## Abstract

Inferring local soot temperature and volume fraction distributions from radiation emission measurements of sooting flames may involve solving nonlinear, ill-posed and high-dimensional problems, which are typically conducted by solving ill-posed problems with big matrices with regularization methods. Due to the high data throughput, they are usually inefficient and tedious. Machine learning approaches allow solving such problems, offering an alternative way to deal with complex and dynamic systems with good flexibility. In this study, we present an original and efficient machine learning approach for retrieving soot temperature and volume fraction fields simultaneously from single-color near-infrared emission measurements of dilute ethylene diffusion flames. The machine learning model gathers information from existing data and builds connections between combustion scalars (soot temperature and volume fraction) and emission measurements of flames. Numerical studies were conducted first to show the feasibility and robustness of the method. The experimental Multi-Layer Perceptron (MLP) neural network model was fostered and validated by the N_{2} diluted ethylene diffusion flames. Furthermore, the model capability tests were carried out as well for CO_{2} diluted ethylene diffusion flames. Eventually, the model performance subjected to the Modulated Absorption/Emission (MAE) technique measurement uncertainties were detailed.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In the context of emission regulations increasingly stringent on all kinds of engines, soot formation has become an important issue, since it affects adversely the environment, human health and radiative heat transfer in the combustion chambers [1]. However, understanding soot particles inception, growth, oxidation and interaction in combustors is still an open field. To address the remaining issues relevant to the soot inception and oxidation processes, the accurate and consistent soot benchmark databases, i.e., soot diameter, temperature and volume fraction are essential. This consensus was further confirmed in the recent International Sooting Flame (ISF) workshop in Germany. A variety of non-intrusive optical-based techniques were developed to obtain high spatial resolution as well as high accuracy and precision of soot information in the laminar target flames. Arana et al. [2] studied a series of partial air premixed ethylene flames with the equivalence ratio from infinite (non-premixed) to 3 via the laser extinction measurements and have provided the soot volume fraction fields. Liu et al. [3] used the spectral soot emission (SSE) technique to probe and document the soot volume fraction and temperature fields variation of the N$_{2}$ and CO$_{2}$ diluted ethylene diffusion flames within the pressure from 5 to 20 atm. Recently, Sun et al. [4] documented and offered the soot volume fraction, gas temperature and primary soot diameter fields with the N$_{2}$ and H$_{2}$ diluted ethylene diffusion flames through the planar laser techniques of laser-induced incandescence (LII), two-line atomic fluorescence (TLAF), and time-resolved LII (TiRe-LII), respectively. Also, Wang et al. [5,6] studied two series of N$_{2}$ and O$_{2}$ diluted ethylene diffusion flames via the Modulated Absorption/Emission (MAE) technique and offered the soot volume fraction and temperature fields with a wider range of dilution fractions.

Except for the planar laser techniques, all the above-mentioned techniques are relevant to the line-of-sight signal collections, and further tedious deconvolution processes, i.e., onion-peeling, Abel inversion, etc., are required in the post-processing to retrieve the local soot information. Although some regularization methods, i.e., Tikhonov regularization is applied to minimize the error susceptibility and propagation, the retrieval accuracy at the flame central region is still limited due to severe ill-posed and non-linear conditions. On the other hand, the machine learning method offers an alternative solution, not only ensures the retrieving accuracy, also substantially reduces the post-processing costs. Machine learning extracts information from data automatically by computational and statistical methods to find relations between inputs and outputs even if the relations between dependent and independent variables are not clear. In the combustion field, the machine learning approaches have been applied for early ignition prediction [7] and turbulent sub-grid scale reaction rates estimation [8]. Most recently, Ren et al. [9] developed an inverse radiation model based on the MLP neural network method to retrieve temperature and gas species volume fraction distributions from infrared spectral emission measurements for combustion gas mixtures. The developed method showed excellent efficiency and capability to retrieve temperature and gas species volume fractions simultaneously for a gas mixture of CO$_2$, H$_2$O, and CO.

