Kaidi Wan

Camille Barnaud

Luc Vervisch
email: vervisch@coria.fr

Pascale Domingo

Chemistry reduction using machine learning trained from non-premixed micro-mixing modeling: Application to DNS of a syngas turbulent oxy-flame with side-wall effects

Keywords: Artificial neural network, Chemistry reduction, Direct numerical simulation, Syngas

A chemistry reduction approach based on machine learning is proposed and applied to direct numerical simulation (DNS) of a turbulent non-premixed syngas oxy-flame interacting with a cooled wall. The training and the subsequent application of artificial neural networks (ANNs) rely on the processing of 'thermochemical vectors' composed of species mass fractions and temperature (ANN input), to predict the corresponding chemical sources (ANN output). The training of the ANN is performed aside from any flow simulation, using a turbulent non-adiabatic non-premixed micro-mixing based canonical problem with a reference detailed chemistry. Heat-loss effects are thus included in the ANN training. The performance of the ANN chemistry is then tested aposteriori in a two-dimensional DNS against the detailed mechanism and a reduced mechanism specifically developed for the operating conditions considered. Then, three-dimensional DNS are performed either with the ANN or the reduced chemistry for additional a-posteriori tests. The ANN reduced chemistry achieves good agreement with the Arrhenius-based detailed and reduced mechanisms, while being in terms of CPU cost 25 times faster than the detailed mechanism and 3 times faster than the reduced mechanism when coupled with DNS. The major potential of the method lies both in its data driven character and in the handling of the stiff chemical sources. The former allows for easy implementation in the context of automated generation of case-specific reduced chemistry. The latter avoids the Arrhenius rates calculation and also the direct integration of stiff chemistry, both leading to a significant CPU time reduction.

Introduction

Accounting for the complexity of chemistry in numerical simulation of flames has been the subject of numerous studies since the 80's. Reduced and skeleton mechanisms were first discussed from the fundamentals of combustion chemistry, sometimes coupled with more or less advanced sensitivity analyses (see [START_REF] Peters | Systematic reduction of flame kinetics: Principles and Details[END_REF][START_REF] Peters | Flame Calculations with Reduced Mechanisms -An Outline[END_REF][START_REF] Griffiths | Reduced kinetic models and their application to practical combustion systems[END_REF][START_REF] Goussis | Model Reduction for Combustion Chemistry[END_REF][START_REF] Turanyi | Analysis of Kinetic Reaction Mechanisms[END_REF] and references therein). Meanwhile, the progress in the understanding of chemistry and transport phenomena has driven the development of detailed and reliable chemistry mechanisms for fuel oxidation, which could now contain up to hundreds of species and thousands of reactions [START_REF] Curran | Developing detailed chemical kinetic mechanisms for fuel combustion[END_REF]. To avoid dealing with too many degrees of freedom, which would need to be solved together with the flow dynamics, these chemical mechanisms are usually downsized before their introduction in computational fluid dynamics software. Thereby, more or less automated reduction techniques of these reference detailed chemical mechanisms have been discussed in the literature. These reduction approaches may combine fuel lumping, time-scale and principal component analysis or a direct numerical treatment of the chemical signals from graph analysis or genetic algorithms, approaches which may also be associated with systematic tabulation techniques [START_REF] Peters | Numerical and asymptotics analysis of systematically reduced reaction schemes for hydrocarbon flames[END_REF][START_REF] Jones | Global reaction schemes for hydrocarbon combustion[END_REF][START_REF] Bradley | Laminar flamelet modeling of recirculating premixed methane and propane-air combustion[END_REF][START_REF] Maas | Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space[END_REF][START_REF] Lam | The CSP method for simplifying kinetics[END_REF][START_REF] Van Oijen | Modeling of complex premixed burner systems by using flamelet-generated manifolds[END_REF][START_REF] Bykov | The extension of the ILDM concept to reaction-diffusion manifolds[END_REF][START_REF] Jones | Reduction of comprehensive chemistry via constraint potentials[END_REF][START_REF] Lu | A directed relation graph method for mechanism reduction[END_REF][START_REF] Pepiot-Desjardins | An efficient error-propagation-based reduction method for large chemical kinetic mechanisms[END_REF][START_REF] Sutherland | Combustion modeling using principal component analysis[END_REF][START_REF] Boivin | An explicit reduced mechanism for H 2 air combustion[END_REF][START_REF] Jaouen | Auto-thermal reforming (ATR) of natural gas: An automated derivation of optimised reduced chemical schemes[END_REF][START_REF] Koniavitis | Reduction of a detailed chemical mechanism for a kerosene surrogate via RCCE-CSP[END_REF][START_REF] Chang | Construction of a skeletal oxidation mechanism of n-pentanol by integrating decoupling methodology, genetic algorithm, and uncertainty quantification[END_REF][START_REF] Scholtissek | A self-contained composition space solution method for strained and curved premixed flamelets[END_REF][START_REF] Wen | A three mixture fraction flamelet model for multi-stream laminar pulverized coal combustion[END_REF] (not exhaustive list).

Considering a detailed chemical mechanism involving a number N S of species interacting according to N R elementary reactions, the reduction procedure consists of expressing the chemical rates of a number N r S < N S of species, so that the thermochemical information relevant for the problem under consideration is preserved. Robust and well-established methods exist to rank species according to their relative importance in the prediction of thermochemical properties, which should be accurately captured by the reduced scheme [START_REF] Lam | The CSP method for simplifying kinetics[END_REF][START_REF] Lu | A directed relation graph method for mechanism reduction[END_REF][START_REF] Pepiot-Desjardins | An efficient error-propagation-based reduction method for large chemical kinetic mechanisms[END_REF][START_REF] Koniavitis | Reduction of a detailed chemical mechanism for a kerosene surrogate via RCCE-CSP[END_REF]. This allows for determining the minimum set of N r S species to be transported with the flow to preserve a given level of accuracy. Once this first step completed, in a second step, the burning rates are usually expressed from a reduced set of N r R < N R elementary reactions, chosen from the detailed scheme. A different approach is attempted in this work for the second step, i.e., after the important species have been determined. With this novel strategy, there is no reduced mechanism produced per se, but instead a specific machine learning procedure is conducted to determine, from the detailed chemical scheme, how to express the burning rates of the reduced number of species transported with the flow. The input of the neural networks is thus the thermochemical reduced vector and the output is either the vector of reaction rates or the new thermochemical vector for a given time increment, depending on the most convenient coupling procedure with the flow solver.

Such combination of the benefits of mechanism reduction and high-dimensional chemistry tabulation approaches is not fully new and has already been achieved by employing artificial neural networks (ANNs). Indeed, ANNs have received extensive attention in tabulation of combustion chemistry, since their first usage back in the 90's [START_REF] Christo | An integrated PDF/neural network approach for simulating turbulent reacting systems[END_REF][START_REF] Blasco | Modelling the temporal evolution of a reduced combustion chemical system with artificial neural network[END_REF], up to recent studies demonstrating that modern ANN architectures offer good predictions with great savings in CPU time and memory requirements [START_REF] Franke | Tabulation of combustion chemistry via Artificial Neural Networks (ANNs): Methodology and application to LES-PDF simulation of Sydney flame L[END_REF][START_REF] Padilha | Solutions for a laminar jet diffusion flame of methly formate using a skeletal mechanism obtained by applying ANNs[END_REF][START_REF] Owoyele | Application of deep artificial neural networks to multi-dimensional flamelet libraries and spray flames[END_REF]. Data driven approaches have also been coupled with principal component analysis for developing closure models in turbulent combustion using experimental multi-scalar measurements [START_REF] Ranade | A framework for data-based turbulent combustion closure: A posteriori validation[END_REF] and for building digital-twins to progress towards furnace control [START_REF] Aversano | Application of reduced-order models based on PCA & Kriging for the development of digital twins of reacting flow applications[END_REF]. Along similar lines, convolutional neural networks (CNN), based on image-like treatment of the thermochemical fields, were shown useful to tackle the modeling of turbulent flames [START_REF] Nikolaou | Progress Variable Variance and Filtered Rate Modelling Using Convolutional Neural Networks and Flamelet Methods[END_REF][START_REF] Lapeyre | Training convolutional neural networks to estimate turbulent sub-grid scale reaction rates[END_REF][START_REF] Seltz | Direct mapping from LES resolved scales to filtered-flame generated manifolds using convolutional neural networks[END_REF] and their control [START_REF] Zhu | Convolutional neural network based combustion mode classification for condition monitoring in the supersonic combustor[END_REF].

