Development of a CFD-Suitable Deep Neural Network Model for Laminar Burning Velocity

Abstract

Hydrogen is a valued resource for today’s industry. As a fuel, it produces large amounts of energy and creates water during the process, unlike most other polluting energy sources. However, the safe use of hydrogen requires reliable tools able to accurately predict combustion. This study presents the implementation of a deep neural network of laminar burning velocity of hydrogen into an open-source CFD solver flameFoam. DNN was developed based on a previously created larger DNN, which was too large for CFD applications since the calculations took around 40 times longer compared to the Malet correlation. Therefore, based on the original model, a faster, but still accurate, DNN was developed and implemented into flameFoam starting with version 0.10. The paper presents the adaptation of the original DNN into a CFD-applicable version and the initial test results of the CFD–DNN simulation.

Full text

Citation: Ambrutis, A.; Povilaitis, M.
Development of a CFD-Suitable Deep
Neural Network Model for Laminar
Burning Velocity. Appl. Sci. 2022,12,
7460. https://doi.org/10.3390/
app12157460
Academic Editor: Talal Yusaf
Received: 30 June 2022
Accepted: 23 July 2022
Published: 25 July 2022
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
published maps and institutional affil-
iations.
Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
applied
sciences
Article
Development of a CFD-Suitable Deep Neural Network Model
for Laminar Burning Velocity
Andrius Ambrutis * and Mantas Povilaitis
Laboratory of Nuclear Installation Safety, Lithuanian Energy Institute, Breslaujos g. 3, LT-44403 Kaunas,
Lithuania; mantas.povilaitis@lei.lt
*Correspondence: andrius.ambrutis@lei.lt
Featured Application: The presented DNN model has been developed for application in the CFD
simulations requiring a simplified estimation of laminar burning velocity in dry hydrogen–air
mixtures. A pair of examples using the progress variable approach, turbulent flame speed closure
model and open-source CFD solver flameFoam are presented at the end of the paper.
Abstract:
Hydrogen is a valued resource for today’s industry. As a fuel, it produces large amounts of
energy and creates water during the process, unlike most other polluting energy sources. However,
the safe use of hydrogen requires reliable tools able to accurately predict combustion. This study
presents the implementation of a deep neural network of laminar burning velocity of hydrogen
into an open-source CFD solver flameFoam. DNN was developed based on a previously created
larger DNN, which was too large for CFD applications since the calculations took around 40 times
longer compared to the Malet correlation. Therefore, based on the original model, a faster, but still
accurate, DNN was developed and implemented into flameFoam starting with version 0.10. The
paper presents the adaptation of the original DNN into a CFD-applicable version and the initial test
results of the CFD–DNN simulation.
Keywords:
turbulent premixed combustion; hydrogen; artificial neural network; CFD; laminar
burning velocity
1. Introduction
The first concepts of non-biological brains were analyzed during World War II by
McCulloch and Pitts [
1
]. Since then, multilayer deep neural networks (DNNs) have been
widely used for numerous research and practical applications. In machine learning, the
algorithm learns from the data, using various transformations of inputs, without a user
or programmer giving it explicit instructions [
2
]. Various other machine learning (ML)
algorithms can also be used as a replacement for neural networks since studies show
that the performance of all ML algorithms depends greatly on the problem itself [
3
,
4
]. In
general, an artificial neural network (ANN) performs better in finding complex behavior
and patterns in large amounts of data. As stated by studies, multi-layer ANNs, such as
deep neural networks (DNN) can reproduce any function with arbitrary precision [5,6].
Simulation of turbulent premixed combustion is challenging numerically due to the
wide range of spatial and temporal scales involved in turbulence and combustion chem-
istry. Therefore, to perform practically relevant simulations, various simplified modeling
approaches are usually employed, e.g., RANS treatment of turbulence or simplified com-
bustion rate estimations instead of direct modeling of chemical kinetics. The most common
approach to avoid chemical simulation is to formulate a used combustion model in a way
that chemistry would be replaced by a simpler estimation of laminar burning velocity
(LBV), e.g., from the correlations based on the experiments.
There are several examples of ANN and other ML methods for the application in the
prediction of combustible mixture properties in the literature. In 2018, Jach et al. [
7
] created
Appl. Sci. 2022,12, 7460. https://doi.org/10.3390/app12157460 https://www.mdpi.com/journal/applsci

Appl. Sci. 2022,12, 7460 2 of 16
three ML models—ANN, SVM and multivariable regression (MR)—for LBVs of mixtures of
air with one of seven hydrocarbons from methane up to n-heptane. The best performance
was obtained by ANN model in terms of R2, RMSE and MAE.
In 2019, Mehra et al. [
8
] were able to create DNN which could predict laminar burning
velocities of HyCONG gas blends with an R-squared value approximately equal to 0.999
and a number of neurons up to 20. Concentrations of blend constituents were used as
inputs. Authors showed that an increase in network weight number decreased the testing
error, which means that DNN with more weights should have higher accuracy. Similar
behavior was noticed for the number of epochs used for model optimization. However,
while more complex networks can show better results, they also are slower and have a
higher risk of overfitting.
In 2020, Pulga et al. [
9
] developed methodology to improve the accuracy of laminar
flame ML simulations, starting from data preparation, model creation and finishing with
model evaluation and results interpretation. Malik et al. presented a light neural network
(as it only had up to five neurons in each layer), which could get good predictions in a
short amount of time [
10
]. Their neural network could be used in real-time and had the
mean squared error equal to 0.3023 (m/s)
2
with hydrogen–air mixtures for laminar burning
velocity predictions. In their work, 577 observations were used for the training of DNN.
Part of them had a high spread and this can lower prediction accuracy. Since the amount
of available experimental data was insufficient to train the neural network, the study
generated new points (up to 7300 observations) for training. Varghese and Kumar [
11
]
analyzed syngas mixtures and created multiple linear regression model to predict their
LBVs with error <10%. Their model was derived partly from the measured velocities, and
partly from the predictions using FFCM-1 kinetic mechanism.
In 2021, Eckart et al. [
12
] compared four different machine learning algorithms pre-
dicting laminar burning velocities of hydrogen–methane mixtures. Their study showed
that the DNN model can give much better predictions than other popular algorithms
such as support vector machines (SVM) or random forest while predicting velocities for
hydrogen–methane mixtures. Correa et al. [
13
] used several ML methods to predict the
research octane numbers of several spark ignition fuels. Study evaluated multiple ML
algorithms using 10-fold cross validation, which ensured that results are less influenced
by noise and more accurately reflects real capabilities of these models. In this study SVM
gave best predictions out of all tested methods. vom Lehn et al. [
14
] created a quantitative
structure–property relationship (QSPR) model for the estimation of LBVs of hydrocarbons
and oxygenated hydrocarbons based on their underlying fuel structures. A set of molecular
groups as well as pressure, temperature and equivalence ratio served as input parameters
to an ANN, which has been trained based on a large database of training data for 124
different compounds.
Recently, Wan et al. [
15
] compared 16 ML algorithms for the prediction of hydrocarbon
and oxygenated fuels LBVs and found out that the Gaussian or Tree-based methods gave
high predictions with most of those models explaining over 90% of all data. Li et al. [
16
]
created ML–QSPR model to screen fuels based on their predicted properties.
In our previous work [
17
], we created a DNN for the prediction of the laminar burning
velocity of hydrogen–air mixtures with a coefficient of determination approximately equal
to 0.985. However, this model was too slow to be used practically (in CFD calculations)—
the testing script could make a million predictions in around 45 s, compared to the Malet
correlation [
18
], with which predictions could be made under the same conditions in less
than a second.
For complex combustion cases, increases in observation quantity can highly improve
predictions of neural networks. However, as the network starts to learn the way to solve
the problem, this effect gets less visible. In the end, if the model overlearns on data, it
can even start to mimic the noise. Therefore, the goal should be to obtain a high enough
amount of well-balanced data. Since there are more data in the specific region, the more
trustworthy prediction can be made by the neural network.