In the present study, a MLP neural network approach is developed and applied to non-smoking axis-symmetric flames to retrieve soot temperature and volume fraction fields simultaneously from single-color near-infrared emission measurements of flames. The robustness of this machine learning approach is demonstrated with simulation data first. Then, this machine learning approach is further experimentally verified by the N$_{2}$ diluted flames data and the approach prediction results of CO$_{2}$ diluted flames are compared with the scalar fields retrieved from the MAE technique (by contrast, the MAE technique requires two-color emission measurements to obtain soot temperatures and volume fractions). The study has demonstrated that the machine learning method can be explored for soot information retrieval, by providing reliable training data from numerical simulations or/and experimental measurements.

## 2. Soot temperature and volume fraction retrievals

In the present study, the target flame is a laminar coflow ethylene/air diffusion flame with emitting, absorbing, but non-scattering medium. As has been previously presented in Ref. [10], the spectral energy $E_\lambda$ along a line-of-sight between $\textit {y}_{\textrm {min}}$ and $\textit {y}_{\textrm {max}}$ and accumulated on the detector’s pixel can be expressed as Eq. (1),

Due to the axis-symmetric nature of these flames, the onion-peeling method can be employed. At any given height $z$ above the burner, half of the flame cross-section is shown in Fig. 1, in which the flame field is divided into $N$ evenly spaced annular elements. The temperatures and soot volume fractions are assumed to be uniform over each element. If one measures flame emission with $N$ number of pixels, each pixel collects emission from one line-of-sight as shown in Fig. 1, a system of nonlinear equation results,

Here $f$ is a nonlinear function of the temperatures and soot volume fractions of all elements. $\textbf {E}_\lambda =\left [E_{\lambda ,i}\right ]$ is the flame spectral emission detected by all pixels and $\textbf {X} = [\textbf {T}_i,\textbf {SVF}_i]$ is the local scalar vector of soot temperature and volume fraction of all annular elements. Equation (2) shows the relationship between detected spectral flame emissions and soot temperatures, as well as soot volume fractions. By solving this equation with the regularization methods, soot temperatures and volume fractions can be decently retrieved from the flame spectral emission measurements, but inefficiently.#### 2.1 Modulated absorption/emission technique

The MAE technique was extensively described in previous studies [5,6,10]. Only the major features are briefly reminded. Instead of retrieving soot temperature and volume fraction directly by solving the nonlinear Eq. (2), the MAE captures the line-of-sight laser extinction and flame radiation images sequentially at red and near-infrared wavelengths, i.e., 645 nm and 785 nm. Then the local absorption coefficient $\kappa _\lambda$ and the local emission rate $\kappa _\lambda I_{b,\lambda }$ at both wavelengths are retrieved by the onion-peeling method with Tikhonov regularization. The soot volume fraction field is calculated from the red absorption coefficient by modeling of the soot refractive index. In contrast, the soot temperature field is retrieved by the ratio of the local spectral emission rates from the two wavelengths, which is free of soot refractive index wavelength-dependent issue. Therefore, soot temperatures and volume fractions are not retrieved simultaneously (at least 30 ms time gap) and the reconstruction processes require careful regularization parameter selection in solving ill-posed problems with big matrices in every diluted flame case.

#### 2.2 Machine learning methodology

To retrieve soot temperature and volume fraction fields simultaneously from the flame emissions, the machine learning method provides solution models for such nonlinear problems when explicit inverse relation, i.e., $f^{-1}$ of Eq. (2), between flame radiation and soot scalars is not available. Inspired by the biological neural network information processing, artificial neural networks are a group of algorithms used for machine learning that model data processing by artificial neurons [11]. By training on a dataset with many inputs and outputs, avoiding solving problems with ill-posedness and huge matrices, a model is generated that can be used to predict new examples from the same type of input features.

