients. The QCT emulator is constructed using three independent components: two deep operator networks (DeepONets) for inelastic and exchange processes and a feed-forward neural network (FNN) for the dissociation reactions. SurQCT is tested on the O2 + O system, showing a computational speed-up of 85%. Furthermore, we carry out a StS master equation analysis of an isochoric, isothermal heat bath simulation at various temperatures to study how the predicted rate coefficients impact the accuracy of multiple quantities of interest (QoIs) at the kinetics level (e.g., global quasi-steady state (QSS) dissociation rate coefficients and energy relaxation times). For all these QoIs, the master equation analysis relying on SurQCT data shows an accuracy within 15% across the entire temperature regime.

Rovibrational internal energy transfer and dissociation of high-temperature oxygen mixture.

This work constructs a rovibrational state-to-state model for the O2 + O2 system leveraging high-fidelity potential energy surfaces and quasi-classical trajectory calculations. The model is used to investigate internal energy transfer and nonequilibrium reactive processes in a dissociating environment using a master equation approach, whereby the kinetics of each internal rovibrational state is explicitly computed. To cope with the exponentially large number of elementary processes that characterize reactive bimolecular collisions, the internal states of the collision partner are assumed to follow a Boltzmann distribution at a prescribed internal temperature. This procedure makes the problem tractable, reducing the computational cost to a comparable scale with the O2 + O system. The constructed rovibrational-specific kinetic database covers the temperature range of 7500-20 000 K. The reaction rate coefficients included in the database are parameterized in the function of kinetic and internal temperatures. Analysis of the energy transfer and dissociation process in isochoric and isothermal conditions reveals that significant departure from the equilibrium Boltzmann distribution occurs during the energy transfer and dissociation phase. Comparing the population distribution of the O2 molecules against the O2 + O case demonstrates a more significant extent of nonequilibrium characterized by a more diffuse distribution whereby the vibrational strands are more clearly identifiable. This is partly due to less efficient mixing of the rovibrational states, which results in more diffuse rovibrational distributions in the quasi-steady-state distribution of O2 + O2. A master equation analysis for the combined O2 + O and O2 + O2 system reveals that the O2 + O2 system governs the early stage of energy transfer, whereas the O2 + O system takes control of the dissociation dynamics. The findings of the present work will provide a strong physical foundation that can be exploited to construct an improved reduced-order model for oxygen chemistry.

Comprehensive Study of HCN: Potential Energy Surfaces, State-to-State Kinetics, and Master Equation Analysis.

Understanding the kinetics of the HCN system is critical to several disciplines in science and engineering, including interstellar chemistry, atmospheric reentry, and combustion, to name a few. This paper constructs a rovibrational state-specific kinetic mechanism for the HCN system, leveraging electronic structure calculations, classical scattering dynamics, and state-to-state kinetics. To this aim, three accurate potential energy surfaces (PESs), 1A', 3A', and 3A″, are constructed using multireference configuration interaction (MRCI) calculations for a comprehensive arrangement of the nuclei. Quasi-classical scattering calculations provide elementary reaction rate constants resulting from the interaction between the CN, CH, and NH molecules with H, N, and C atoms, respectively. The rovibrational collisional model developed comprises 50 million bound-bound and free-bound collisional processes. This model is used to study the dynamics of energy transfer and dissociation in an isochoric and isothermal chemical reactor via the solution of the master equation for a wide temperature range from 1000 to 10,000 K. This study unravels the dynamics of dissociation of the molecules in the HCN system, which the PESs primarily control via the formation of short-lived intermediates that shortcut the dissociation pathway. The exchange processes in CH and NH enhance the dissociation by over 80%. The importance of exchange processes is also highlighted in comparing the quasi-steady state and thermal dissociation rates with state-of-the-art rate models and experimental fits.

Rovibrational-Specific QCT and Master Equation Study on N2(X1Σg+) + O(3P) and NO(X2Π) + N(4S) Systems in High-Energy Collisions.