Appl. Sci. 2022,12, 7460 3 of 16
To achieve more accurate predictions, aberrant outliers may need to be removed from
the data. As suggested by J. Heaton, this can be carried out with the so-called ‘double-D’
algorithm [
19
]. The algorithm itself has many names, however, the idea behind it is to use
a double standard deviation as a removal condition. However, it is more useful when the
database has a very large number of observations. Since the amount of experimental data
on hydrogen–air LBV in the literature is not that large, this method might not significantly
improve predictions as the neural network might not have enough data to learn well. On
the other hand, it could be a direction for future work. Moreover, outliers have a significant
influence on the mean value as it is one of the neural network’s measures based on which it
decides how the final model will look like. Luckily, the model can also be evaluated using
mean absolute error (MAE) instead. The influence on a model based on MAE compared
to a model based on mean squared error (MSE) was analyzed by J. Qi [
20
]. It is much
less affected by outliers but error can be influenced by the number of observations. As
pointed out by T. Chai [
21
], each metric has its advantages and disadvantages. Due to
this, it would be wise to have the MSE as a side evaluator as well and not trust in a single
measure (MAE) blindly.
As the presented research overview shows, most of the related studies to date have
focused on the development of stand-alone ML models of combustible mixtures’ properties.
While there are works considering the implementation of such a model into a CFD code [
10
],
this has not been performed up to now. Consequentially, when developing, there is less
pressure to create lean and fast, but sufficiently accurate models, and their suitability is
not tested in CFD frameworks. In addition, the last, but main step, producing a working,
research or application suitable CFD–ANN code is never performed. To fill this gap,
we have developed a CFD-suitable deep neural network model for hydrogen LBV and
implemented it into an actual OpenFOAM-based CFD solver flameFoam.
The purpose of the current paper is to, based on the literature and our previous
work [
17
], create a fast and light, suitable for CFD applications, DNN model for LBV in dry
hydrogen–air mixtures, implement the developed model into a CFD solver, and perform
testing CFD–DNN simulations of turbulent premixed combustion.
2. Developed Deep Neural Network Model for Laminar Burning Velocity
The presented model was based on a previously developed (further referred to as
the original) large DNN [17], which was too slow for CFD application. The further model
development aimed to improve the calculation speed without a significant accuracy loss.
The original DNN model was trained on the dataset of experimental hydrogen–air mixtures
LBV at various temperatures, pressures and hydrogen concentrations, collected from the
open literature [17].
For the training of the new DNN model, the original database was expanded to 2871
data points [
22
–
47
]. Around 33% of all of them are experimental values; however, the
number of pure experimental values was insufficient for the reliable training of DNN.
Therefore, additional values were interpolated from the experimental values, by adding
points from the correlation curve produced from the experimental points of the given
experiment, close to the actual experimental points. Since the results of the newest studies
are most likely to have a higher accuracy due to technological development, they were
prioritized for the interpolation, in this way giving them more ‘weight’ in the final database,
since more data points have been added based on the newest observations and fewer points
in the area near old observations.
As can be noticed from Figure 1, most of the data are around 1 bar pressure or
298 K temperature, because most experimental works start by taking normal conditions
and changing only a single parameter. However, the expansion of the database was
performed to make the model more reliable while predicting LBVs at higher temperatures
and higher pressures, since most of the new data are under non-normal conditions. These
changes should have made the model more reliable in an entire applicability range. The

Appl. Sci. 2022,12, 7460 4 of 16
used experimental data model is applicable in the range of 295–600 K temperature and
0.1–5 bar pressure.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 18
should have made the model more reliable in an entire applicability range. The used ex-
perimental data model is applicable in the range of 295–600 K temperature and 0.1–5 bar
pressure.
(a) (b)
Figure 1. Gathered data statistics [22–47]: (a) count against pressure and (b) count against temper-
ature.
Data were split randomly into training and testing sets with around 75% of data used
to train the model and the remaining data used for validation. Further modifications such
as outlier removal were focused on the training set, avoiding modification of the testing
set, this way trying to obtain as reliable data as possible for validation.
Four hidden layers were left in the new model (same as in the original); however, to
ensure faster calculations, the number of neurons was changed to (7,10,7,5) (Figure 2) from
(40,50,30,10) in previous work [17]. The number of neurons was reduced by trying various
combinations and balancing accuracy against prediction speed.
In general, reducing the number of neurons increased the risk that the model will not
be as robust to data as the previous model. Due to this, we had to re-train the model mul-
tiple times as changes in weights could affect the predictions. Overall, this meant that the
smaller neural network required more human monitoring during the training to lessen
accuracy loss. In addition, we increased the risk that fewer neurons will lack data to learn
in some areas; to counter this, we had to increase our database. In the end, the main draw-
back related to the reduced number of neurons was that we had to put in substantial ad-
ditional effort to obtain the model with results comparable to those of larger DNN.
Figure 1.
Gathered data statistics [
22
–
47
]: (
a
) count against pressure and (
b
) count against temperature.
Data were split randomly into training and testing sets with around 75% of data used
to train the model and the remaining data used for validation. Further modifications such
as outlier removal were focused on the training set, avoiding modification of the testing set,
this way trying to obtain as reliable data as possible for validation.
Four hidden layers were left in the new model (same as in the original); however, to
ensure faster calculations, the number of neurons was changed to (7,10,7,5) (Figure 2) from
(40,50,30,10) in previous work [
17
]. The number of neurons was reduced by trying various
combinations and balancing accuracy against prediction speed.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 18
Figure 2. Architecture of the improved model.
Several different sets of activation functions were also tested and the best predictions
were obtained when using tanh activation function for the first (dense) hidden layer
(Equation (1)) and ReLU for the rest (Equation (2)):
𝑡𝑎𝑛ℎ󰇛𝑧󰇜𝑒𝑒
𝑒+𝑒 (1)
𝑅𝑒𝐿𝑈󰇛𝑜𝑢𝑡𝑝𝑢𝑡󰇜=𝑜𝑢𝑡𝑝𝑢𝑡,𝑖𝑓𝑜𝑢𝑡𝑝𝑢𝑡0;
0,𝑒𝑙𝑠𝑒 (2)
The original model used only ReLU functions [18]; however, the switch to tanh func-
tion in the first layer gave accuracy improvement to the new model. Further inclusions of
tanh functions did not produce significant further improvement, even though in the liter-
ature there is an example of LBV–DNN with all tanh activation functions [11]. Since tanh
function takes longer to calculate and did not give a significant advantage when used in
deeper layers, we left the ReLU activation function in those layers. Result improvement
might have been achieved since the tanh activation function managed to give primary
predictions which later were adjusted by layers with ReLU bending curve so that it would
fit data better.
However, since the number of neurons is not large, it is likely that with small laminar
burning velocities (0 m/s or close to it), the model might predict values below zero. To
avoid negative predictions, the Equation (2) was also applied to the final output function.
For model optimization RMSprop optimizer was used [48]. As studies show, it is one
of the best methods for optimization [49]. In addition, the regularization parameters were
changed compared to the original model. The first layer had Ridge (L2) regularization
parameter equal to 0.000002 and other hidden layers had Lasso (L1) regularization equal
to 0.0004. These values were selected after multiple tests with varied values. The effects of
the decrease in errors due to L1 and L2 are well documented in [50,51]. In general, they
prevent DNN from overfitting, which makes the network more stable and reliable.
To improve the reliability of the new model, MAE and MSE errors were combined
and minimized as weighted errors (equal weights were selected after several test cases)
for the training of the model as a loss function. Most of the influence was displayed by
MAE since velocity was in meters per second and lower than 1 m/s. On the other hand,
MSE helped in cases when MAE became almost constant. We believe this approach could
Figure 2. Architecture of the improved model.
In general, reducing the number of neurons increased the risk that the model will
not be as robust to data as the previous model. Due to this, we had to re-train the model
multiple times as changes in weights could affect the predictions. Overall, this meant that