Recently, chemistry reduction based on CNN was discussed with a training of the CNN performed from the targeted flow simulation with detailed chemistry [START_REF] Wan | Machine learning for detailed chemistry reduction in DNS of a syngas turbulent oxy-flame with side-wall effects[END_REF]. However, for machine learning to be fully efficient and of practical interest, the training should be achieved on a canonical problem, as generic as possible, and ideally aside from the targeted flow simulation, as it was envisioned in [START_REF] Franke | Tabulation of combustion chemistry via Artificial Neural Networks (ANNs): Methodology and application to LES-PDF simulation of Sydney flame L[END_REF][START_REF] Chatzopoulos | A chemistry tabulation approach via Rate Controlled Constrained Equilibrium (RCCE) and Artificial Neural Networks (ANNs), with application to turbulent non-premixed CH 4 /H 2 /N 2 flames[END_REF]. To this end, training from a non-premixed turbulent stochastic micromixing problem is discussed in this paper. This canonical problem was previously introduced in the context of automated chemistry reduction [START_REF] Jaouen | Auto-thermal reforming (ATR) of natural gas: An automated derivation of optimised reduced chemical schemes[END_REF], with a simple pairwise micro-mixing closure.

Instead, the Euclidean minimum spanning tree mixing modeling [START_REF] Subramaniam | A mixing model for turbulent reactive flows based on Euclidean minimum spanning trees[END_REF] is here chosen to secure a better accurate detailed chemistry response in mixture fraction space. Non-adiabatic effects are also accounted for through added enthalpy losses.

The operating conditions retained for the study are those of a syngas non-premixed oxy-flame interacting with a cooled wall. This configuration covers a wide range of combustion and turbulence conditions in terms of equivalence ratio, progress of reaction and enthalpy including thermal and viscous boundary layers effects, making it a relatively generic test case. The syngas fuel ANNs are trained successfully, the burning rates of the reduced set of thermochemical variables are readily obtained from the knowledge of temperature and the retained species mass fractions. An Arrhenius-type reduced chemical scheme previously derived is also available for these operating conditions [START_REF] Wan | Machine learning for detailed chemistry reduction in DNS of a syngas turbulent oxy-flame with side-wall effects[END_REF].

Direct numerical simulation (DNS) of the non-premixed syngas oxy-flame with side wall effects are then performed. First, two-dimensional DNS with the detailed chemical scheme serves as reference to test both the Arrhenius-type reduced scheme and the ANNs trained aside from DNS. Then, three-dimensional DNS of the same flame configuration are conducted with both the Arrhenius-type reduced scheme and the ANN-type chemistry. The comparison between results confirm the robustness of the ANNs for chemistry reduction along with the opportunity to train neural networks for chemistry reduction prior to flow simulations.

Non-premixed flame configuration and numerical modeling

A turbulent non-premixed oxy-flame under an elevated pressure of 341.3 kPa with side-wall effects is considered, as shown in Fig. 1. A representative syngas composition is injected from the upper inlet over a width of 2 mm, while pure oxygen is supplied at the lower inlet over 10 mm.

A 0.2 mm splitter plate separates these fuel and oxidizer inlets. The corresponding boundary conditions (velocity, temperature and species mass fractions) are summarized in Table 1.

A preliminary simulation of a turbulent periodical 3D channel flow initialized with the oxygen bulk velocity serves as a database to impose a proper turbulent inflow for the lower oxygen stream [START_REF] Pierce | Method for generating equilibrium swirling inflow conditions[END_REF][START_REF] Schlüter | Large-eddy simulation inflow conditions for coupling with Reynolds-averaged flow solvers[END_REF]. A laminar channel flow inlet profile is imposed for the close-to-wall fuel stream. A non-slip wall is placed at the top of the computational domain, while a symmetrical boundary condition is employed at the bottom. Convective boundary condition is used for the outlet of the domain. Periodic boundary conditions are used in the spanwise z direction (Fig. 1).

The temperature of the upper wall T w (x, t) is non-uniform and calculated so that the heat flux locally reaching the wall is evacuated mimicking water cooling, with a convective heat transfer

h = 3300 W • m -2 • K -1 and a water temperature T o = 353 K (i.e., h(T w -T o ) = λ(∂T/∂n w ) gas ,
where n w is the wall normal direction toward the gas).

The physical dimensions of the computational domain are L x = 100 mm, L y = 12 mm and L z = 7.68 mm. One-dimensional canonical strained non-premixed flames simulations were conducted to calibrate the grid resolution. It was also verified through a non-reacting simulation that the structure of the wall boundary layer agrees with what is expected (viscous sub-layer, buffer layer and log-law). The mesh composed of 66.44M cells is uniform in the streamwise x and spanwise z directions, with the grid spacing h x = h z = 60 µm. In the vertical y direction, the mesh keeps a fine resolution of 20 µm for the upper region close to the wall (y > 10 mm), where the flame is located. As y decreases from 10 to 0 mm, the mesh gets coarser to reach a grid spacing of 100 µm at the bottom of the domain. The first point off the top wall is at y + = 0.55, where the superscript '+' denotes usual dimensionless near-wall quantities. There are 9 grid points within y + = 10 to resolve the viscous layer near the wall [START_REF] Gruber | Turbulent flamewall interaction: a direct numerical simulation study[END_REF]. The time step is fixed at 0.3 µs (CFL≈0.8)

with sub-iterations in a stiff-chemistry ODE solver (CVODE).

The governing equations for the gas mixture are solved with a low-Mach number in-house code [START_REF] Wan | Alkali metal emissions in an early-stage pulverized-coal flame: DNS analysis of reacting layers and chemistry tabulation[END_REF][START_REF] Wan | Numerical study of HCl and SO2 impact on sodium emissions in pulverized-coal flames[END_REF][START_REF] Wan | Numerical study of HCl and SO2 impact on potassium emissions in pulverized-biomass combustion[END_REF][START_REF] Wan | Modelling alkali metal emissions in large-eddy simulation of a preheated pulverised-coal turbulent jet flame using tabulated chemistry[END_REF][START_REF] Wan | Large-eddy Simulation of Pilot-assisted Pulverized-coal Combustion in a Weakly Turbulent Jet[END_REF][START_REF] Wan | Online-CPD-coupled large-eddy simulation of pulverized-coal pyrolysis in a hot turbulent nitrogen jet[END_REF]. The conservation equations for mass, momentum, species and temperature of the gas phase are cast as follows:

∂ρ ∂t + ∂ρu j ∂x j = 0, (1)
∂ρu i ∂t + ∂ρu i u j ∂x j = - ∂p ∂x i + ∂τ i j ∂x j , (2)
∂ρY n ∂t + ∂ρu j Y n ∂x j = ∂ ∂x j ρD ∂Y n ∂x j + ωY n , (3)
∂ρT ∂t + ∂ρu j T ∂x j = ∂ ∂x j λ C P ∂T ∂x j + λ C 2 P ∂C P ∂x j ∂T ∂x j + ωT , ( 4
)
where t is the time [s], ρ is density of the gas mixture [kg

• m -3 ], u is the gas velocity [m • s -1 ],
Y n is the mass fraction of the n-th chemical species

• s -1 ], thermal conductivity [W • m -1 • K -1 ] and specific heat capacity [J • kg -1 • K -1
] of the gas mixture, respectively. In the present study, the Lewis number is assumed to be unity for all the species, which leads to D = λ/(ρC p ); while µ and λ are calculated using Wilke's type mixing rules. Finally, ωY n and ωT are the species and temperature source terms due to chemical reaction, respectively.

The numerical scheme is based on an approach previously employed for both DNS and largeeddy simulation (LES) [START_REF] Pierce | Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion[END_REF][START_REF] Desjardins | High order conservative finite difference scheme for variable density low Mach number turbulent flows[END_REF]. The time advancement relies on a second-order Crank-Nicolson scheme. However, due to the splitting method for chemistry integration, the overall order of accuracy of the solver in time advancement may drop below second-order. A bounded quadratic upstream interpolation for convective kinematics (BQUICK) scheme is applied to the scalar advection terms in the species and temperature equations, while a second-order central difference scheme is used for the scalar diffusion terms in the species and temperature equations and all terms in the momentum equation. An alternating direction implicit (ADI) approach has been used, in which semi-implicit tridiagonal/pentadiagonal equations are solved separately for each direction.

Machine learning chemistry reduction methodology

Stochastic micro-mixing problem

The training database for the ANN chemistry is built from a turbulent non-adiabatic nonpremixed micro-mixing canonical problem, which was first discussed for process control of selective non-catalytic reduction (SNCR) [START_REF] Farcy | Reduced-order modeling for the control of selective non-catalytic reduction (SNCR)[END_REF][START_REF] Locci | Selective Non-Catalytic Reduction (SNCR) of nitrogen oxide emissions: A perspective from numerical modeling[END_REF] and then extended to automated chemistry reduc-tion [START_REF] Jaouen | Automatic reduction and optimisation of chemistry for turbulent combustion modelling: Impact of the canonical problem[END_REF].

As in the DNS of the non-premixed oxy-flame, two inlets are considered and stochastic particles evolve from initial conditions characterized by these inlets given in Table 1. The number of stochastic particles per inlet (i.e., the mass flow rates of the inlets [START_REF] Farcy | Reduced-order modeling for the control of selective non-catalytic reduction (SNCR)[END_REF]) are organized so that the overall system operates under the stoichiometric condition. Here, a total number of 1000 stochastic particles are employed in the reactor, at initial time 660 particles are assigned to the composition and enthalpy of the fuel inlet and 340 particles to that of the oxygen inlet.