This work presents a detailed investigation of the energy-transfer and dissociation mechanisms in N2(X1Σg+) + O(3P) and NO(X2Π) + N(4S) systems using rovibrational-specific quasiclassical trajectory (QCT) and master equation analyses. The complete set of state-to-state kinetic data, obtained via QCT, allows for an in-depth investigation of the Zel'dovich mechanism leading to the formation of NO molecules at microscopic and macroscopic scales. The master equation analysis demonstrates that the low-lying vibrational states of N2 and NO have dominant contributions to the NO formation and the corresponding extinction of N2 through the exchange process. For the considered temperature range, it is found that nearly 50% of the dissociation processes for N2 and NO molecules occur in the quasi-steady-state (QSS) regime, while for the Zel'dovich reaction, the distribution of the reactants does not reach the QSS conditions. Furthermore, using the QSS approximation to model the Zel'dovich mechanism leads to overestimating NO production by more than a factor of 4 in the high-temperature range. The breakdown of this well-known approximation has profound consequences for the approaches that heavily rely on the validity of QSS assumption in hypersonic applications. Finally, the investigation of the rovibrational state population dynamics reveals substantial similarities among different chemical systems for the energy-transfer and the dissociation processes, providing promising physical foundations for the use of reduced-order strategies in other chemical systems without significant loss of accuracy.

Carbon Clusters: Thermochemistry and Electronic Structure at High Temperatures.

This paper studies the thermochemistry and electronic structure of small carbon clusters and hydrocarbons, which are major constituents of pyrolysis gases released into the boundary layer of ablating heat shields. Our focus lies on clusters of up to four carbon atoms. Among other molecules, thermochemistry data for molecules such as C3H and C4H have been determined using the Weizmann-1 (W1) method. These molecules have very limited thermochemistry data recorded in the literature, thereby necessitating new and accurate computations of required properties such as electronic energies of low-lying states, heats of formation, harmonic frequencies, and rotational constants. A study of electronically excited states of these molecules computed using the equations of motion coupled cluster singles doubles method revealed C4 and C4H to be potential sources of radiation absorption in the boundary layer. The excited electronic states of interest are studied further to obtain their optimum geometries, rotational constants, and vibrational frequencies. Moreover, we also study the effect of low-lying excited electronic states on the partition function to assess their effect on the thermodynamics of these pyrolysis gases in the high-temperature regime. Neglecting the excited electronic states records a maximum difference of 12% in the computed specific heat capacity values, Cp values. Finally, comparisons of the equilibrium mole fractions obtained using the thermodynamics computed in this paper with the existing state-of-the-art tables used for hypersonic applications (e.g., JANAF and Gurvich Tables) show an order of magnitude difference in the mixture compositions. It is shown that the rhombic isomer of C4 (1Ag), which is energetically close to the ground state (3Σg-) and usually neglected in composition calculations, contributes to a 28% increase in the equilibrium mole fraction of the C4 molecule.

State-to-State Master Equation and Direct Molecular Simulation Study of Energy Transfer and Dissociation for the N2-N System.

We present a detailed comparison of two high-fidelity approaches for simulating non-equilibrium chemical processes in gases: the state-to-state master equation (StS-ME) and the direct molecular simulation (DMS) methods. The former is a deterministic method, which relies on the pre-computed kinetic database for the N2-N system based on the NASA Ames ab initio potential energy surface (PES) to describe the evolution of the molecules' internal energy states through a system of master equations. The latter is a stochastic interpretation of molecular dynamics relying exclusively on the same ab initio PES. It directly tracks the microscopic gas state through a particle ensemble undergoing a sequence of collisions. We study a mixture of nitrogen molecules and atoms forced into strong thermochemical non-equilibrium by sudden exposure of rovibrationally cold gas to a high-temperature heat bath. We observe excellent agreement between the DMS and StS-ME predictions for the transfer rates of translational into rotational and vibrational energy, as well as of dissociation rates across a wide range of temperatures. Both methods agree down to the microscopic scale, where they predict the same non-Boltzmann population distributions during quasi-steady-state dissociation. Beyond establishing the equivalence of both methods, this cross-validation helped in reinterpreting the NASA Ames kinetic database and resolve discrepancies observed in prior studies. The close agreement found between the StS-ME and DMS methods, whose sole model inputs are the PESs, lends confidence to their use as benchmark tools for studying high-temperature air chemistry.