Appl. Sci. 2022,12, 7460 5 of 16
the smaller neural network required more human monitoring during the training to lessen
accuracy loss. In addition, we increased the risk that fewer neurons will lack data to learn in
some areas; to counter this, we had to increase our database. In the end, the main drawback
related to the reduced number of neurons was that we had to put in substantial additional
effort to obtain the model with results comparable to those of larger DNN.
Several different sets of activation functions were also tested and the best predic-
tions were obtained when using tanh activation function for the first (dense) hidden layer
(Equation (1)) and ReLU for the rest (Equation (2)):
tanh(z)ez−e−z
ez+e−z(1)
ReLU(out put)=output,i f output >0;
0, else (2)
The original model used only ReLU functions [
18
]; however, the switch to tanh function
in the first layer gave accuracy improvement to the new model. Further inclusions of tanh
functions did not produce significant further improvement, even though in the literature
there is an example of LBV–DNN with all tanh activation functions [
11
]. Since tanh function
takes longer to calculate and did not give a significant advantage when used in deeper
layers, we left the ReLU activation function in those layers. Result improvement might
have been achieved since the tanh activation function managed to give primary predictions
which later were adjusted by layers with ReLU bending curve so that it would fit data better.
However, since the number of neurons is not large, it is likely that with small laminar
burning velocities (0 m/s or close to it), the model might predict values below zero. To
avoid negative predictions, the Equation (2) was also applied to the final output function.
For model optimization RMSprop optimizer was used [
48
]. As studies show, it is one
of the best methods for optimization [
49
]. In addition, the regularization parameters were
changed compared to the original model. The first layer had Ridge (L2) regularization
parameter equal to 0.000002 and other hidden layers had Lasso (L1) regularization equal to
0.0004. These values were selected after multiple tests with varied values. The effects of
the decrease in errors due to L1 and L2 are well documented in [
50
,
51
]. In general, they
prevent DNN from overfitting, which makes the network more stable and reliable.
To improve the reliability of the new model, MAE and MSE errors were combined
and minimized as weighted errors (equal weights were selected after several test cases) for
the training of the model as a loss function. Most of the influence was displayed by MAE
since velocity was in meters per second and lower than 1 m/s. On the other hand, MSE
helped in cases when MAE became almost constant. We believe this approach could give
predictions closer to real values, as performed analysis shows that averaging MAE and
MSE indeed gives values in between for the data gathered for this research. This can be
helpful in cases with areas lacking observations. Also, it is worth noticing that in cases of
unique (single) observations, the MAE and MSE will point to the same location. A simple
explanation of this could be the idea that while MAE minimizes the model towards median
and MSE—towards mean, the combination of them would predict the value in between as
visualized in Figure 3.
To obtain predictions closer to original experimental values, the study did not modify
the testing set; however, the training set for improved DNN was adjusted by making some
outliers equal to the mean. Outliers were considered values that are different by over 10%
from the median value when results from three or more different experimental sources
under the same or similar conditions were available. These values were considered outliers
since the experimental spread of more than 10% shows higher experimental uncertainties,
and given several sources, aberrant points are the most suspect. Around 3–4% of all points
were adjusted. Yet, considering that this can detect many data points near zero, those
values were ignored. Due to the way in how the model was trained, outliers still could
have influenced the final model. However, due to the mean value, this influence is reduced.

Appl. Sci. 2022,12, 7460 6 of 16
This allowed the creation of a model which is stable and can perform well on data sets that
have outliers.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 18
give predictions closer to real values, as performed analysis shows that averaging MAE
and MSE indeed gives values in between for the data gathered for this research. This can
be helpful in cases with areas lacking observations. Also, it is worth noticing that in cases
of unique (single) observations, the MAE and MSE will point to the same location. A sim-
ple explanation of this could be the idea that while MAE minimizes the model towards
median and MSE—towards mean, the combination of them would predict the value in
between as visualized in Figure 3.
Figure 3. Predicted loss function behavior tested by comparing DNN models with different loss
functions.
To obtain predictions closer to original experimental values, the study did not modify
the testing set; however, the training set for improved DNN was adjusted by making some
outliers equal to the mean. Outliers were considered values that are different by over 10%
from the median value when results from three or more different experimental sources
under the same or similar conditions were available. These values were considered outli-
ers since the experimental spread of more than 10% shows higher experimental uncertain-
ties, and given several sources, aberrant points are the most suspect. Around 3–4% of all
points were adjusted. Yet, considering that this can detect many data points near zero,
those values were ignored. Due to the way in how the model was trained, outliers still
could have influenced the final model. However, due to the mean value, this influence is
reduced. This allowed the creation of a model which is stable and can perform well on
data sets that have outliers.
To use this model in the CFD solver, it was exported by saving the weights 𝑤,,,
biases bi,j and performing matrix multiplication with them. Here, j denotes a layer number,
and ij denotes a weight number in the j-th layer. j can obtain values from 1 to k and ij can
obtain values from 1 to nj. Then the i-th value in the j-th hidden layer hi,j is obtained from
the Equations (3) and (4):
ℎ,
∗=𝑤,,ℎ, +𝑤,,ℎ,+⋯+𝑤,,ℎ, (3)
ℎ, =𝑎𝑐𝑡𝐹ℎ,
∗+𝑏,, (4)
where actF is an activation function given by the Equations (1) and (2) for the first and
other layers, respectively. By applying Equations (3) and (4), the model can be expressed
and implemented using weights (Wj) and biases (Bj) matrixes and inputs vector (I).
Figure 3.
Predicted loss function behavior tested by comparing DNN models with different loss functions.
To use this model in the CFD solver, it was exported by saving the weights
wij−1,ij,j
,
biases b
i,j
and performing matrix multiplication with them. Here, jdenotes a layer number,
and i
j
denotes a weight number in the j-th layer. jcan obtain values from 1 to kand i
j
can
obtain values from 1 to n
j
. Then the i-th value in the j-th hidden layer h
i,j
is obtained from
the Equations (3) and (4):
h∗
i,j=wi,0,jh0,j−1+wi,1, jh1,j−1+. . . +wi,nj−1,jhnj−1,j−1(3)
hi,j=actFh∗
i,j+bi,j, (4)
where actF is an activation function given by the Equations (1) and (2) for the first and other
layers, respectively. By applying Equations (3) and (4), the model can be expressed and
implemented using weights (W
j
) and biases (B
j
) matrixes and inputs vector (I). Weights
and biases can also be expressed as follows by Equations (5) and (6) in which nand kmark
the number of weights in interacting layers.
Wj=


w1,1 . . . wnj,1
. . . . . . . . .
w1,nj−1. . . wnj,nj−1


(5)
Bj=b1, .., bnj(6)
The output becomes the new input for the next layer. This process is repeated until all
layers are calculated and the final output—value of LBV is predicted.
3. Comparison with Other Methods
To test the quality of the developed DNN model and to check whether some simpler
method could be sufficient, other models using popular machine learning methods were
created based on the same database. All models were optimized to make a million predic-
tions in under 2 s with a testing script. Final models were tested and compared with DNN
in terms of prediction density and the most widely used metrics such as MAE, MSE and