The MLP neural network is one of the most popular types of artificial neural networks in machine learning [12,13]. Figure 2 shows a representative MLP neural network architecture for soot temperatures and volume fractions retrieval from spectral emission measurements. The MLP consists of an input layer, one or more hidden layers, and one output layer. Each layer comprises several nodes called neurons. Neurons of one layer are directly connected to the next layer by weights. As shown in Fig. 2, the leftmost layer, known as the input layer, consists of a set of neurons representing the input features (near-infrared spectral emissions). Each neuron in the hidden layers transforms the values from the previous layer with a weighted linear summation, followed by a nonlinear activation function (The readers are referred to Ref. [9] for more details about how data are processed within neurons). The output layer receives the values from the last hidden layer and transforms them into output values (soot temperatures and volume fractions). The numbers of neurons in the input and output layers are determined by the input and output dimensions, respectively. However, there is no specific approach to determine the number of hidden layers and their neurons for different problems, the optimal choice is usually made by trial and error [14].

An important part of modeling with the neural network is the so-called training of the network (learning). Training neural networks is done by adjusting appropriate weights $\textbf {W}$ between neurons to minimize the error of the cost function so that the output values generated by the network are compared with the actual corresponding values. The cost function here is,

For machine learning using the artificial neural network, a large set of training data has to be routinely available [9,17]. These data can be either from experimental measurements or numerical simulations, or both. The training inputs are flame near-infrared emission data along a row of pixels at any height $z$ above the burner and the training outputs are soot temperature and volume fraction distribution data along the center of the flame at the same height above the burner, as shown in Fig. 2. In the present study, numerical investigations are conducted first with training data of machine learning model from simulated combustion fields and calculated spectral emissions, which is followed by training and predicting soot temperature and volume fraction from measured near-infrared spectral emission for N$_2$ and CO$_2$ diluted ethylene diffusion flames at wavelength $\lambda$ of 785 nm and the scalar values obtained from the MAE technique.

## 3. Robustness of the machine learning method

In order to show the feasibility and robustness of the machine learning model, a numerical study is conducted first. The numerical fields of temperature and soot volume fraction provided by Blacha et al. [18] are used to compute the theoretical flame emission signals captured by a matrix of pixels of a camera, as shown in Ref. [10] and outlined as Eq. (1) here. Without calibration, here we simply assume that the calibration constant $K$ is unity. The training data of temperatures, soot volume fractions and the calculated near-infrared spectral emission are perturbed with two sets of Gaussian random noises of 5% and 10%. After training of MLP neural network, the calculated spectral emission with the same level of noises added is fed into the neural network to predict new soot temperature and volume fraction distributions. For these cases, 4 hidden layers are used for the MLP neural networks and each layer has 400 neurons, and the regularization parameter $\alpha$ is 1.0. Figure 3 demonstrates the comparison between ideal (without noise) temperature and soot volume fraction fields and the retrieved ones from the MLP neural network predictions. As shown in the figure, even with 10% of random noises in the training and testing data, temperature and soot volume fraction fields are reconstructed very well.

The correlation between the MLP predicted scalar values and the ideal ones for the 5% case are shown in the upper two frames of Fig. 4, and the retrieved scalar distributions at the height above burner $z$ = 40 mm are shown in the lower two frames in Fig. 4. The noisy training data is also present for comparison, where the shadow areas represent 5% standard deviation intervals. As indicated in the figure, the discrepancies between the retrieved and ideal values do not depend on the location within the flame for both temperature and soot volume fractions. Despite relatively large noises in the training and testing data, almost all the retrieved temperatures and soot volume fractions are within 50 K and 0.3 ppm discrepancies from the ideal values. When increasing the noise level in the training and testing data to 10%, the training temperatures and soot volume fractions can be as high as 400 K and 2 ppm away from the ideal values, respectively. Even though the MLP neural network model recovers the ideal temperatures and soot volume fractions quite well and the discrepancies from the ideal values are well within 100 K and 0.5 ppm, as indicated in Fig. 5.

These numerical tests by using artificial synthetic numerical data show that the machine learning approach of MLP neural networks is capable of retrieving temperatures and soot volume fractions simultaneously from near-infrared spectral emission measurement for axis-symmetric sooting flames and is significantly robust.