Coarse-grained modeling of thermochemical nonequilibrium using the multigroup maximum entropy quadratic formulation.

This work addresses the construction of a reduced-order model based on a multigroup maximum entropy formulation for application to high-enthalpy nonequilibrium flows. The method seeks a piecewise quadratic representation of the internal energy-state populations by lumping internal energy levels into groups and by applying the maximum entropy principle in conjunction with the method of moments. The use of higher-order polynomials allows for an accurate representation of the logarithm of the distribution of the low-lying energy states, while preserving an accurate description of the linear portions of the logarithm of the distribution function that characterize the intermediate- and high-energy states. A comparison of the quadratic and the linear reconstructions clearly demonstrates how the higher-order reconstruction provides a more accurate representation of the internal population distribution function at a modest increase in the computational cost. Numerical simulations carried out under conditions relevant to hypersonic flight reveal that the proposed model is able to capture the dynamics of the nonequilibrium distribution function using as few as three groups, thereby reducing the computational costs for simulations of nonequilibrium flows.

Comparison of potential energy surfaces and computed rate coefficients for N2 dissociation.

Comparisons are made between potential energy surfaces (PES) for N2 + N and N2 + N2 collisions and between rate coefficients for N2 dissociation that were computed using the quasiclassical trajectory method (QCT) on these PESs. For N2 + N we compare the Laganà's empirical LEPS surface with one from NASA Ames Research Center based on ab initio quantum chemistry calculations. For N2 + N2 we compare two ab initio PESs (from NASA Ames and from the University of Minnesota). These use different methods for computing the ground state electronic energy for N4, but give similar results. Thermal N2 dissociation rate coefficients, for the 10,000K-30,000K temperature range, have been computed using each PES and the results are in excellent agreement. Quasi-stationary state (QSS) rate coefficients using both PESs have been computed at these temperatures using the Direct Molecular Simulation of Schwartzentruber and coworkers. The QSS rate coefficients are up to a factor of 5 lower than the thermal ones and the thermal and QSS values bracket the results of shock-tube experiments. We conclude that the combination of ab initio quantum chemistry PESs and QCT calculations provides an attractive approach for the determination of accurate high-temperature rate coefficients for use in aerothermodynamics modeling.

General multi-group macroscopic modeling for thermo-chemical non-equilibrium gas mixtures.

This paper opens a new door to macroscopic modeling for thermal and chemical non-equilibrium. In a game-changing approach, we discard conventional theories and practices stemming from the separation of internal energy modes and the Landau-Teller relaxation equation. Instead, we solve the fundamental microscopic equations in their moment forms but seek only optimum representations for the microscopic state distribution function that provides converged and time accurate solutions for certain macroscopic quantities at all times. The modeling makes no ad hoc assumptions or simplifications at the microscopic level and includes all possible collisional and radiative processes; it therefore retains all non-equilibrium fluid physics. We formulate the thermal and chemical non-equilibrium macroscopic equations and rate coefficients in a coupled and unified fashion for gases undergoing completely general transitions. All collisional partners can have internal structures and can change their internal energy states after transitions. The model is based on the reconstruction of the state distribution function. The internal energy space is subdivided into multiple groups in order to better describe non-equilibrium state distributions. The logarithm of the distribution function in each group is expressed as a power series in internal energy based on the maximum entropy principle. The method of weighted residuals is applied to the microscopic equations to obtain macroscopic moment equations and rate coefficients succinctly to any order. The model's accuracy depends only on the assumed expression of the state distribution function and the number of groups used and can be self-checked for accuracy and convergence. We show that the macroscopic internal energy transfer, similar to mass and momentum transfers, occurs through nonlinear collisional processes and is not a simple relaxation process described by, e.g., the Landau-Teller equation. Unlike the classical vibrational energy relaxation model, which can only be applied to molecules, the new model is applicable to atoms, molecules, ions, and their mixtures. Numerical examples and model validations are carried out with two gas mixtures using the maximum entropy linear model: one mixture consists of nitrogen molecules undergoing internal excitation and dissociation and the other consists of nitrogen atoms undergoing internal excitation and ionization. Results show that the original hundreds to thousands of microscopic equations can be reduced to two macroscopic equations with almost perfect agreement for the total number density and total internal energy using only one or two groups. We also obtain good prediction of the microscopic state populations using 5-10 groups in the macroscopic equations.