Appl. Sci. 2022,12, 7460 7 of 16
R-squared. While density estimates are not the most optimal method of comparison, we
reduced the Gaussian kernel and estimation for all methods which was carried out with
the same kernel; therefore, we consider it to be sufficient for comparison purposes. For
all models, the same training and testing datasets were used as for the developed DNN
model (75%/25%).
Density plots in Figure 4explain the high spread (Table 1) of some models (RF, MARS).
While DNN and k-NN both show good agreement with the density of data until LBV
reaches 350 cm/s, k-NN shows worse results at higher velocities. Even more, only SVM
and DNN models predict higher than 700 cm/s laminar burning velocities. On the other
hand, evaluation of SVM shows that this model is most likely to make predictions too high
at over 280 cm/s and otherwise, guesses too low. As stated by this study earlier, DNN
was trained by minimizing MAE and MSE at the same time, which should give optimal
prediction between median and mean values. It means that the model might not have the
lowest MSE and MAE values; therefore, it would be wise to evaluate it based on how well it
explained the data. However, the results show (see Table 1) that DNN has the lowest MEA
and MSE values of all tested methods with the highest being R-Squared at approximately
0.997. The second place would go to SVM regression which shows less spread than other
models (RF, k-NN, MARS). The remaining solutions performed well by showing the ability
to explain most of the data; however, their main drawback is the limited ability to predict
higher burning velocities.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 8 of 18
Figure 4. Prediction densities by multiple machine learning algorithms.
Table 1. Comparison of machine learning algorithms (values calculated against testing set).
Model MSE, (cm/s2) MAE, (cm/s) R-Squared
DNN 73.07057 6.094699 0.9972631
SVM (SVR) 843.9467 23.36927 0.9687515
Random Forest 1100.299 23.68055 0.9650517
K-Nearest Neighbors (k-NN) 1164.565 18.31963 0.9541389
MARS 1324.741 24.40525 0.9487737
The good performance of the developed DNN model can be visible in Figure 5 below,
which shows the comparison of experimental and predicted LBV values. The red line,
which shows linear regression of DNN predictions plotted against reference velocities,
almost perfectly covers the perfect match line (light blue).
Figure 4. Prediction densities by multiple machine learning algorithms.
Table 1. Comparison of machine learning algorithms (values calculated against testing set).
Model MSE, (cm/s2)MAE, (cm/s) R-Squared
DNN 73.07057 6.094699 0.9972631
SVM (SVR) 843.9467 23.36927 0.9687515
Random Forest 1100.299 23.68055 0.9650517
K-Nearest Neighbors (k-NN) 1164.565 18.31963 0.9541389
MARS 1324.741 24.40525 0.9487737
The good performance of the developed DNN model can be visible in Figure 5below,
which shows the comparison of experimental and predicted LBV values. The red line,

Appl. Sci. 2022,12, 7460 8 of 16
which shows linear regression of DNN predictions plotted against reference velocities,
almost perfectly covers the perfect match line (light blue).
Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 18
Figure 5. DNN predictions versus experimental data.
LBV has high uncertainty in the equivalence ratio range [0.2, 0.5]. Malet [18] investi-
gated the burning behavior of hydrogen mixtures and offered an expression to calculate
LBV at low equivalence ratios. A comparison of the DNN model and Malet formula pre-
dictions (Figure 6) shows that the DNN model gives predictions closer to reality with
MAE equal to 0.0589 m/s, while the Malet correlation has MAE of 0.1002 m/s.
Figure 6. Malet and DNN comparison.
Figure 5. DNN predictions versus experimental data.
LBV has high uncertainty in the equivalence ratio range [0.2, 0.5]. Malet [
18
] investi-
gated the burning behavior of hydrogen mixtures and offered an expression to calculate
LBV at low equivalence ratios. A comparison of the DNN model and Malet formula predic-
tions (Figure 6) shows that the DNN model gives predictions closer to reality with MAE
equal to 0.0589 m/s, while the Malet correlation has MAE of 0.1002 m/s.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 18
Figure 5. DNN predictions versus experimental data.
LBV has high uncertainty in the equivalence ratio range [0.2, 0.5]. Malet [18] investi-
gated the burning behavior of hydrogen mixtures and offered an expression to calculate
LBV at low equivalence ratios. A comparison of the DNN model and Malet formula pre-
dictions (Figure 6) shows that the DNN model gives predictions closer to reality with
MAE equal to 0.0589 m/s, while the Malet correlation has MAE of 0.1002 m/s.
Figure 6. Malet and DNN comparison.
Figure 6. Malet and DNN comparison.

Appl. Sci. 2022,12, 7460 9 of 16
4. CFD Simulations
This chapter presents a couple of flameFoam-DNN calculation examples showing that
the developed DNN model is usable with CFD simulation and provides results in line with
the Malet correlation or experimental data.
The developed model was implemented in the open-source turbulent premixed com-
bustion solver flameFoam [
52
] v0.10. flameFoam is an OpenFOAM-based solver using
a progress variable approach and turbulent flame speed closure model to simulate com-
bustion. The turbulent flame speed closure model requires turbulent flame speed values,
which in turn require LBV values. In the flameFoam-DNN simulations, presented be-
low, these LBV values were estimated on-the-fly in the flameFoam using the DNN model
programmed according to Equations (3)–(6) with the weights exported from the DNN
training. The solver source code, including the DNN model, is available on GitHub:
https://github.com/flameFoam/flameFoam (accessed on 29 June 2022).
To compare flameFoam calculation results obtained using the DNN model versus
Malet correlation, simulation of ENACCEF2 facility (France, CNRS-ICARE) TEST1 from
ETSON-MITHYGENE benchmark [53] was performed.
ENACCEF2 is a closed vertical steel tube of 7.65 m height and 0.23 m inner diameter
(Figure 7a). Flame acceleration is achieved in the simulated experiment by 9.2 mm thin
annular obstacles situated in the lower part of the facility (Figure 7b). The flame is ignited
at the bottom center and propagates upwards.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 18
4. CFD Simulations
This chapter presents a couple of flameFoam-DNN calculation examples showing
that the developed DNN model is usable with CFD simulation and provides results in
line with the Malet correlation or experimental data.
The developed model was implemented in the open-source turbulent premixed com-
bustion solver flameFoam [52] v0.10. flameFoam is an OpenFOAM-based solver using a
progress variable approach and turbulent flame speed closure model to simulate combus-
tion. The turbulent flame speed closure model requires turbulent flame speed values,
which in turn require LBV values. In the flameFoam-DNN simulations, presented below,
these LBV values were estimated on-the-fly in the flameFoam using the DNN model pro-
grammed according to Equations (3)–(6) with the weights exported from the DNN train-
ing. The solver source code, including the DNN model, is available on GitHub:
https://github.com/flameFoam/flameFoam (accessed on 29 June 2022).
To compare flameFoam calculation results obtained using the DNN model versus
Malet correlation, simulation of ENACCEF2 facility (France, CNRS-ICARE) TEST1 from
ETSON-MITHYGENE benchmark [53] was performed.
ENACCEF2 is a closed vertical steel tube of 7.65 m height and 0.23 m inner diameter
(Figure 7a). Flame acceleration is achieved in the simulated experiment by 9.2 mm thin
annular obstacles situated in the lower part of the facility (Figure 7b). The flame is ignited
at the bottom center and propagates upwards.
(a) (b) (c)
Figure 7. ENACCEF2 facility and computational grid: (a) overall scheme, (b) obstacle region close-
up and (c) computational mesh around the obstacle.
The hydrogen concentration in the homogenous combustible mixture with air was
13%. The experiment was performed at 23 °C temperature and 100,000 Pa pressure.
Figure 7.
ENACCEF2 facility and computational grid: (
a
) overall scheme, (
b
) obstacle region close-up
and (c) computational mesh around the obstacle.
The hydrogen concentration in the homogenous combustible mixture with air was
13%. The experiment was performed at 23 ◦C temperature and 100,000 Pa pressure.
The computational mesh (Figure 7c) of the facility was composed of the structured
orthogonal grids of the facility-free volume (fluid region, blue in Figure 7) and facility wall