## 4. Validation with N$_{2}$ diluted flames data

As motioned before, the training data for machine learning model can be either from experimental measurements or numerical simulations, or both. In order to further validate the proposed method, in this section, the experimental measured near-infrared emission of N$_{2}$ diluted ethylene diffusion flames with the soot temperature and volume fraction fields obtained by Wang et al. [5] with the MAE technique, are provided as the inputs and outputs for training and testing MLP neural network models, since both flame emission and scalar fields data are available. In Ref. [5], the N$_{2}$ was gradually added to the fuel stream of a basic Santoro flame with an increasing N$_{2}$ volume fraction from 0 to 56% and the flame emissions were measured at each condition. The soot temperatures and volume fractions were then inferred with the MAE technique. A total of eight sets of experiments have been conducted, and each flame measurement is with a 2-D field of 915$\times$80 pixel in dimension, which could provide 915 sets of independent radial distributed data for training a machine learning model. Therefore, the total number of training data available is $915\times 8= 7320$.

In order to assess how much data are good enough for training a machine learning model, six cases of machine learning models were trained independently with different sets of flame data. As shown in Table 1, for example, Case-A only used the flame data without N$_2$ dilution for training; and Case-F applied data from flames with N$_2$ volume fractions of 0%, 8%, 28%, 32%, 46% and 56%. Here a Bayesian optimization [19] process was applied to select the optimal MLP neural network hyperparameters (the number of hidden layers, the number of neurons in each of the hidden layers and the regularization parameter $\alpha$) for each of the cases listed in Table 1. The Bayesian optimization maximizes the testing score of each machine learning model, and all the eight flames data were used for testing. The training and testing scores are both the $R^2$ score, also know as the coefficient of determination, is defined as,

During training of the machine learning models when the cost function of Eq. (3) is not improving by at least 0.01%, convergence is considered to be reached and training stops. After training, six machine learning models were obtained. In order to quantitatively show its flexibility to noise and performance to never-seen-before data for each of the trained models, Gaussian random noises of 5% were added to all the emission data for the eight flames; these data were then fed into these models to predict temperature and soot volume fraction distributions, and compared against the soot temperatures and volume fractions retrieved by the MAE technique. The MLP predicted temperatures and soot volume fractions from the 6 trained models for the eight flames are presented in Figs. 7 and 8, where the 2-D temperature and soot volume fraction distributions are shown, respectively. For these flames, the soot is constantly formatted at the flame annular region, while the soot existing domain and volume fraction were significantly decreased. The soot temperature inferred by the MAE technique was limited to the ratio confidence region [10], where the soot volume fraction is relatively high and correspondingly resulted in shrinkage of measurable soot temperature region with the increasing of diluted N$_2$ volume fraction. Therefore, the temperatures inferred with the MAE technique were only within the ratio confidence region and the temperatures elsewhere were not obtained. As indicated in Figs. 7, because that is the feature of the training data and the MLP models learned from that feature and also preserve that feature.

Training machine learning models require a minimum number of samples, which should also spread evenly over the entire range of interest, since it essentially makes interpolations. For Case-A, it only used the flame data without N$_2$ dilution for training, which only has 915 sets of independent near-infrared emission radial distributions along with the corresponding MAE temperatures and soot volume fractions. As indicated in Figs. 7 and 8, the trained model for Case-A only works well for the two flames where N$_2$ dilutions are 0% and 8%. After adding the $x$N$_2=$ 56% flame data for training, as in Case-B, the prediction performance becomes much better comparing to Case-A, but soot temperatures and volume fractions are still over-predicted for most part of the flames. When more flame data were used for training, the performance becomes much better. It is indicated in Figs. 7 and 8 that the models obtained in Cases-D, E and F show almost the same level of performance for predicting soot temperatures and volume fractions for all the flames. However, Case-D only used four sets of flame data for training, as shown in Table 1. Therefore, the model trained from Case-D can be considered to be a converged case and is good enough for soot temperatures and volume fractions predictions.