From the initial condition at time t = 0, the species mass fraction Y p i (t) and sensible enthalpy h p s (t) of each stochastic particle evolve according to the following equations:

dY p i (t) dt = MIX p i (τ T ) + ωp i , (5)
dh p s (t) dt = MIX p h s (τ T ) + ωp h s -α loss (T -T o ) , (6)
where 'MIX' denotes the stochastic turbulent micro-mixing closure for the diffusive budget, which is here the well-established Euclidean minimum spanning tree (EMST) mixing model [START_REF] Subramaniam | A mixing model for turbulent reactive flows based on Euclidean minimum spanning trees[END_REF][START_REF] Ren | Implementation of the EMST mixing model[END_REF]. τ T denotes the micro-mixing time and ωp i and ωp h s are the species and enthalpy chemical sources, respectively. The linear sink term α loss (T -T o ) in Eq. ( 6) represents convective heat loss effects, as previously done in [START_REF] Bioche | Premixed flamewall interaction in a narrow channel: impact of wall thermal conductivity and heat losses[END_REF][START_REF] Bioche | Simulating upstream flame propagation in a narrow channel after wall preheating: Flame analysis and chemistry reduction strategy[END_REF] in the derivation of reduced chemical schemes devoted to flame wall/interaction. T o = 353 K is the temperature of the cooling water in the DNS configuration.

α loss is the heat loss coefficient (W • kg -1 • K -1 ).
The value of the micro-mixing time is chosen in line with the global turbulent mixing characteristics of the shear layer. According to the inlet conditions, T , the integral length scale in the fully developed zone can be expected of the order of a few millimeters. Taking T ≈ 3 mm and velocity fluctuations u of the order of a few meters per second, τ T ≈ T /u ∼ 0.3 ms. Accordingly, the micro-mixing time is set at 0.3 ms while the time step is 0.3 µs, with sub-iterations to solve for the stiff-chemistry part of the problem using CVODE. In fact, as soon as it does not prevent ignition or favor very fast chemistry, varying the value of τ T was not found to have a major impact on the results (see [START_REF] Jaouen | Automatic reduction and optimisation of chemistry for turbulent combustion modelling: Impact of the canonical problem[END_REF] for a discussion on the impact of τ T value on chemistry reduction using a similar stochastic formulation). In the present case, the most crucial point is to cover with Eqs. ( 5) and ( 6) the full range of chemical responses starting from frozen flow mixing, followed by ignition and fast oxy-flame burning, to end up with burnt products subsequently subjected to heat-loss. Equations ( 5) and ( 6) are solved using the CANTERA package [START_REF] Goodwin | Cantera: An Object-oriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes[END_REF]. The reference GRI-3.0 methane/air detailed mechanism [START_REF] Smith | [END_REF] without NO x (35 chemical species and 217 elementary reactions) serves as reference to test the reduction methodology.

Reference reduced-scheme

Combining a directed relation graph analysis (DRGEP) [START_REF] Pepiot-Desjardins | An efficient error-propagation-based reduction method for large chemical kinetic mechanisms[END_REF] with a genetic algorithm [START_REF] Jaouen | Automatic reduction and optimisation of chemistry for turbulent combustion modelling: Impact of the canonical problem[END_REF], an 11-species reduced scheme was previously derived with the simple pairwise Curl mixingmodel [START_REF] Curl | Dispersed phase mixing: I. Theory and effects in simple reactors[END_REF] and validated for the syngas oxy-flame considered here [START_REF] Wan | Machine learning for detailed chemistry reduction in DNS of a syngas turbulent oxy-flame with side-wall effects[END_REF] (Table 2). Both the GRI-3.0 and this reduced-scheme will serve as references thereafter.

Non-premixed database for ANN training

At the start of the simulation with the detailed scheme (GRI-3.0), α loss is set to zero in Eq. ( 6), i.e., no heat loss is present. Because the particles of the fuel inlet are preheated to 1223 K, rapid ignition occurs with the detailed chemistry mechanism at 81 µs after mixing with the particles from the oxygen inlet. Figure 2 shows representative scatter plots of the instantaneous temperature of the stochastic particles against mixture fraction. The instantaneous values at a given time (red) are plotted along with the time history of all particles up to this time, for three representative times.

The mixture fraction is defined as Z = Y N 2 /0.1549. Because of fuel preheating (Table 1), ignition first appears on the fuel-rich side for Z > 0.9 (Fig. 2(a)). After 0.3 ms, i.e., 1000 time steps, the micro-mixing simulation reaches a stable burning status. The peak temperature is 3113 K around the stoichiometric condition at Z = 0.66 (Fig. 2(b)). Then, to model the interaction of burnt gases with a cold wall, the heat loss is turned on by setting α loss = 10 7 W • kg -1 • K -1 , and the simulation proceed for another 0.6 ms until all the stochastic particles are cooled down (Fig. 2(c)).

By saving the 35 species mass fractions and temperature of each stochastic particle for every time step, 3000 snapshots each containing thermochemical information from 1000 particles are available to built a training database. To augment the database up to 9000 snapshots and avoid over-fitting, as it is usually done in ANN training procedure, the 3,000,000 data are perturbed twice applying 2% and 3% of uncorrelated random noise, still securing mass conservation.

The chemical sources of intermediate radicals featuring the fastest evolution (i.e., those associated to the smallest time scales) are known to be difficult to interpolate, because the concentrations of these species jump from zero to their maximum levels over very small characteristic times. The issue appeared in the present case for the evolution of H 2 O 2 , a species which was found mandatory to be kept in the reduced scheme to properly capture ignition. To secure an accurate interpolation of the H 2 O 2 source with the ANN, the neural network needs to be exposed to numerous data points, where the H 2 O 2 concentration lies between zero and its maximum. A brute force way to overcome this difficulty would be to train the ANN at all the sub-iterations of the time integration performed by the stiff-chemistry solver (CVODE), which returns the proper evolutions of the radical species entering the detailed scheme. This option would result in a huge computing effort during the learning phase.

Instead, an alternative data driven approach has been followed, which is valid only for a single variable affected by an overly quick evolution and whose contribution to the overall mass stays small (the max value of H 2 O 2 mass fraction is 5.3×10 -4 ). To populate the training database with The database now have 18000 snapshots each containing 1000 data. 2.7% of the data have a temperature below 320 K, these points with negligible sources are dropped out, which leads to a final database of 17.5×10 6 data. Organizing the sources in this form is convenient for the subsequent coupling with flow simulation following two options: In the case where a specific time integration method for stiff differential systems is preferred, the source can be used directly. If sub-iterations take place with the time step δt > δt o , interpolations will be needed to estimate ωi (t). A second option for the coupling with the flow, is to directly read from the ANNs φ i (t + δt). Here again, if δt > δt o , sub-iterations will be required to secure accuracy. The second option is used thereafter and since δt = δt o in our flow simulation, sub-iterations are not performed.

ANN training procedure

The chemical sources of the reduced set of N r S = 11 species (source of N 2 is set to zero) are computed from the detailed chemical scheme with the N S = 35 species. Therefore these sources do not preserve the atomic budgets computed over the N r S < N S species. Even if at its maximum, the mass default is only 0.016% of the total mass of these atoms, in order to fully conserve mass, corrective sources are added according to the relative contribution of the species to atomic budgets. Let us denote α A,i the number of atom A (i.e., A ≡ C, H or O) in the i-th species.

The mass imbalance in A source reads for the reduced set of species

∆ ωA = N r s i=1 α A,i W A W i ωi , (7)
∆ ẇi = - Y i Y A ∆ ωA , (8)
with the atom mass fraction

Y A = N r s i=1 α A,i (W A /W i )Y i .
For oxygen, Eq. ( 8) is applied only to O 2 and with the already corrected sources of C and H containing species. This correction is applied during the ANN-DNS coupled simulations (therefore after ANN training). Then, the chemical sources perfectly preserve atomic budgets, as chemical sources computed from elementary reactions with Arrhenius laws would. In practice this correction stays marginal, a source correction (Eq. ( 8)) significantly contributing to the burning rates would mean that too many species were removed, jeopardizing the quality of the ANN prediction.

Within the information from the stochastic particles containing 17.5×10 6 data, 3.5×10 6 (20%) of them are randomly selected to form the testing database, while the remaining 14×10 6 (80%) constitute the training database. All data contributing to the "input vectors" and to the "targets" are normalized by their respective maximum absolute value over the entire database, and these maximum levels are stored.

ANN structure

The exact structure of the sequential ANNs1 is summarized in Table 3. Five fully connected dense layers are employed between the input and output layers. Each neuron of the dense layers2 .

(see green circles in Fig. 3) performs the following two operations: a weighted sum of inputs from all the neurons of the previous layer and the subsequent calculation of a non-linear activation function. In the present study, the popular rectified linear unit (ReLU) activation function is used:

ReLU(x) = max(x, 0) . (9)
Various options were attempted (even using a different network for each species and temperature).