Appl. Sci. 2022,12, 7460 10 of 16
(solid region, yellow in Figure 7). 2D axisymmetric mesh with 1 mm cell sizes was used in
the presented simulations. In the fluid region, flameFoam solves Navier-Stokes, turbulence
and combustion equations, while in the solid region heat conductivity equation is solved.
The regions are coupled through standard OpenFOAM inter-region boundary condition
for temperature compressible::turbulentTemperatureCoupledBAffleMixed.
Initial conditions were set according to the experiment. Turbulence was modeled using
the k-
ε
RANS model. Negligible values for initial turbulence parameters were selected. The
boundary condition for temperature on the external side of the solid grid was set to 23
◦
C.
Figure 8presents the flame propagation velocity profile (obtained from flame arrival
times) obtained by both simulations. Results are very similar, showing that the DNN
model hasn’t introduced any regressions compared to the Malet correlation around the
experimental conditions [
52
]. There was no significant difference of computation time
between Malet and DNN calculations.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 18
The computational mesh (Figure 7c) of the facility was composed of the structured
orthogonal grids of the facility-free volume (fluid region, blue in Figure 7) and facility wall
(solid region, yellow in Figure 7). 2D axisymmetric mesh with 1 mm cell sizes was used
in the presented simulations. In the fluid region, flameFoam solves Navier-Stokes, turbu-
lence and combustion equations, while in the solid region heat conductivity equation is
solved. The regions are coupled through standard OpenFOAM inter-region boundary
condition for temperature compressible::turbulentTemperatureCoupledBAffleMixed.
Initial conditions were set according to the experiment. Turbulence was modeled us-
ing the k-ε RANS model. Negligible values for initial turbulence parameters were se-
lected. The boundary condition for temperature on the external side of the solid grid was
set to 23 °C.
Figure 8 presents the flame propagation velocity profile (obtained from flame arrival
times) obtained by both simulations. Results are very similar, showing that the DNN
model hasn’t introduced any regressions compared to the Malet correlation around the
experimental conditions [52]. There was no significant difference of computation time be-
tween Malet and DNN calculations.
Figure 8. Vertical flame propagation velocity profiles in ENACCEF2 facility.
Figure 9 presents pressure evolutions at three different heights of the facility, time-
shifted to facilitate comparison. Pressure evolutions corresponding to the turbulent com-
bustion phase (accelerated flame) are also very similar in both cases. Differences between
evolutions obtained with Malet and DNN seem to be mostly related to a slower (quasi-
)laminar combustion phase and, consequentially, a relatively stronger impact of turbulent
acceleration, which results in a higher peak pressure wave value. At a 4 m height, the peak
value obtained with DNN is slightly lower; however, this seems to be a local result caused
by coincidental interference, since further, at 6.2 m (Figure 10), DNN results already show
higher pressure values.
Figure 8. Vertical flame propagation velocity profiles in ENACCEF2 facility.
Figure 9presents pressure evolutions at three different heights of the facility, time-
shifted to facilitate comparison. Pressure evolutions corresponding to the turbulent com-
bustion phase (accelerated flame) are also very similar in both cases. Differences between
evolutions obtained with Malet and DNN seem to be mostly related to a slower
(quasi-)
laminar combustion phase and, consequentially, a relatively stronger impact of turbulent
acceleration, which results in a higher peak pressure wave value. At a 4 m height, the peak
value obtained with DNN is slightly lower; however, this seems to be a local result caused
by coincidental interference, since further, at 6.2 m (Figure 10), DNN results already show
higher pressure values.
Figure 10 presents pressure evolutions at a height of 6.227 m. Initial shockwave
propagation and reflected shockwave arrival are visible. Results are unshifted in time to
show the delay in the laminar phase simulation obtained with the DNN model compared to
the Malet correlation. The DNN model provides a longer duration of quasi-laminar phase
(overall slower flame propagation rate until the first obstacle). However, flameFoam with
the TFC model does not simulate a quasi-laminar regime correctly [
52
]. The assumption is
made that in turbulent cases, the main transient is controlled by the turbulent regime and
the influence of the laminar phase is limited to a larger or smaller shift in time of the onset
of the turbulent phase. This is partially illustrated in this case in Figures 9and 10 since
different methods of laminar flame speed estimation led to different evolution and duration
of the quasi-laminar regime; however, when flame accelerates due to turbulence, pressure
evolutions become very similar, even including the shockwave propagation and reflection.

Appl. Sci. 2022,12, 7460 11 of 16
Appl. Sci. 2022, 12, x FOR PEER REVIEW 12 of 18
Figure 9. Pressure evolutions at different heights of ENACCEF2 (time-shifted).
Figure 10 presents pressure evolutions at a height of 6.227 m. Initial shockwave prop-
agation and reflected shockwave arrival are visible. Results are unshifted in time to show
the delay in the laminar phase simulation obtained with the DNN model compared to the
Malet correlation. The DNN model provides a longer duration of quasi-laminar phase
(overall slower flame propagation rate until the first obstacle). However, flameFoam with
the TFC model does not simulate a quasi-laminar regime correctly [52]. The assumption
is made that in turbulent cases, the main transient is controlled by the turbulent regime
and the influence of the laminar phase is limited to a larger or smaller shift in time of the onset
of the turbulent phase. This is partially illustrated in this case in Figures 9 and 10 since different
methods of laminar flame speed estimation led to different evolution and duration of the
quasi-laminar regime; however, when flame accelerates due to turbulence, pressure evolu-
tions become very similar, even including the shockwave propagation and reflection.
Figure 9. Pressure evolutions at different heights of ENACCEF2 (time-shifted).
Appl. Sci. 2022, 12, x FOR PEER REVIEW 13 of 18
Figure 10. Pressure evolution at 6.227 m height of ENACCEF2.
Figure 11 presents flow streamlines and laminar burning velocity values calculated
in both cases when the flame is in a similar location. Only values on the flame brush (pro-
gress variable source term larger than 1000 1/s) are shown. There is a clear difference in
the laminar flame velocity values obtained with both methods. The influence of pressure
and temperature are far more pronounced in the Malet correlation case, where the burnt-
mixture side of the flame brush exhibits two to three times higher velocity values than the
leading side or whole brush in the case of the DNN model. Consequently, a very thin
flame brush is obtained with Malet correlation, since combustion is completed at a higher
rate. However, at the leading side of the brush, both methods seem to produce similar
values of laminar burning velocity, corresponding to the similar overall flame propaga-
tion rate in both cases (Figure 8); higher values at the trailing side mostly only impact the
flame brush thickness.
Figure 11. Simulated LBV distribution in ENACCEF2 at selected moments.
To validate the DNN model at a somewhat higher concentration, where the Malet
correlation becomes not valid, a simulation of 22.65% hydrogen–air mixture combustion
in the vented laboratory-scale chamber of the University of Sydney (US) [54] was per-
formed.
The US chamber is a rectangular box of 5 × 5 × 25 cm dimensions with an open top.
Flame acceleration can be achieved with various configurations of obstacles. In the
Figure 10. Pressure evolution at 6.227 m height of ENACCEF2.
Figure 11 presents flow streamlines and laminar burning velocity values calculated
in both cases when the flame is in a similar location. Only values on the flame brush
(progress variable source term larger than 1000 1/s) are shown. There is a clear difference
in the laminar flame velocity values obtained with both methods. The influence of pressure
and temperature are far more pronounced in the Malet correlation case, where the burnt-
mixture side of the flame brush exhibits two to three times higher velocity values than the
leading side or whole brush in the case of the DNN model. Consequently, a very thin flame
brush is obtained with Malet correlation, since combustion is completed at a higher rate.
However, at the leading side of the brush, both methods seem to produce similar values
of laminar burning velocity, corresponding to the similar overall flame propagation rate
in both cases (Figure 8); higher values at the trailing side mostly only impact the flame
brush thickness.