Communication: Surface-to-bulk diffusion of isolated versus interacting C atoms in Ni(111) and Cu(111) substrates: A first principle investigation.

Using first principle calculations, we study the surface-to-bulk diffusion of C atoms in Ni(111) and Cu(111) substrates, and compare the barrier energies associated with the diffusion of an isolated C atom versus multiple interacting C atoms. We find that the preferential Ni-C bonding over C-C bonding induces a repulsive interaction between C atoms located at diagonal octahedral voids in Ni substrates. This C-C interaction accelerates C atom diffusion in Ni with a reduced barrier energy of ∼1 eV, compared to ∼1.4-1.6 eV for the diffusion of isolated C atoms. The diffusion barrier energy of isolated C atoms in Cu is lower than in Ni. However, bulk diffusion of interacting C atoms in Cu is not possible due to the preferential C-C bonding over C-Cu bonding, which results in C-C dimer pair formation near the surface. The dramatically different C-C interaction effects within the different substrates explain the contrasting growth mechanisms of graphene on Ni(111) and Cu(111) during chemical vapor deposition.

Rovibrational internal energy transfer and dissociation of N2(1Σg+)-N(4S(u)) system in hypersonic flows.

A rovibrational collisional model is developed to study energy transfer and dissociation of N(2)((1)Σ(g)(+)) molecules interacting with N((4)S(u)) atoms in an ideal isochoric and isothermal chemical reactor. The system examined is a mixture of molecular nitrogen and a small amount of atomic nitrogen. This mixture, initially at room temperature, is heated by several thousands of degrees Kelvin, driving the system toward a strong non-equilibrium condition. The evolution of the population densities of each individual rovibrational level is explicitly determined via the numerical solution of the master equation for temperatures ranging from 5000 to 50,000 K. The reaction rate coefficients are taken from an ab initio database developed at NASA Ames Research Center. The macroscopic relaxation times, energy transfer rates, and dissociation rate coefficients are extracted from the solution of the master equation. The computed rotational-translational (RT) and vibrational-translational (VT) relaxation times are different at low heat bath temperatures (e.g., RT is about two orders of magnitude faster than VT at T = 5000 K), but they converge to a common limiting value at high temperature. This is contrary to the conventional interpretation of thermal relaxation in which translational and rotational relaxation timescales are assumed comparable with vibrational relaxation being considerable slower. Thus, this assumption is questionable under high temperature non-equilibrium conditions. The exchange reaction plays a very significant role in determining the dynamics of the population densities. The macroscopic energy transfer and dissociation rates are found to be slower when exchange processes are neglected. A macroscopic dissociation rate coefficient based on the quasi-stationary distribution, exhibits excellent agreement with experimental data of Appleton et al. [J. Chem. Phys. 48, 599-608 (1968)]. However, at higher temperatures, only about 50% of dissociation is found to take place under quasi-stationary state conditions. This suggest the necessity of explicitly including some rovibrational levels, when solving a global kinetic rate equation.