In the end, the above ANN topology, which is a quite standard one, including the number of layers and neurons per layer, appears as the simplest efficient network for the prediction of the chemical sources. More complex networks would slow down the training process, with an increased risk of overfitting. In total, the neural network contains 179,851 weights to be adjusted.

To improve the precision of the ANNs for small values of the chemical sources, a second network with the exact same structure is trained for a subset of the database defined by Y O 2 > 0.9, i.e., where the overall burning rates stay very low. A sufficiently large number of data points are involved in the training so that continuity is secured between the two networks. The subset-database is normalized by its own maximum absolute values. In the following, the networks are referred to as the main ANN and the second ANN (subset for lower burning rates).

The training of the ANNs is performed using the TensorFlow Python library with GPU support (www.tensorflow.org). The neural weights are adjusted during the training process until a satisfying minimal error is obtained between the "target" sources and the ones predicted by the ANNs.

The Adam optimizer [START_REF] Kingma | A Method for Stochastic Optimization[END_REF] is used with a mean-squared-error (MSE) loss function comparing the predictions to the target. A training step is performed with a batch of 10240 "input vectors", leading to 1369 steps per "epoch"/iteration for the main ANN. The learning rate [START_REF] Aghdam | Guide to convolutional neural networks[END_REF] is initially set to 0.0003, to decay versus the steps (n) according to

l r = 0.0003 1.0 + 0.0001 • n . ( 10
)
The network is trained for 500 epochs to reach convergence, which requires for the main ANN about 3.2 hours on an Nvidia GeForce GTX 1080 Ti GPU.

ANN testing

The performance of the ANNs is first evaluated a-priori on the testing database containing 3.5×10 6 data unseen during training. The mean-square error between the ANN predictions and the targets is 9.2×10 -7 for the main ANN and 3.6×10 -7 for the second ANN. Figure 4 shows the predicted chemical sources against their target values for both the main ANN and the second ANN, confirming in a preliminary manner the accuracy of the ANN reduced chemistry approach.

ANN-DNS full coupling

The ANN reduced chemistry is now coupled with the flow solver to simulate the syngas nonpremixed oxy-flame (only CPU is used in this part with ANNs exported in C language, see supplementary material). To evaluate the performance of the ANN reduced chemistry, simulations are performed with the reference detailed GRI-3.0 chemistry (2D), ANN chemistry (2D and 3D) and the reduced chemistry of Table 2 (2D and 3D).

Detailed, Arrhenius-reduced and ANN-reduced chemistry in 2D simulations

The comparison of the 2D DNS results between the detailed GRI-3.0, the 11-species reduced mechanism of Table 2 and the ANN chemistry are shown in Figs. 5 and6 (streamwise and spanwise distributions). Statistical averages have been collected for 10 flow-through times after the flow is fully established.

Until the position x ≈ 8 mm along the planar fuel jet centerline (Fig. 5), the averaged streamwise velocity distribution decreases. At this streamwise location, the non-premixed syngas/air mixture starts to ignite, as indicated by the increase in temperature, while CO and H 2 decrease to produce CO 2 and H 2 O (Fig. 5(a)). At 15 mm from the nozzle inlet, approaching the cooled wall, the chemistry slows down (x = 0 in Fig. 6(a)). From this streamwise position the flame progressively spreads, to approach the close vicinity of the wall and the plate progressively heats up (Fig. 6(c)). Overall, the system is fuel-lean and the mass fraction of O 2 increases along the centerline. O and OH are the two major radicals while the mass fractions of HO 2 and H 2 O 2 are well below 10 -4 (Fig. 5(b)).

Comparing against the simulation with the reference detailed GRI-3.0 chemistry mechanism (circles in Figs. 5 and6), velocity, temperature and all the major species are well captured by the ANN-DNS (dashed-lines). Considering the radicals and minor species, the ANN-DNS shows a good prediction on the two major radicals O and OH, while it tends to over-predict HO 2 and under-predict H 2 O 2 .

The statistics from the simulation with the reduced mechanism of Table 2 (lines) are also very close to the detailed chemistry DNS and will serve as reference to test the ANN in 3D-DNS. For the H radical, the ANN-DNS returns a closer agreement against the reference detailed mechanism (Figs. 5(b) and 6). Because of the large number of sub-steps involved in the generation of both the ANN chemistry and the reduced mechanism of Table 2, it is difficult to identify the exact underlying reason for this behavior of the H radical. This is one of the weak point of data driven approaches, in the forest of neural connections and weights, it is barely feasible to track back specific relations between the species which lead to the behavior of intermediate minor species.

However, it is likely that learning from the large number of information available during the training phase, was helpful for the prediction of radicals. Notice also that the ANNs were trained with the detailed mechanism and the errors are due to the machine learning procedure (training dataset, training method, etc.). On the other hand, the reduced mechanism has been derived by dropping some species and reactions and therefore its discrepancies are for totally different reasons.

Arrhenius-reduced and ANN chemistry in 3D simulations

Vortex stretching was missing in the two-dimensional simulations, therefore the statistics collected in the three-dimensional flow are different from those of the planar flow. In particular, slightly smoother gradients are observed in 3D, as seen in Figs. 7 and8, compared to Figs. 5 and6. Overall, the three-dimensional turbulent flow cascade enhances the turbulence-chemistry interaction, leading to different scalar averages.

The ANN-chemistry provides statistics which are very close to those obtained with the reduced mechanism, for all the major species (Figs. 7 and8). A departure is observed for H and also for HO 2 and H 2 O 2 . The ANN chemistry predicts a higher level for HO 2 and a lower level for H and H 2 O 2 . According to the comparison against the GRI-3.0 reported above in the two-dimensional case, the scalar structure of H radical obtained with the ANN chemistry is actually expected to be closer than the reduced scheme to the one that would be seen in 3D with the detailed mechanism, which would be too expensive with available computing power. At stage, it is concluded that for the limited amount of species involved, the ignition phenomenon and the responses of reactants and major products are well captured, which is the main objective of most reduced chemical schemes used in computational fluid dynamics of complex systems. However as it can be expected, much more degrees of freedom would be needed to fully predict radicals and intermediate species. (Fig. 9(a)). The heat flux profiles feature similar characteristics in 3D, with however turbulent mixing in the third direction leading to a delayed increase of the mean heat flux and also a lower peak value at about 900 kW • m -2 . This behavior is confirmed in the mean temperature distribution shown in Fig. 10, as expected the thermal boundary layer in 2D case is much thinner than that in 3D case.

The computational cost of the coupled ANN-DNS approach is given in Fig. 11 for the 2D simulations, which include all three approaches (detailed, reduced and ANN). For solving the chemical source part of the problem, simulations with ANN chemistry are 25 times faster than those with the GRI-3.0 detailed mechanism, and 3 times faster than the 11-species reduced mechanism.

This speed-up results mainly from the fact that Arrhenius rates calculation are not needed and the time integration of stiff chemical sources is also avoided when the option of directly reading the species increments from the ANN is chosen. The computational time for solving the convectivediffusive part of the turbulent flame mainly depends on the number of species to be transported, it is therefore similar for the ANN-DNS and reduced chemistry DNS. (Same remarks apply to the 3D-DNS.)

Conclusions

One of the major stumbling block to the systematic introduction of artificial neural networks (ANNs) in flame simulation lies in the definition of generic training procedures, so that actually performing the simulation of the target flow geometry is not mandatory to complete the training of the ANN. Canonical problems, associated to given network architectures and training procedures, which could be applied prior to flame simulations, are likely to help popularizing the application of machine learning to computational combustion. Indeed, aside from a few pioneer works [START_REF] Franke | Tabulation of combustion chemistry via Artificial Neural Networks (ANNs): Methodology and application to LES-PDF simulation of Sydney flame L[END_REF][START_REF] Chatzopoulos | A chemistry tabulation approach via Rate Controlled Constrained Equilibrium (RCCE) and Artificial Neural Networks (ANNs), with application to turbulent non-premixed CH 4 /H 2 /N 2 flames[END_REF],

in most of previously published applications, the training and the subsequent use of the neural networks have been demonstrated on the very same flame configuration, thus limiting the generic character of the approach.

In the present work, a non-adiabatic non-premixed canonical problem, based on a stochastic micro-mixing closure, was introduced to train artificial neural networks from a reference detailed chemistry (GRI-3.0), including heat-loss effects. The trained ANNs provide directly the reaction rates of a reduced set of chemical species from the knowledge of their mass fractions and temperature, thus reducing drastically the cost of chemistry simulation. The training, performed aside from the flow simulation, is based on a reference detailed mechanism involving a large number of intermediates species and elementary reactions. This secures that the ANNs benefit from the fully detailed thermochemical information to predict the burning rate of a limited number of chemical species, which will be transported with the flow in the three-dimensional unsteady simulation.