Appl. Sci. 2022,12, 7460 12 of 16
Appl. Sci. 2022, 12, x FOR PEER REVIEW 13 of 18
Figure 10. Pressure evolution at 6.227 m height of ENACCEF2.
Figure 11 presents flow streamlines and laminar burning velocity values calculated
in both cases when the flame is in a similar location. Only values on the flame brush (pro-
gress variable source term larger than 1000 1/s) are shown. There is a clear difference in
the laminar flame velocity values obtained with both methods. The influence of pressure
and temperature are far more pronounced in the Malet correlation case, where the burnt-
mixture side of the flame brush exhibits two to three times higher velocity values than the
leading side or whole brush in the case of the DNN model. Consequently, a very thin
flame brush is obtained with Malet correlation, since combustion is completed at a higher
rate. However, at the leading side of the brush, both methods seem to produce similar
values of laminar burning velocity, corresponding to the similar overall flame propaga-
tion rate in both cases (Figure 8); higher values at the trailing side mostly only impact the
flame brush thickness.
Figure 11. Simulated LBV distribution in ENACCEF2 at selected moments.
To validate the DNN model at a somewhat higher concentration, where the Malet
correlation becomes not valid, a simulation of 22.65% hydrogen–air mixture combustion
in the vented laboratory-scale chamber of the University of Sydney (US) [54] was per-
formed.
The US chamber is a rectangular box of 5 × 5 × 25 cm dimensions with an open top.
Flame acceleration can be achieved with various configurations of obstacles. In the
Figure 11. Simulated LBV distribution in ENACCEF2 at selected moments.
To validate the DNN model at a somewhat higher concentration, where the Malet
correlation becomes not valid, a simulation of 22.65% hydrogen–air mixture combustion in
the vented laboratory-scale chamber of the University of Sydney (US) [
54
] was performed.
The US chamber is a rectangular box of 5
×
5
×
25 cm dimensions with an open
top. Flame acceleration can be achieved with various configurations of obstacles. In the
simulated experiment, three rows of baffles and a small upper obstacle were used, the
so-called configuration BBBS (Figure 12a). The flame is ignited at the bottom center and
propagates upwards.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 14 of 18
simulated experiment, three rows of baffles and a small upper obstacle were used, the so-
called configuration BBBS (Figure 12a). The flame is ignited at the bottom center and propa-
gates upwards.
The computational mesh (Figure 12b) of the facility was composed of the structured
orthogonal grid of the chamber-free volume and additional volume outside the top open-
ing to simulate surroundings. 2D axisymmetric mesh with 0.25 mm cell sizes was used in
the presented simulations. Around the obstacles, the mesh was graded up to 0.03125 mm
(Figure 12c).
(a) (b) (c)
Figure 12. Vented chamber and computational grid: (a) chamber scheme, (b) overall mesh view and
(c) computational mesh details below the upper obstacle.
Initial conditions were set according to the experiment—20 °C temperature and
100,000 Pa pressure. Turbulence was modeled using the k-ω-SST RANS model. Negligible
values for initial turbulence parameters were selected. The adiabatic boundary condition
for temperature was set.
Figure 13 shows overpressure evolution at the facility’s bottom. Results obtained by
flameFoam–ANN simulation are close to the experimental, including the shape of the
pressure curve. However, overpressure values are slightly over-predicted. A possible ex-
planation for the over-prediction could be missing support for quenching in the flame-
Foam, which, if implemented, could slightly decrease the average combustion rate in tur-
bulent conditions and by the obstacles and wall surfaces. Nevertheless, obtained results
indicate that the developed DNN model and its implementation in flameFoam are suita-
ble for the simulation of turbulent combustion in similar conditions.
Figure 12.
Vented chamber and computational grid: (
a
) chamber scheme, (
b
) overall mesh view and
(c) computational mesh details below the upper obstacle.
The computational mesh (Figure 12b) of the facility was composed of the structured
orthogonal grid of the chamber-free volume and additional volume outside the top opening
to simulate surroundings. 2D axisymmetric mesh with 0.25 mm cell sizes was used in
the presented simulations. Around the obstacles, the mesh was graded up to 0.03125 mm
(Figure 12c).
Initial conditions were set according to the experiment—20
◦
C temperature and
100,000 Pa
pressure. Turbulence was modeled using the k-
ω
-SST RANS model. Neg-
ligible values for initial turbulence parameters were selected. The adiabatic boundary
condition for temperature was set.
Figure 13 shows overpressure evolution at the facility’s bottom. Results obtained
by flameFoam–ANN simulation are close to the experimental, including the shape of the

Appl. Sci. 2022,12, 7460 13 of 16
pressure curve. However, overpressure values are slightly over-predicted. A possible expla-
nation for the over-prediction could be missing support for quenching in the flameFoam,
which, if implemented, could slightly decrease the average combustion rate in turbulent
conditions and by the obstacles and wall surfaces. Nevertheless, obtained results indicate
that the developed DNN model and its implementation in flameFoam are suitable for the
simulation of turbulent combustion in similar conditions.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 15 of 18
Figure 13. Numerical and experimental [54] overpressure evolution.
Figure 14 presents flow streamlines and laminar burning velocity values after the
flame has passed the upper obstacle. Only values on the flame brush are shown. In this
case, DNN–LBV predictions display similar characteristics as in the ENACCEF2 case—
the predicted values do not significantly vary across the flame brush width. Obtained LBV
values are not large and a thicker flame brush is also obtained. However, different LBV
values are obtained in different locations of the facility, though the variation is not large.
This is explained through the dependency on the pressure, which also varies in the facility
as shown by the color of the streamlines. Due to venting at the facility top, there is an
overall pressure gradient in the facility with the largest pressure at the bottom, and high-
est values reaching around 166,000 Pa. This pressure variation results in different LBV
values predicted by the DNN at different heights of the facility.
Figure 14. Simulated LBV and pressure distributions in the vented chamber at a selected moment.
5. Conclusions
A developed DNN able to predict hydrogen LBV in dry air was developed based on
a larger DNN. The accuracy of fast DNN was tested against a testing set of data and values
from the literature. Additionally, obtained LVB curve smoothness was checked. Predic-
tions of developed DNN showed proper behavior and an R-squared value of approximately
0.997. Speed improvement of around 30 times compared to the original DNN was obtained.
Figure 13. Numerical and experimental [54] overpressure evolution.
Figure 14 presents flow streamlines and laminar burning velocity values after the
flame has passed the upper obstacle. Only values on the flame brush are shown. In this
case, DNN–LBV predictions display similar characteristics as in the ENACCEF2 case—the
predicted values do not significantly vary across the flame brush width. Obtained LBV
values are not large and a thicker flame brush is also obtained. However, different LBV
values are obtained in different locations of the facility, though the variation is not large.
This is explained through the dependency on the pressure, which also varies in the facility
as shown by the color of the streamlines. Due to venting at the facility top, there is an
overall pressure gradient in the facility with the largest pressure at the bottom, and highest
values reaching around 166,000 Pa. This pressure variation results in different LBV values
predicted by the DNN at different heights of the facility.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 15 of 18
Figure 13. Numerical and experimental [54] overpressure evolution.
Figure 14 presents flow streamlines and laminar burning velocity values after the
flame has passed the upper obstacle. Only values on the flame brush are shown. In this
case, DNN–LBV predictions display similar characteristics as in the ENACCEF2 case—
the predicted values do not significantly vary across the flame brush width. Obtained LBV
values are not large and a thicker flame brush is also obtained. However, different LBV
values are obtained in different locations of the facility, though the variation is not large.
This is explained through the dependency on the pressure, which also varies in the facility
as shown by the color of the streamlines. Due to venting at the facility top, there is an
overall pressure gradient in the facility with the largest pressure at the bottom, and high-
est values reaching around 166,000 Pa. This pressure variation results in different LBV
values predicted by the DNN at different heights of the facility.
Figure 14. Simulated LBV and pressure distributions in the vented chamber at a selected moment.
5. Conclusions
A developed DNN able to predict hydrogen LBV in dry air was developed based on
a larger DNN. The accuracy of fast DNN was tested against a testing set of data and values
from the literature. Additionally, obtained LVB curve smoothness was checked. Predic-
tions of developed DNN showed proper behavior and an R-squared value of approximately
0.997. Speed improvement of around 30 times compared to the original DNN was obtained.
Figure 14. Simulated LBV and pressure distributions in the vented chamber at a selected moment.