Figures 9 shows the results along the radial direction at a height above burner $z= 40$ mm by the machine learning model from Case-D. The MLP predicted soot volume fractions match very well with the MAE retrieved values, while the temperature fields predicted by the MLP model are not fully smooth and have some discrepancies from the MAE values at locations where temperature profiles are discontinued. That is because the current MLP model applied regularization to avoid overfitting, which assumes that a regression function is smooth [20]. However, the training temperature distributions have discontinuities, that makes the predicted temperatures less accurate, especially at where the discontinuities are. Nevertheless, the prediction accuracy from the MLP model is considered as the same level as that of MAE technique. The results in Fig. 9 show that the trained machine learning model from the experimental data is able to predict soot temperature and volume fraction fields very well from similar flame emission measurements, even for never-seen-before data (these testing emission data were perturbed with noises and the flame data for $x$N$_2=$ 8%, 20%, 38% and 46% were never included in the training data in Case-D).

## 5. CO$_{2}$ diluted flames predictions

It should be noted that after training of the MLP neural networks, “global" inverse models were obtained, which can predict soot temperature and volume fraction fields within similar structure flames (same burner with different combustion conditions). Ideally, the previously trained MLP model from the N$_{2}$ diluted flames data for Case-D should also be able to predict soot temperature and volume fraction distributions for similar flames. It was demonstrated by retrieving the scalars capability from the flame emission measurements at different conditions, as indicated in Fig. 10, where predictions were conducted for the CO$_{2}$ diluted flames. Four extra CO$_{2}$ diluted flames experiments are investigated based on the basic Santoro flame again, in which the flow rates of ethylene and coflowing air streams, are fixed at the constant values of 0.231 L/min and 43 L/min, respectively. Then CO$_{2}$ are gradually added to the central fuel stream, resulting in the CO$_{2}$ volume fractions ranging from 8% to 32% of four different flames. And the soot temperature and volume fraction fields for all the flames are probed by the MAE technique for comparison as well. The MAE technique has been briefly reminded in section 2.1, and the detailed post-processing procedures refer to [10], which is same to that of N$_{2}$ diluted flames [5].

Figure 10 shows the comparison of MLP (the machine learning model from Case-D) and MAE retrieved soot temperature and volume fraction distributions for the four CO$_{2}$ diluted flames, i.e., ${x\textrm {CO}_{2}}$ = 8%, 20%, 28% and 38%, and the predicted radial directions at height above burner $z= 40$ mm are presented in Fig. 11. The input emission data for the machine learning model were artificially perturbed with 5% of Gaussian random noises as well. As indicated in these figures, the machine learning model trained with the N$_2$ diluted flame data is also able to predict soot temperature and volume fraction distributions for the CO$_2$ diluted flame from the same burner. The overall predictions are quantitatively well; however, soot volume fractions are slightly overpredicted for the ${x\textrm {CO}_{2}}$ = 8% and 20% flames, and soot temperature is slightly underpredicted for the other two flames. This may be caused by the uncertainties in the original MAE experiments or slightly different calibration constant $K$ between N$_{2}$ and CO$_{2}$ diluted flame emission measurements.

## 6. MLP model performance to noise data

The previous two sections demonstrated that a machine learning model trained based on some limited available experimental data could be used for combustion diagnostics for other similar flames. However, the prediction accuracy highly depended on the reliability of the experimental data used for model training. Provided by [10], the uncertainties for the optical emission measurements and soot temperature and volume fraction retrievals by the MAE technique were up to $\pm$5%. Here a new machine learning model was trained again with the same data in Case-D but with 5% of Gaussian random noises added to all emission, soot temperature and volume fraction data. The hyperparameters for the artificial neural network remain exactly the same as in Table 1. After training, the model (refer to as “noise model") was used to predict soot temperature and volume fraction distributions for all the eight N$_2$ diluted flames and the four CO$_2$ diluted flames again, the testing input emission data were perturbed with 5% of Gaussian random noises as well. The results are presented in Figs. 12–15.