The ANN reduced chemistry is applied to the direct numerical simulation of a non-premixed turbulent syngas-flame interacting with a cooled-wall. To assess the quality of the ANN reduced chemistry, two-dimensional simulations are simultaneously conducted with the reference fully detailed scheme (GRI-3.0) and an 11-species chemical mechanism specifically reduced for the syngas operating conditions considered. The three simulations return very similar results, with the ANN chemistry providing a significant reduction in CPU cost (25 times faster than detailed chemistry and 3 time faster than the reduced scheme). Then, the ANN chemistry and the reduced chemistry are applied to a three-dimensional direct numerical simulation of the syngas-flame with side wall effects. The statistical analysis of the non-premixed turbulent flame confirms the good performance of the ANN reduced chemistry trained prior and aside from the flow simulation. The scalar flame structure, in terms of reactants and major products, and the wall heat-flux are well predicted by the ANN reduced chemistry. Most of the expected trends are also recovered for the intermediate radicals. The potential of the discussed data driven approach lies in the automated character of the procedure to derive the reduced scheme, which can easily be implemented as a pre-processing tool. Another major advantage is that both the Arrhenius rates calculation and the direct integration of the stiff chemical system are avoided when choosing the option of reading the species increments directly from the ANN, which therefore leads to a significant reduction in CPU cost.

More physics could be introduced in the non-premixed canonical problem, as liquid fuel injection [START_REF] Farcy | Reduced-order modeling for the control of selective non-catalytic reduction (SNCR)[END_REF][START_REF] Bouaniche | Vitiated High Karlovitz n-decane/air Turbulent Flames: Scaling Laws and Micro-mixing Modeling Analysis[END_REF] or multiple inlets [START_REF] Jaouen | Automatic reduction and optimisation of chemistry for turbulent combustion modelling: Impact of the canonical problem[END_REF], however the extension to turbulent premixed combustion would require additional work to account for flame propagation. 2), and ANN. (Normalization by GRI-3.0.)

a

  sufficient variety of H 2 O 2 chemical sources, fields are manufactured by adding a copy of the 9000 snapshots in which the H 2 O 2 mass fraction is replaced by a random value between zero and its peak level. The mass fractions of others species are scaled according to the new total mass fractions, to conserve mass. These species vectors with specific H 2 O 2 mass fractions then enter the GRI-3.0 mechanism, to compute their corresponding exact chemical sources. For these additional species vectors, the detailed scheme returns chemical source rates which fulfill all the expected physical properties imposed by the detailed scheme and which can be used for the neural network to learn the relation between species mass fraction and temperature, as input, and sources, as output. Hence, the training of the ANN to reproduce chemical sources with intermediate values of H 2 O 2 becomes easily tractable.

Figure 3

 3 Figure3shows the overall ANN training procedure from the micro-mixing database. The input of the ANNs are the vectors φ(t) of the 10 species mass fractions and temperature (N 2 not included). The "target" (or "label") of each input vector is the corresponding reaction rates ωi = (φ i (t + δt o ) -φ i (t))/δt o , with δt o = 0.3 µs, the time step of flow simulations in the present work (close to the shortest chemical time scale observed for the intermediate species retained in the mechanism). Organizing the sources in this form is convenient for the subsequent coupling with

Figure 9

 9 Figure9shows the statistics of wall heat flux at the upper wall for both 2D and 3D cases, good agreement is found between ANN chemistry and reduced chemistry. Right after the splitter plate, the wall heat flux shows a high value of about 800 kW • m -2 at x = 0, which is due to the injection of the preheated fuel at 1223 K. As the fuel gets cooled down by the water-cooled wall, the mean heat flux decreases rapidly to the order of 200 kW • m -2 at x = 30 mm in the 2D cases (Fig.9(a)). Then, the mean heat flux starts to increase until x = 80 mm, where it reaches its maximum of 1250 kW • m -2 , because of the flame-wall interaction. Further downstream, the mean heat flux decreases again when the flame losses heat at the wall. The intermittency of the

Figure 1 :Figure 2 :

 12 Figure 1: Instantaneous iso-surface of Q criterion [38] predicted by ANN-DNS: Q = 10 7 s -2 , colored by gas temperature. Top graph: distribution of instantaneous wall temperature along the centerline.

Figure 3 : 9 Figure 4 :

 394 Figure 3: Sketch of the ANN training process from the micro-mixing database.

Figure 5 :

 5 Figure 5: Averaged distributions along the planar fuel-jet centerline (y = 11 mm). Symbols: GRI-3.0. Solid line: Reduced mechanism (Table 2). Dashed line: ANN chemistry. (a) Streamwise velocity and major species mass fractions: CO 2 , CO, O 2 , H 2 and H 2 O. (b) Temperature and radicals and minor species mass fractions: O, OH, H, HO 2 , and H 2 O 2 . 2D case.

Figure 6 :

 6 Figure 6: Spanwise averaged distributions of CO 2 , O 2 , CO, OH, H mass fractions and temperature. Symbols: GRI-3.0. Solid line: Reduced mechanism (Table 2). Dashed line: ANN chemistry. (a): 15 mm from the nozzle inlet. (b): 50 mm from the nozzle inlet. (c): 80 mm from the nozzle inlet. 2D case.

Figure 7 :

 7 Figure 7: Averaged distributions along the planar fuel-jet centerline (y = 11 mm). Solid line: Reduced mechanism (Table 2). Dashed line with symbols: ANN chemistry. (a) Streamwise velocity and major species mass fractions: CO 2 , CO, O 2 , H 2 and H 2 O. (b) Temperature and radicals and minor species mass fractions: O, OH, H, HO 2 , and H 2 O 2 . 3D case.

Figure 8 : 2 )

 82 Figure 8: Spanwise averaged distributions of CO 2 , O 2 , CO, OH, H mass fractions and temperature. Solid line: Reduced mechanism (Table 2). Dashed line with symbols: ANN chemistry. (a) 15 mm from the nozzle inlet. (b) 50 mm from the nozzle inlet. (c) 80 mm from the nozzle inlet. 3D case.

Figure 9 :

 9 Figure 9: Averaged distribution of wall heat flux at the upper wall. Mean ± RMS. 2D (a) Symbols: GRI-3.0. Solid line and orange: Reduced mechanism (Table 2). Dashed line and green: ANN chemistry. 3D (b) Dash dotted line and blue: Reduced mechanism (Table 2). Dotted line and red: ANN chemistry.

Figure 10 :

 10 Figure 10: Averaged distribution of temperature predicted by ANN chemistry in 2D and 3D cases. The zoomed-in sub-figures show the thermal boundary layers near the upper wall.

Figure 11 :

 11 Figure 11: Normalized average wall-clock-time per time step for solving reaction sources and the convective-diffusive part of the equations in 2D DNS coupled with GRI-3.0, 11-species reduced mechanism (Table2), and ANN. (Normalization by GRI-3.0.)

Table 1 :

 1 Operating conditions of the syngas oxy-flame (IGAR ADEME project).

		Upper inlet Lower inlet
	Bulk velocity [m/s] 50	100
	Temperature [K]	1223	298
	Mass fraction [-]
	H 2	0.0085	0
	CO	0.7852	0
	CO 2	0.0514	0
	N 2	0.1549	0
	O 2	0	1.0
	composition is representative of a CO

2 free blast-furnace exhaust flux, as found in modern recycling combustion processes. With the most important species determined, the response of the stochastic particles of the canonical problem enters the training and the testing of ANNs. After the

  [-], and T is the gas temperature [K]. p is the pressure [Pa] and the viscous stress tensor τ i j = µ(∂u i /∂x j + ∂u j /∂x i -(2/3)(∂u k /∂x k )δ i j ). µ, D, λ and C P are the molecular viscosity [Pa • s], molecular mass diffusivity coefficient [m 2

  The most important species found: O, O 2 , H, H 2 , OH, HO 2 , H 2 O, H 2 O 2 , CO and CO 2 , N 2 and the temperature are retained to constitute the reduced set of thermochemical variables.