Appl. Sci. 2022,12, 7460 14 of 16
5. Conclusions
A developed DNN able to predict hydrogen LBV in dry air was developed based on a
larger DNN. The accuracy of fast DNN was tested against a testing set of data and values
from the literature. Additionally, obtained LVB curve smoothness was checked. Predictions
of developed DNN showed proper behavior and an R-squared value of approximately
0.997. Speed improvement of around 30 times compared to the original DNN was obtained.
Developed DNN was implemented into a combustion solver flameFoam for the
prediction of LBV values. Testing CFD–DNN simulations were performed and compared
with simulations using the established Malet correlation of LBV. The comparison showed
similar results with the DNN model predicting more uniform and lower values of LBV.
Comparison against a selected experiment also showed the validity of the implemented
DNN model.
Further planned work is to perform validation of developed CFD–DNN modeling
against the experimental data.
Author Contributions:
Conceptualization, A.A. and M.P.; methodology, A.A. and M.P.; software,
A.A. and M.P.; validation, A.A. and M.P.; formal analysis, A.A. and M.P.; investigation, A.A. and M.P.;
resources, M.P.; data curation, A.A.; writing—original draft preparation, A.A.; writing—review and
editing, M.P.; visualization, A.A. and M.P.; supervision, M.P.; project administration, M.P. All authors
have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
McCulloch, W.S.; Pitts, W. A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophys.
1943
,5, 115–133.
[CrossRef]
2.
Phillipson, F. Quantum Machine Learning: Benefits and Practical Examples. In Proceedings of the International Workshop on
QuANtum SoftWare Engineering & PRogramming (QANSWER), Talavera, Spain, 11–12 February 2020.
3.
Islam, M.D.; Wu, C.; Poly, T.; Yang, H.; Li, Y. Applications of Machine Learning in Fatty Live Disease Prediction. Stud. Health
Technol. Inform. 2018,247, 166–170. [PubMed]
4.
Zhou, J.; Zhou, Z.; Zhao, Q.; Han, Z.; Wang, P.; Xu, J.; Dian, Y. Evaluation of Different Algorithms for Estimating the Growing
Stock Volume of Pinus massoniana Plantations Using Spectral and Spatial Information from a SPOT6 Image. Forests
2020
,11, 540.
[CrossRef]
5.
Artrith, N.; Urban, A. An implementation of artificial neural-network potentials for atomistic materials simulations: Performance
for TiO2.Comput. Mater. Sci. 2016,114, 135–150. [CrossRef]
6.
Artrith, N.; Urban, A.; Ceder, G. Efficient and Accurate Machine-Learning Interpolation of Atomic Energies in Compositions with
Many Species. Phys. Rev. B. 2017,96, 014112. [CrossRef]
7.
Jach, A.; ˙
Zbikowski, M.; Teodorczyk, A. Laminar burning velocity predictions of single-fuel mixtures of C1-C7 normal hydrocar-
bon and air. J. KONES 2018,25, 227–235.
8.
Mehra, R.K.; Duan, H.; Luo, S.; Ma, F. Laminar burning velocity of hydrogen and carbon-monoxide enriched natural gas
(HyCONG): An experimental and artificial neural network study. Fuel 2019,246, 476–490. [CrossRef]
9.
Pulga, L.; Bianchi, G.M.; Falfari, S.; Forte, C. A machine learning methodology for improving the accuracy of laminar flame
simulations with reduced chemical kinetics mechanisms. Combust. Flame 2020,216, 72–81. [CrossRef]
10.
Malik, K.; ˙
Zbikowski, M.; Teodorczyk, A. Laminar Burning Velocity Model Based on Deep Neural Network for Hydrogen and
Propane with Air. Energies 2020,13, 3381. [CrossRef]
11.
Varghese, R.J.; Kumar, S. Machine learning model to predict the laminar burning velocities of H
2
/CO/CH
4
/CO
2
/N
2
/air
mixtures at high pressure and temperature conditions. Int. J. Hydrog. Energy 2020,45, 3216–3232. [CrossRef]
12.
Eckart, S.; Prieler, R.; Hochenauer, C.; Krause, H. Application and comparison of multiple machine learning techniques for the
calculation of laminar burning velocity for hydrogen-methane mixtures. Therm. Sci. Eng. Prog. 2022,32, 101306.
13.
Correa Gonzalez, S.; Kroyan, Y.; Sarjovaara, T.; Kiiski, U.; Karvo, A.; Toldy, A.I.; Larmi, M.; Santasalo-Aarnio, A. Prediction of
Gasoline Blend Ignition Characteristics Using Machine Learning Models. Energy Fuels 2021,35, 9332–9340. [CrossRef]
14.
vom Lehn, F.; Cai, L.; Cáceres, B.C.; Pitsch, H. Exploring the fuel structure dependence of laminar burning velocity: A machine
learning based group contribution approach. Combust. Flame 2021,232, 111525. [CrossRef]
15.
Wan, Z.; Wang, Q.D.; Wang, B.Y.; Liang, J. Development of machine learning models for the prediction of laminar flame speeds of
hydrocarbon and oxygenated fuels. Fuel Commun. 2022,12, 100071. [CrossRef]