Figure 12 shows the predicted soot temperature and volume fraction fields for the eight N$_2$ diluted flames, and Fig. 13 shows the radial profiles at a height above burner $z= 40$ mm. The corresponding results for the four CO$_2$ diluted flames are shown in Figs. 14 and 15. By comparing these results with the model from Case D (without noises in the training data), the prediction accuracy from the new model (with noises in the training data) barely changes: the machine learning model still performances similarly well with noisy training data.

## 7. Discussion

In fact, we could consider the MLP prediction errors come from two stages (parts) of model data stream. In the first training stage, the model prediction errors inherit from original training data that is a more intrinsic one. It cannot be removed, except for the original data improvement from regularization methods, i.e., Tikhonov regularization or other advanced diagnostic techniques. In the present model (Case-D), such ill-posed incurred errors are significantly reduced in the present Tikhonov regularization MAE technique, in turn, are trivial in the MLP model predictions. In the second MLP model prediction stage, The MLP model immunization to training data uncertainties and input flame near-infrared emission noises (the potential real experimental noise) are demonstrated in Section 6. It indicates that the MLP model barely increases the prediction errors at this stage. Thus, the proposed MLP model is almost immune to the training data uncertainties, i.e., flame near-infrared emission, soot temperature and volume fraction from MAE technique. Thus, we believe, the MLP model prediction errors mainly come from the discontinued locations below the height of 50 mm in the flames. This error may be resolved by MLP model structure optimization or suitable regression function selection. It deserves further investigation in the future works.

In addition, the MLP model offers several following unique benefits, which may not be achieved by other conventional diagnostics.

- (1) The MLP model significantly boosts experimental data post-processing in a decent way. In the same computer configuration (Intel Xeon Gold 6130 processor with 2.10 GHz of processor base frequency), the MLP model simultaneously offers the soot temperature and volume fraction fields (915$\times$80 pixel) in about 2.08s of CPU time. On the other hand, approximately 180s of CPU time is required for obtaining the soot temperature and volume fraction fields in the MAE technique. That is because the local soot extinction field, the red emission field and the infrared emission field are processed separately for the reconstructions of soot volume fraction and temperature fields for the MAE technique. In addition, during the above three deconvolution procedures, the Tikhonov regularization parameters have to be selected manually.
- (2) The MLP approach simplifies the experimental requirements and experimental costs. As shown in Section 4 of the paper, once the model has been trained, one model (the Case-D model) could predict other N$_2$ diluted cases with good prediction accuracy.
- (3) The N$_2$ diluted MLP model offers a reasonable way to predict temperature and soot volume fraction for similar flames, i.e., the CO$_2$ diluted flames, with acceptable prediction accuracy (Section 5).

## 8. Conclusion

A machine learning method based on the Multi-Layer Perceptron neural network for soot temperature and volume fraction fields retrieval from near-infrared flame emission measurements was demonstrated for axis-symmetric sooting laminar flames. The robustness of the MLP method prediction was demonstrated by the numerical combustion simulations with two levels of Gaussian random noise perturbations. The MLP method was experimentally established and validated by the N$_{2}$ diluted series Santoro flames dataset. It was found that using four sets of flame data for training (Case-D) MLP was good enough for soot temperature and volume fractions predictions. The MLP model prediction capability was further tested by the CO$_{2}$ diluted Santoro flames and the similar prediction accuracy of MLP model to that of MAE technique was confirmed via extra MAE measurement comparisons.

The purpose of the present study is to show the applicability of machine learning method for helping combustion diagnostics. The trained model is not globally applicable to other type flames, but the study has shown that with the assistant of this proposed machine learning process, it could reduce the experimental procedures and post-processing costs in simultaneous obtaining the soot temperature and volume fraction fields of soot-eliminated diluted Santoro flames. Furthermore, the information about burner size, calibration constant $K$, correlations between soot temperature and volume fractions may also be used to train machine learning models. That would significantly extend the applicability of the method and a “global" model maybe be obtained.

## Funding

National Natural Science Foundation of China (51706140); Natural Science Foundation of Shanghai (20ZR1426900).

## Disclosures

The authors declare no conflicts of interest.

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