Table 2 :

 2 Reduced chemistry for syngas oxy-flame. Units are mol, s, cm3 , cal and K. The Chaperon efficiencies of the GRI-3.0 mechanism are preserved for both three-body and fall-off reactions (see also supplementary material).

	n o	Reaction			A	β	E a
	1	2 O +M	O 2 +M	1.20e+17 -1.00	0
	2	O + H 2	H + OH	3.87e+04	2.70	6260
	3	O + HO 2	OH + O 2	2.00e+13	0	0
	4	O + CO (+M)	CO 2 (+M)	1.80e+10	0	2385
	5	O 2 + CO	O + CO 2	2.50e+12	0	47800
	6	H + O 2 +M	HO 2 +M	2.80e+18 -0.86	0
	7	H + O 2 + N 2	HO 2 + N 2	2.60e+19 -1.24	0
	8	H + 2 O 2	HO 2 + O 2	2.08e+19 -1.24	0
	9	H + O 2 + H 2 O	HO 2 + H 2 O 1.13e+19 -0.76	0
	10 H + O 2	O + OH	2.65e+16 -0.67	17041
	11 H + OH +M	H 2 O +M	2.20e+22 -2.00	0
	12 H + HO 2	O 2 + H 2	4.48e+13	0	1068
	13 H + HO 2	2OH	8.40e+13	0	635
	14 OH + H 2	H + H 2 O	2.16e+08	1.51	3430
	15 2 OH (+M)	H 2 O 2 (+M)	7.40e+13 -0.37	0
	16 OH + HO 2	O 2 + H 2 O	1.45e+13	0	-500
	17 OH + H 2 O 2	HO 2 + H 2 O	2.00e+12	0	427
	18 OH + H 2 O 2	HO 2 + H 2 O	1.70e+18	0	29410
	19 OH + CO	H + CO 2	4.76e+07	1.23	70
	20 2 HO 2	O 2 + H 2 O 2	1.30e+11	0 -1630
	21 2 HO 2	O 2 + H 2 O 2	4.20e+14	0	12000
	22 HO 2 + CO	OH + CO 2	1.50e+14	0	23600
	23 OH + HO 2	O 2 + H 2 O	5.00e+15	0	17330

  where W A and W i are molar weights. Sources of carbon containing species of the syngas (CO and CO 2 ) and hydrogen containing species (H 2 , H, H 2 O, OH, HO 2 and H 2 O 2 ) are corrected adding

Table 3 :

 3 Structure of the sequential ANNs used.

	Layer type	Activation Output Shape
		function
	Input	-	(None, 11)
	Dense	ReLU	(None, 512)
	Dense	ReLU	(None, 256)
	Dense	ReLU	(None, 128)
	Dense	ReLU	(None, 64)
	Dense	ReLU	(None, 16)
	Dense (Output) -	(None, 11)

In a sequential ANN, each layer has exactly one input tensor and one output tensor.

A dense layer is driven by the same formulas as the linear layers, but the end result is passed through a non-linear function called 'Activation function'.

Acknowledgments

This work is supported by ADEME (Agence de l'Environnement et de la Maîtrise de l'Energie) under the project IGAR (Injection de Gaz de Recycle) Grant Nb 1882C001. Computing resources were provided by CRIANN (http://www.criann.fr).


Kaidi Wan

Camille Barnaud

Luc Vervisch
email: vervisch@coria.fr

Pascale Domingo

Machine learning for detailed chemistry reduction in DNS of a syngas turbulent oxy-flame with side-wall effects

Keywords: Convolutional neural network, Chemistry reduction, Direct numerical simulation, Syngas

come

Machine learning for detailed chemistry reduction in DNS of a syngas turbulent oxy-flame with side-wall effects

Introduction

The chemical mechanisms for hydrocarbon fuel combustion can involve up to hundreds of species and thousands of reactions [START_REF] Curran | Developing detailed chemical kinetic mechanisms for fuel combustion[END_REF]. Because of the very large number of degrees of freedom to be solved, such fully detailed mechanisms can hardly be introduced in threedimensional (3D) numerical simulations of flames. In this context, the downsizing of chemical kinetics, still securing the prediction of given major thermochemical properties, is mandatory for virtual prototyping of real combustion systems, particularly those involving heavy hydrocarbon liquid or solid fuels [START_REF] Cheng | Development and validation of a generic reduced chemical kinetic mechanism for CFD spray combustion modelling of biodiesel fuels, Combust[END_REF].

Numerous successful strategies have been discussed in the literature to reduce detailed chemical kinetics. They are based mainly on fuel lumping, time-scale and principal component analysis or direct numerical treatment of the chemical responses from graph analysis or genetic algorithms, sometimes associated with tabulation methods [START_REF] Peters | Numerical simulation of combustion phenomena[END_REF][START_REF] Jones | Global reaction schemes for hydrocarbon combustion[END_REF][START_REF] Bradley | Laminar flamelet modeling of recirculating premixed methane and propane-air combustion[END_REF][START_REF] Maas | Simplifying chemical kinetics: Intrinsic lowdimensional manifolds in composition space[END_REF][START_REF] Lam | The CSP method for simplifying kinetics[END_REF][START_REF] Van Oijen | Modeling of complex premixed burner systems by using flamelet-generated manifolds[END_REF][START_REF] Bykov | The extension of the ILDM concept to reaction-diffusion manifolds[END_REF][START_REF] Jones | Reduction of comprehensive chemistry via constraint potentials[END_REF][START_REF] Pepiot-Desjardins | An efficient error-propagationbased reduction method for large chemical kinetic mechanisms[END_REF][START_REF] Boivin | An explicit reduced mechanism for H 2 air combustion[END_REF][START_REF] Jaouen | Automatic reduction and optimisation of chemistry for turbulent combustion modelling: Impact of the canonical problem[END_REF][START_REF] Koniavitis | Reduction of a detailed chemical mechanism for a kerosene surrogate via RCCE-CSP[END_REF][START_REF] Chang | Construction of a skeletal oxidation mechanism of n-pentanol by integrating decoupling methodology, genetic algorithm, and uncertainty quantification[END_REF][START_REF] Scholtissek | A selfcontained composition space solution method for strained and curved premixed flamelets[END_REF][START_REF] Wen | A three mixture fraction flamelet model for multi-stream laminar pulverized coal combustion[END_REF].

Learning from a database to then reproduce the chemical response has received extensive attention in tabulation of combustion chemistry. Artificial neural networks (ANNs) were found accurate, and both CPU and memory efficient [START_REF] Christo | An integrated PDF/neural network approach for simulating turbulent reacting systems[END_REF][START_REF] Franke | Tabulation of combustion chemistry via Artificial Neural Networks (ANNs): Methodology and application to LES-PDF simulation of Sydney flame L[END_REF]. Along similar lines, convolutional neural networks (CNNs), which were originally developed for analyzing visual representations [START_REF] Lecun | Deep learning[END_REF], have been introduced in turbulent combustion modeling for direct deconvolution of the filtered progress variable [START_REF] Nikolaou | Progress Variable Variance and Filtered Rate Modelling Using Convolutional Neural Networks and Flamelet Methods[END_REF], for modeling the unresolved flame surface wrinkling [START_REF] Lapeyre | Training convolutional neural networks to estimate turbulent sub-grid scale reaction rates[END_REF] and for the direct reconstruction of unresolved sources and fluxes from mesh-resolved quantities in large-eddy simulation (LES) [START_REF] Seltz | Direct mapping from LES resolved scales to filtered-flame generated mani-folds using convolutional neural networks[END_REF]. Compared to ANNs, CNNs reduce the number of connections per layer to stack more of them efficiently, thus increasing the depth of the neural network.

In the present study, a novel chemistry reduction approach based on CNNs is proposed and also coupled with a three-dimensional direct numerical simulation (DNS) for validation. A turbulent non-premixed oxy-flame interacting with a cooled wall is considered. This configuration covers a wide range of combustion and turbulence conditions in terms of equivalence ratio, progress of reaction and enthalpy including thermal and viscous boundary layers effects, making it a relatively generic test case. The syngas fuel composition is representative of a CO 2 free blast-furnace exhaust flux, as found in a recycling combustion process. The input of the CNN are two-dimensional (2D) images of the species mass fractions and temperature, while the output are the corresponding chemical burning rates at the center point of the image. The training is performed from a simplified two-dimensional planar DNS, with detailed chemistry and input images built from a subset of the full species vector. This subset of 11 species is chosen to match the major and minor species retained in a reduced chemical mechanism optimized for syngas. Once the CNN trained, this reduced set of species becomes the input to get the corresponding chemical rates. Then, only this reduced set of species needs to be transported with the flow, saving much CPU time and reducing memory requirement.

The objective is to test the feasibility of the novel method, which should be applicable to any reference detailed chemical mechanism. The generic GRI-3.0 detailed mechanism [24] without NO x (217 elementary reactions and 35 chemical species) is used in this first attempt. Once trained for reproducing the chemical sources of a reduced set of 11 species, the CNN for chemistry reduction is tested a priori against the detailed chemistry DNS. Then, the obtained neural network is fully coupled with the flow solution for two-and threedimensional DNS of the non-premixed flame-wall interaction. The two-dimensional cases are with the detailed chemistry, the reduced chemical mechanism and the CNN chemistry. The three-dimensional cases are with the CNN and the reduced chemistry.