Appl. Sci. 2022,12, 7460 15 of 16
16.
Li, R.; Herreros, J.M.; Tsolakis, A.; Yang, W. Integrated machine learning-quantitative structure-property relationship (ML-QSPR)
and chemical kinetics for high throughput fuel screening toward internal combustion engine. Fuel
2022
,307, 121908. [CrossRef]
17.
Ambrutis, A.; Povilaitis, M. Laminar burning velocity estimation using deep neural network. In Proceedings of the 17th
International Conference of Young Scientists on Energy and Natural Sciences Issues, Kaunas, Lithuania, 24–28 May 2021.
18.
Malet, F. Numerical and Experimental Study of Premixed Turbulent Hydrogen Flame Propagation in Lean and Wet Atmosphere.
Ph.D. Thesis, Orleans University, Orléans, France, 2005.
19. Heaton, J. Applications of Deep Neural Networks with Keras. arXiv 2022, arXiv:2009.05673.
20.
Qi, J.; Du, J.; Siniscalchi, S.M.; Ma, X.; Lee, C.-H. On Mean Absolute Error for Deep Neural Network Based Vector-to-Vector
Regression. IEEE Signal Process. Lett 2020,27, 1485–1489. [CrossRef]
21.
Chai, T.; Draxler, R. Root mean square error (RMSE) or mean absolute error (MAE)? Geosci. Model Dev.
2014
,7, 1247–1250.
[CrossRef]
22.
Bradley, D.; Lawes, M.; Liu, K.; Verhelst, S.; Woolley, R. Laminar burning velocities of lean hydrogen-air mixtures at pressures up
to 1.0 MPa. Combust. Flame 2007,149, 162–172. [CrossRef]
23.
Burke, M.P.; Chen, Z.; Ju, Y.; Dryer, F.L. Effect of cylindrical confinement on the determination of laminar flame speeds using
outwardly propagating flames. Combust. Flame 2009,156, 771–779. [CrossRef]
24.
Dahoe, A.E. Laminar burning velocities of hydrogen-air mixtures from closed vessel gas explosions. J. Loss Prev. Process. Ind.
2005,18, 152–166. [CrossRef]
25.
Das, A.K.; Kumar, K.; Sung, C.-J. Laminar flame speeds of moist syngas mixtures. Combust. Flame
1957
,158, 345–353. [CrossRef]
26.
Dayma, G.; Halter, F.; Dagaut, P. New insights into the peculiar behavior of laminar burning velocities of hydrogen-air flames
according to pressure and equivalence ratio. Combust. Flame 2014,161, 2235–2241. [CrossRef]
27.
Egolfopoulos, F.N.; Law, C.K. An experimental and computational study of the burning rates of ultra-lean to moderately-rich
H2/O2/N2 laminar flames with pressure variations. Symp. Int. Combust. 1991,23, 333–340. [CrossRef]
28.
Hu, E.; Huang, Z.; He, J.; Miao, H. Experimental and numerical study on laminar burning velocities and flame instabilities of
hydrogen-air mixtures at elevated pressures and temperatures. Int. J. Hydrogen Energy 2009,34, 8741–8755. [CrossRef]
29.
Karpov, V.P.; Lipatnikov, A.N.; Wolanski, P. Finding the Markstein number using the measurements of expanding spherical
laminar flames. Combust. Flame 1997,109, 436–448. [CrossRef]
30.
Koroll, G.W.; Kumar, R.K.; Bowles, E.M. Burning velocities of hydrogen-air mixtures. Combust. Flame
1993
,94, 330–340. [CrossRef]
31.
Krejci, M.C.; Mathieu, O.; Vissotski, A.J.; Ravi, S.; Sikes, T.G.; Petersen, E.L.; Kérmonès, A.; Metcalfe, W.; Curran, H.J. Laminar
Flame Speed and Ignition Delay Time Data for the Kinetic Modeling of Hydrogen and Syngas Fuel Blends. J. Eng. Gas Turbines
Power 2013,135, 021503. [CrossRef]
32.
Kuznetsov, M.; Kobelt, S.; Grune, J.; Jordan, T. Flammability limits and laminar flame speed of hydrogen-air mixtures at
sub-atmospheric pressures. Int. J. Hydrogen Energy 2012,37, 17580–17588. [CrossRef]
33.
Kwon, O.C.; Faeth, G.M. Flame/stretch interactions of premixed hydrogen-fueled flames: Measurements and predictions.
Combust. Flame 2001,124, 590–610. [CrossRef]
34.
Lamoureux, N.; Djebaıli-Chaumeix, N.; Paillard, C.E. Laminar flame velocity determination for H
2
-air-He-CO
2
mixtures using
the spherical bomb method. Exp. Therm. Fluid Sci. 2003,27, 385–393. [CrossRef]
35.
Alekseev, V. Laminar Burning Velocity of Hydrogen and Flame Structure of Related Fuels for Detailed Kinetic Model Validation.
Ph.D. Thesis, Lund University, Lund, Sweden, 2015.
36.
Pareja, J.; Burbano, H.J.; Ogami, Y. Measurements of the laminar burning velocity of hydrogen-air premixed flames. Int. J.
Hydrogen Energy 2010,35, 1812–1818. [CrossRef]
37.
Park, O.; Veloo, P.S.; Burbano, H.; Egolfopoulos, F.N. Studies of premixed and non-premixed hydrogen flames. Combust. Flame
2015,162, 1078–1094. [CrossRef]
38.
Sabard, J.; Chaumeix, N.; Bentaib, A. Hydrogen explosion in ITER: Effect of oxygen content on flame propagation of H
2
/O
2
/N
2
mixtures. Fusion Eng. Des. 2013,88, 2669. [CrossRef]
39.
Sun, Z.-Y.; Li, G.X. Propagation characteristics of laminar spherical flames within homogeneous hydrogen-air mixtures. Energy
2016,116, 116–127. [CrossRef]
40. Taylor, S.C. Burning Velocity and the Influence of Flame Stretch. Ph.D. Thesis, University of Leeds, Leeds, UK, 1991.
41.
Tse, S.D.; Zhu, D.L.; Law, C.K. Morphology and burning rates of expanding spherical flames in H
2
/O
2
/inert mixtures up to
60 atmospheres. Proc. Combust. Inst. 2000,28, 1793–1800. [CrossRef]
42.
Vagelopoulos, C.; Egolfopoulos, F.; Law, C. Further considerations on the determination of laminar flame speeds with the
counterflow twin-flame technique. Symp. Int. Combust. 1994,25, 1341–1347. [CrossRef]
43.
Varea, E.; Beeckmann, J.; Pitsch, H.; Chen, Z.; Renou, B. Determination of burning velocities from spherically expanding H
2
/air
flames. Proc. Combust. Inst. 2015,35, 711–719. [CrossRef]
44.
Verhelst, S.; Woolley, R.; Lawes, M.; Sierens, R. Laminar and unstable burning velocities and Markstein lengths of hydrogen-air
mixtures at engine-like conditions. Proc. Combust. Inst. 2005,30, 209–216. [CrossRef]
45.
Wu, C.K.; Law, C.K. On the determination of laminar flame speeds from stretched flames. Symp. Int. Combust.
1985
,20, 1941–1949.
[CrossRef]
46.
Aung, K.; Hassan, M.; Faeth, G. Flame stretch interactions of laminar premixed hydrogen/air flames at normal temperature and
pressure. Combust. Flame 1997,109, 1–24. [CrossRef]

Appl. Sci. 2022,12, 7460 16 of 16
47.
Kuznetsov, M.; Czerniak, M.; Grune, J.; Jordan, T. Effect of Temperature on Laminar Flame Velocity for Hydrogen-Air Mixtures
at Reduced Pressures. In Proceedings of the International Conference on Hydrogen Safety, Progress in Safety of Hydrogen
Technologies and Infrastructure: Enabling the Transition to Zero Carbon Energy, Brussels, Belgium, 9–11 September 2013.
48. RMSprop. Available online: Keras.io/api/optimizers/rmsprop/ (accessed on 29 June 2022).
49.
Taqi, A.M.; Awad, A.; Al-Azzo, F.; Milanova, M. The Impact of Multi-Optimizers and Data Augmentation on TensorFlow
Convolutional Neural Network Performance. In Proceedings of the IEEE Conference on Multimedia Information Processing and
Retrieval (MIPR), Miami, FL, USA, 10–12 April 2018.
50.
Zhang, T. Some sharp performance bounds for least squares regression with L1 regularization. Ann. Statist.
2009
,37, 2109–2144.
[CrossRef]
51.
Gupta, S.; Gupta, R.; Ojha, M.; Singh, K.P. A Comparative Analysis of Various Regularization Techniques to Solve Overfitting
Problem in Artificial Neural Network. In Data Science and Analytics, Proceedings of the International Conference on Recent Developments
in Science, Engineering and Technology, Gurgaon, India, 15–16 November 2017; Panda, B., Sharma, S., Roy, N., Eds.; Springer:
Singapore, 2018.
52.
Povilaitis, M.; Jaseli
¯
unait
˙
e, J. FlameFoam: An open-source CFD solver for turbulent premixed combustion. Nucl. Eng. Des.
2021
,
383, 111361. [CrossRef]
53.
Bentaib, A.; Chaumeix, N.; Grosseuvres, R.; Alexandre, B.; Gastaldo, L.; Ludovic, M.; Jallais, S.; Vyazmina, E.; Kudriakov, S.;
Studer, E.; et al. ETSON-MITHYGENE benchmark on simulations of upward flame propagation experiment in the ENACCEF2
experimental facility. In Proceedings of the 12th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics, Operation
and Safety (NUTHOS-12), Qingdao, China, 14–18 October 2018.
54.
Elshimy, M.; Ibrahim, S.; Malalasekera, W. Numerical studies of premixed hydrogen/air flames in a small-scale combustion
chamber with varied area blockage ratio. Int. J. Hydrogen Energy 2020,45, 14979–14990. [CrossRef]