Turbulent flame configuration and numerics

A turbulent non-premixed oxy-flame flame under a pressure of 341.3 kPa and featuring side-wall effects is simulated (Fig. 1). A representative syngas composition is injected through the upper inlet (width of 2 mm) at 1223 K and bulk velocity of 50 m/s, with H 2 /0.0085, CO/0.7852, CO 2 /0.0514, N 2 /0.1549 in mass fraction. Below the splitter plate of thickness 0.2 mm, the lower inlet (width of 10 mm) supplies pure oxygen at 298 K and 100 m/s. A preliminary simulation of an oxygen turbulent periodical 3D channel flow at the oxygen bulk velocity serves as a database to impose a proper turbulent inlet to the oxygen stream. A laminar channel flow profile is imposed for the fuel stream, between the wall and the splitter plate. A non-slip wall is placed at the top of the domain while a symmetrical boundary condition is used at the bottom. Periodic boundary conditions are employed in the spanwise z direction (Fig. 1) and a convective boundary condition is used for the outflow. The wall temperature T w (x, t) is nonuniform and computed assuming that the heat flux locally reaching the wall is evacuated on its other side by an efficient water cooling with a convective heat transfer h = 3300 W • m -2 • K -1 , with a water temperature T o = 353 K (i.e., h(T w -T o ) = λ(∂T/∂n w ) gas , n w is the wall normal direction). The physical dimensions of the computational domain composed of 66.44M cells are L x = 100 mm, L y = 12 mm and L z = 7.68 mm. The mesh is uniform in x (streamwise) and z (spanwise, not considered in 2D) directions, with the grid spacing h x = h z = 60 µm. In the y (vertical) direction, the mesh keeps a fine resolution of 20 µm for the upper region close to the wall (y > 10 mm) where the flame is located. 1 The mesh gets coarser as y decreases from 10 to 0 mm to reach a grid spacing of 100 µm at the bottom of the computational domain. The first point off the 1 Simulations of one-dimensional canonical strained non-premixed flames were conducted to determine the grid resolution for the present free stream conditions. top wall is at y + = 0.55 where the superscript + denotes the usual non-dimensionalization by the viscous length scales. There are 9 grid points within y + = 10 to resolve the viscous layer near the wall [START_REF] Gruber | Turbulent flamewall interaction: a direct numerical simulation study[END_REF]. It was verified from a non-reactive simulation that the structure of the wall boundary layer agrees with what is expected (Fig. 2). The time step is fixed at 0.3 µs (CFL≈0.8) with subiteration in a stiff-chemistry ODE solver.

The governing equations for the gas mixture are solved with a low-Mach number in-house code [START_REF] Wan | Alkali metal emissions in an early-stage pulverized-coal flame: DNS analysis of reacting layers and chemistry tabulation[END_REF][START_REF] Wan | Online-CPD-coupled large-eddy simulation of pulverized-coal pyrolysis in a hot turbulent nitrogen jet[END_REF][START_REF] Wan | Large-eddy Simulation of Pilot-assisted Pulverized-coal Combustion in a Weakly Turbulent Jet[END_REF][START_REF] Wan | Modelling alkali metal emissions in large-eddy simulation of a preheated pulverised-coal turbulent jet flame using tabulated chemistry[END_REF][START_REF] Wan | Numerical study of HCl and SO2 impact on sodium emissions in pulverized-coal flames[END_REF]. The conservation equations for mass, momentum, species and temperature of the gas phase are those of [START_REF] Wan | Numerical study of HCl and SO2 impact on sodium emissions in pulverized-coal flames[END_REF], here without the source terms due to coal particles. The flow viscosity and the thermal conductivity of the gas mixtures are computed from Wilke's type mixing rules. The Lewis number is assumed to be unity for all the species. The numerics is based on an approach previously used for both DNS and LES [START_REF] Pierce | Progress-variable approach for largeeddy simulation of non-premixed turbulent combustion[END_REF][START_REF] Desjardins | High order conservative finite difference scheme for variable density low Mach number turbulent flows[END_REF]. The time advancement relies on a second-order Crank-Nicolson scheme. A bounded quadratic upstream interpolation for convective kinematics (BQUICK) scheme is used for the scalar advection terms in the species and temperature equations, while a second-order central difference scheme is applied to the scalar diffusion terms in the species and temperature equations and all terms in the momentum equation. An alternating direction implicit (ADI) method has been employed, in which semiimplicit tridiagonal/pentadiagonal equations are solved Because the syngas is preheated to 1223 K, rapid ignition occurs after mixing with pure oxygen and a nonpremixed flame develops in the computational domain (Fig. 1). After the flame is established, the simulation proceeds for 2 flow-through times (2 ms) based on the bulk velocity of the O 2 stream.

The training database is built saving every 50 µs the 0.52 million 2D DNS fields of species mass fractions and temperature together with their sources, leading to 40 snapshots each containing 520,000 data. To augment the database up to 80 snapshots, the input data are perturbed applying 2% of uncorrelated random noise. Then, over the 80 snapshots, one of every 5 is selected to form the subsequent testing database (16 snapshots), while the left 64 snapshots constitute the training database.

In order to train the CNN with the two-dimensional simulations, a 2D test-box is constructed around every i-th DNS node with its neighboring 8 nodes (Fig. 3). Selecting the 8 neighboring nodes with the target node at the center, to form a 3x3 2D image, appears as the most efficient procedure with the smallest size of CNN input. This test-box is used to built the i-th image composed of 9 points from the 2D distributions of the vector φ(x ,

t) = (Y 1 (x , t), • • • , Y N (x , t), T (x , t)) for = 1, • • • , 9
, which contains information on the N = 10 species mass fractions and temperature (N 2 not included). The "target" (or "label") of each image is the vector of the reaction rates at the x-centred i-th point, ω[i] = (φ(x, t + δt) -φ (x, t))/δt, with δt = 0.3µs and where φ denotes the solution before the ODE stiffchemistry solver is applied. Casting the "targets" (i.e., the sources) in this form presents many advantages. Indeed, the output of the CNN can be used either as a chemical source ω[i] for integration, or directly to read from the CNN φ(t + δt) = ω[i] × 0.3 • 10 -6 + φ (t) in a flow solver relying on splitting for chemistry integration where φ (t) is known. In the latter option, sub-iterations may be needed for δt > 0.3µs. In our simulations, the time step stays the same between training and usage of the CNN, therefore no linear approximation is actually performed. However, care should be taken that splitting method for chemistry integration can decrease the overall order of accuracy of the solver.

To secure the atomic mass balance over a reduced set of species whose sources ωi are learned from a detailed mechanism, the following procedure is adopted. (In practice, over the computational domain, the total default mass to be corrected represents only 0.016% of the total mass.) First, the atomic sources mass default are computed for carbon, hydrogen and oxygen. Let us denote α A,i the number of atom A (i.e., A ≡ C, H or O) in the i-th species. The mass imbalance in A source may be written

∆ ωA = N i=1 [(α A,i W A )/W i ]
A = N i=1 α A,i (W A /W i )Y i .
For oxygen atom, the correction is applied only to O 2 and with the already corrected sources of C and H containing species. Then, atomic budgets are perfectly preserved by the chemical sources learned by the CNN. 46% of the images have a central node temperature below 320 K, these points with negligible sources are dropped out from the database. This leads to a final training database of 14.6 GB and a testing database of 3.7 GB. All data contributing to the "images" and to the "targets" are normalized by their respective maximum absolute value over all the database, these 22 maximum levels are stored (10 species mass fractions, temperature and sources -N 2 not involved). The complete structure of the sequential CNNs is summarized in Table 1. The CNN architecture is inspired from [START_REF] Seltz | Direct mapping from LES resolved scales to filtered-flame generated mani-folds using convolutional neural networks[END_REF], with two additional dense layers to secure the performance of the regression task required here. In total, the neural network contains 237,579 weights, which need to be adjusted.

To improve the precision of the CNNs for small values of the chemical sources, a second network with exactly the same structure is trained for a subset of the database defined by Y O 2 > 0.9 at the central node (i.e., where overall the burning rates stay very low). The subset-database is normalized based on its maximum absolute values. In the following, the networks are referred to as the main CNN and the second CNN (subset for lower burning rates).

The training of the CNNs is performed using the TensorFlow Python library with GPU support (www.tensorflow.org).

A series of convolution/sampling operations are performed iteratively, in which the neural weights are adjusted until a satisfying minimal error is achieved between the "target" sources and the ones predicted by the CNNs. The Adam optimizer [START_REF] Kingma | A Method for Stochastic Optimization[END_REF] is employed with a meansquared-error (MSE) loss function comparing the predictions to the target. A training step is performed with a batch of 10240 "images", leading to 1740 steps per "epoch"/iteration for the main CNN. The learning rate [START_REF] Aghdam | Guide to convolutional neural networks[END_REF] is initially set to 0.001 to decay versus the steps (n) according to l r = 0.001/(1.0 + 0.0001 • n). The network converges in approximatively 500 epochs, requiring for the main CNN 15 hours on an Nvidia GeForce GTX 1080 Ti GPU.

Testing phase

The performance of the machine learning reduced chemistry is first examined in a-priori manner by comparing chemical sources from CNN and GRI-3.0. The part of the database unseen during training is probed. The mean-square error between the predictions and the targets is 1.5×10 -7 for the main CNN and 4.5×10 -8 for the second CNN. Figure 4 displays the predicted chemical sources against their reference values for the main CNN, confirming the accuracy of the approach. The second CNN achieves similar performance in this a-priori validation test (not shown for brevity).

CNN-DNS full coupling

The CNN reduced chemistry is now employed in the simulation of the syngas non-premixed flame (only CPU is used in this part with CNNs exported in C language, see supplementary material). Simulations are then performed with detailed chemistry (2D), CNN chemistry (2D and 3D) and a reduced chemistry (2D and 3D) given in Table 2. This reduced mechanism was derived from GRI-3.0 for the present syngas operating condition including wall heat loss effects, with the automated approach reported in detail in [START_REF] Jaouen | Automatic reduction and optimisation of chemistry for turbulent combustion modelling: Impact of the canonical problem[END_REF][START_REF] Jaouen | Auto-thermal reforming (ATR) of natural gas: An automated derivation of optimised reduced chemical schemes[END_REF] 2). Dashed line: CNN chemistry. 2D case.

Table 2: Reduced chemistry for syngas oxy-flame. Units are mol, s, cm 3 , cal and K. The Chaperon efficiencies of the GRI-3.0 mechanism are preserved for both three-body and fall-off reactions (see also supplementary material). -2D simulations: The results of CNN-DNS are compared to that of the detailed GRI-3.0 and the 11-species reduced mechanims of Table 2 in Figs. 5 and6 (streamwise and spanwise distributions). Statistical means are computed after the flow is established (10 flow through times). The averaged streamwise velocity distribution along the planar fuel jet centerline decreases until x ≈ 8 mm (Fig. 5), location where the fuel ignites and agation [START_REF] Pepiot-Desjardins | An efficient error-propagationbased reduction method for large chemical kinetic mechanisms[END_REF][START_REF] Lu | A directed relation graph method for mechanism reduction[END_REF] and a genetic algorithm [START_REF] Holland | Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control and Artificial Intelligence[END_REF]. The reaction rates of the reduced mechanism are optimized against the response of the detailed mechanism in non-adiabatic non-premixed canonical problems involving turbulent micro-mixing as in [START_REF] Jaouen | Auto-thermal reforming (ATR) of natural gas: An automated derivation of optimised reduced chemical schemes[END_REF].

n o Reaction A β E a 1 2 O +M O 2 +M 1.20e+17 -1.00 0 2 O + H 2 H + OH 3.
the mass fractions of CO and H 2 decrease. The nonpremixed turbulent flame scalar structure is as expected, the syngas fuel composed of CO and H 2 is mainly transformed into CO 2 and H 2 O (Fig. 5), with a slowing down of chemistry close to the cooled wall. Because of the overall fuel-lean character of the system, the mass fraction of O 2 increases along the centreline. According to Figs. 5 and6, the coupled CNN-DNS reproduces very well the scalar structure of the nonpremixed planar jet flame, with a close agreement against the same DNS with the detailed GRI-3.0 mechanism. Interestingly, for the same set of species transported with the flow, the reduced mechanism tends to over-predicts the mass fraction of H radical while the CNN-DNS keeps a close agreement with the reference detailed mechanism (Fig. 6). The prediction of the radical H chemical source may actually benefit from the image-type treatment by CNN, in which the 2D distribution of species is used as input. The averaged departure of wall heat flux between detailed and CNN (respectively reduced) chemistry is 3% (respectively -1%). With the CNN, the standard deviation of the departure represents 6.5% of the mean heat flux.

-3D simulations: The statistics of the 3D run with CNN and the reduced mechanism are initialized from scratch and collected from 1 flow though time, with additional averaging over the 128 spanwise planes, after the initial condition has been evacuated from the computational domain and the turbulence is fully developed. Thereby the memory of the initial condition cannot affect the results. Even though it is sufficient for training, the twodimensional channel turbulence lacks vortex stretching and the three-dimensional turbulent flow cascade differs in 3D, modifying the turbulence/chemistry interaction and the scalar averages, as seen comparing Figs. 6 and7. The predictions of the CNN-DNS still match those by the reduced mechanism DNS, but for H mass fraction, the reduced mechanism predicting again a higher level than the CNN. Based on above comparisons against GRI-3.0 in 2D, it is legitimate to assume that the scalar flame structure obtained with the CNN chemistry is actually closer to the one that would be seen in 3D with the detailed mechanism (out of reach with actual computing power). The averaged departure of wall heat flux between CNN and reduced chemistry is 14%, with a standard deviation of 13.8% of the mean flux.

The comparison of the computational costs due to chemistry for advancing the thermochemical variables, thus excluding the flow transport part, shows that CNNchemistry is 11 times faster than the GRI-3.0 detailed mechanism, and 1.35 times faster than the 11-species reduced mechanism, the latter mainly because the Ar- rhenius rates calculation is avoided.

Summary

A new paradigm grounded on machine learning is discussed to introduce complex chemistry in turbulent flame simulation and implemented in three-dimensional direct numerical simulation of a syngas non-premixed flame/wall interaction. The training of a convolutional neural network (CNN) is performed on a simplified twodimensional simulation with a reference detailed chemistry. The obtained reduced chemistry is then applied to a three-dimensional turbulent flame simulation.

The comparisons against the detailed chemistry response, a reduced mechanism and the CNN confirm the validity of the approach, which allows for conveniently developing reduced chemistry specialized for given operating conditions from a generic detailed scheme and even save CPU time compared to using reduced mechanisms based on Arrhenius rates.

One major drawback of the presented approach lies in the need to simulate a simplified two-dimensional version of the target three-dimensional flow geometry to train the CNN. However, in future works, generic training procedures will certainly emerge, so that actually performing a flow simulation will not be mandatory to optimize the neural networks. Along these lines, the use of non-premixed stochastic micro-mixing model problems appears as a promising alternative to perform CNN training in a pre-processing mode. Because it has already been successfully used as a canonical problem, prior to any flow simulation, in the context of automated chemistry reduction [START_REF] Jaouen | Automatic reduction and optimisation of chemistry for turbulent combustion modelling: Impact of the canonical problem[END_REF].

Figure 1 :

 1 Figure 1: Instantaneous iso-surface of Q criterion [25] predicted by CNN-DNS: Q = 10 7 s -2 , colored by gas temperature. Top graph: distribution of instantaneous wall temperature along the centreline.

Figure 2 :

 2 Figure 2: Velocity distribution in wall units for 3D non-reactive case (mesh verification).

Figure 3 :

 3 Figure 3: CNN training from 2D database.

Figure 4 :

 4 Figure 4: Comparison of the main CNN prediction on normalized sources of the 10 species and temperature against the GRI-3.0 target values for the testing database unseen during training.

  .

2

 2

Figure 5 :

 5 Figure 5: Averaged distributions of streamwise velocity and major species mass fractions: CO 2 , CO, O 2 , H 2 and H 2 O along the planar fuel-jet centreline (y 11 mm). Symbols: GRI-3.0. Solid line: Reduced mechanism (Table2). Dashed line: CNN chemistry. 2D case.

Figure 6 :

 6 Figure 6: Spanwise averaged distributions of CO 2 , O 2 , CO, OH, H mass fractions and temperature. Symbols: GRI-3.0. Solid line: Reduced mechanism (Table 2). Dashed line: CNN chemistry. (a): 15 mm from the nozzle inlet. (b): 50 mm from the nozzle inlet. (c): 80 mm from the nozzle inlet. 2D case.

Figure 7 :

 7 Figure 7: Averaged distributions of CO 2 , O 2 , CO, OH, H mass fractions and temperature. Solid line: Reduced mechanism (Table 2). Line with symbols: CNN chemistry. (a): BFG jet centerline (streamwise). (b): 15 mm from the nozzle inlet (spanwise). (c): 80 mm from the nozzle inlet (spanwise). 3D case.

Table 1 :

 1 Structure of the sequential CNNs used (see[START_REF] Aghdam | Guide to convolutional neural networks[END_REF] for details on functions).Figure3sketches the overall CNN training process from the 2D database, which is applied to the species O, O 2 , H, H 2 , OH, HO 2 , H 2 O, H 2 O 2 , CO and CO 2 (NO x being not included, no source is computed for N 2 ).

	Layer type	Activation Output Shape
		function
	Input	-	(None, 3, 3, 11)
	Conv2D	ReLU	(None, 3, 3, 64)
	MaxPooling2D -	(None, 2, 2, 64)
	Conv 2D	ReLU	(None, 2, 2, 256)
	MaxPooling2D -	(None, 1, 1, 256)
	Flatten	-	(None, 256)
	Dense	ReLU	(None, 256)
	Dense	ReLU	(None, 64)
	Dense	ReLU	(None, 16)
	Dense (Output) -	(None, 11)
	separately for each direction.
	3. CNN chemistry reduction methodology
	3.1. Training phase

  ωi , where W A and W i are molar weights. Second, sources of carbon containing species of the syngas (CO and CO 2 ) and hydrogen containing species (H 2 , H, H 2 O, OH, HO 2 and H 2 O 2 ) are corrected adding ∆ ẇi = -(Y i /Y A )∆ ωA , with the atom mass fraction Y

This method combines a stochastic probing of the detailed chemistry response with directed relation graphs analysis with error prop-

Acknowledgments

This work is supported by ADEME (Agence de l'Environnement et de la Maîtrise de l'Energie) under the project IGAR (Injection de Gaz de Recycle) Grant Nb 1882C001. Computing resources were provided by CRIANN (http://www.criann.fr).