Yann Grégoire
email: yann.gregoire@ineris.fr

Jérôme Daubech

Christophe Proust

Emmanuel Leprette

Vented gas explosion overpressure calculation based on a multi-layered neural network

Keywords: prevention, mitigation, industrial explosions, …

The case of a gas explosion occurring in a geometrically simple enclosure, equipped with a vent is considered. It is well known in the gas explosion scientific community that the calculation of the reduced explosion overpressure, determinant in safety studies, is not trivial. Not only there is a strong dependency on the chemical kinetics of the combustible but also on the enclosure geometry, the fluid flow, the vent mechanical behaviour, shape, etc… As a result, the modelling of the physics at stake is challenging, a wide range of models are proposed in the scientific literature and this reference situation is still the object of extensive research. A new simulation approach ignoring a large part of the underlying physics is investigated. It is based on the use of an artificial neural network (ANN). The focus is given on the method of use and results obtained with the ANN rather than on the neural network itself. Our observations are discussed within the scope of industrial safety problems. Calculations performed with the relatively simple ANN proposed in the official TensorFlow tutorial, on a vented explosion database containing 268 tests, led to surprisingly good results considering the ANN implementation efforts. The tool might look promising but is also far from being as trivial as it seems at a first glance: not only the results of simulations obtained with this type of model must be examined with the greatest care but also the initial data base must be very well controlled. Routes are proposed to enhance the initial database and perform relevant analyses of the neural network predictions.

Introduction

From automotive technologies to financial forecasting or climate modelling, AI based algorithms are progressively emerging practically in all scientific fields. They can provide quick estimations or classifications with an acceptable accuracy. Above all, they promise an extreme capability to perform regressions on practically any kind of data. Example can be found relatively easily on internet on neural networks-based algorithms that reproduce efficiently the structure and presentation of a given random text that can be either a Shakespeare's work or a mathematics book including figures [START_REF] Karpathy | The Unreasonable Effectiveness of Recurrent Neural Networks, Hacker's guide to Neural Andrej Karpathy blog[END_REF]. In that example the neural network can built grammatically correct sentences, in a text that makes no sense. More generally, such models rely on data interpretation that may have been obtained through calculations or experiments. In various applications, these calculation methods completed or even entirely replaced the complex mathematical models used before. Current works investigate the possibility for such models to provide estimations of the overpressures following a gas explosion in a vented enclosure. The physics of gas explosion venting involve tremendously complex phenomena, of flame propagation in fluid structure coupled problem in which not only the chemical reaction has an importance, but also the enclosure geometry, ignition location, vent characteristics, internal as well as external flow conditions, among others. In this paper, INERIS experimental data is used together with published experimental data (see references) gas explosions on (hydrogen, methane, ethylene and propane) in vented vessels to calibrate a multi-layered neural network. The TensorFlow library [START_REF] Abadi | TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems[END_REF] and its sub-library Keras are used to build and run the model. TensorFlow was released by Google under the Apache License 2.0 on November 9, 2015. The Keras library is barely older and was initially released on the 27 th March 2015 by Chollet.

Our investigation on the neural network capabilities and performance is taken under the angle of a regression problem. Various parameters are used as input data, such as the type of gas, concentration, turbulent velocity u' before ignition, mixture expansion ratio, enclosure or vent dimensions. The selected data only consist in variables that depend on the test setup and can be accessed before conducting the explosion venting test in each configuration. The neural network is used to estimate the reduced explosion pressure. It is trained using older INERIS data test and/or published data. Then, it is confronted to other tests that:

• are not part of the training data,

• are performed on a vessel different from those listed in the training data.

The underlying idea is to examine if this tool that relies on a purely mathematic model, entirely ignoring the physics of gas explosion venting, can perform predictive calculations and under which conditions.

Neural network

Paradigm of supervised learning

As this work is dedicated to explosion science experts rather than neural network specialists, the paradigm of supervised learning with neural networks, which is used here, is briefly described. If not purely mathematical, the modelling of a physical phenomenon in the traditional approach consists in building a model from the physical understanding of the phenomenon then confronting it to a reference: an experimental or numerical observation. Obtaining a comprehensive generalisable model is the final purpose.

With artificial neural networks and supervised learning, the paradigm is changed as the model structure is chosen at the beginning and initiated partly randomly. It may consist in a large and complex structure of several interconnected functions and comprising a large number of adjustable coefficients. The network is fed with data D, and deliver a result R. Following this step, the model result is compared to the expected measurement M and the inner coefficients of the neural network are adjusted to minimize the error between R and M.

Because it is a statistical approach, this inductive process is to be repeated several hundred or thousands of times ideally on a large dataset to obtain the best possible fit. This process is called the training or learning phase. At the end of the learning process, the model is not directly interpretable and despite all the inner structure of the neural network is accessible, it may be seen as a black box from a physicist point of view. To use it, a new sample of data with the same format at that used for the training phase, is fed to the network, which gives a new result. This is called the prediction phase.

Structure and function of the MLP

In the current study, a specific type of artificial neural network (ANN) is used referred as "multilayer perceptron" (MLP) to estimate the overpressure obtained during a confined gas explosion in a vented enclosure. It consists in several densely connected layers of artificial neurons converging to a single output This neural network architecture is chosen as it is particularly well suited for regression problems (Cybenko, 1989). Analogies can be made between artificial neural networks functioning and the behaviour of neurons in a brain. In the present context, the neurons can be considered as mathematical functions that maps a given input to a desired output.

An example is given is the next figure:

Fig. 1. Schematic of a 3 layers MLP network, with 3 input features and a single output

Thus, the first step is to select the network architecture: that is the numbers of layers and neurons in each layer. This is unfortunately entirely empirical. Despite some specific architecture have been identified to function well in certain cases, there is absolutely no rule to dimension the neural network, except possibly a very general recommendation to build a network as small as possible for a given task.

Then in such network, an activation function σ is selected, in most cases, it is a continuous (derivable) functions yielding outputs between 0 and 1 or -1 and 1. A famous one is the sigmoid function:

𝜎(𝑧) = 1 1+𝑒 -𝑧 (1)
To use the neural network, it must firstly be trained, to set its internal weights (wi) and biases (bi). This is performed through a feedforward, loss estimation and back propagation phase. To put it simply, the user needs to know the real output for a given set of inputs. This first set of information is referred as the training data.

In the current example, the input features are the conditions in which a given experimental test is performed and the "real output" is the Pred obtained. Then the input features values are passed through the network, and an output estimation is obtained. For a single test, with the scheme of Fig. 1, this operation takes the form: (2)

{ ℎ 11 = 𝜎(𝑖 1 • 𝑤 1 + 𝑖 2 • 𝑤 2 + 𝑖 3 • 𝑤 3 + 𝑏 1 ) ℎ 12 = 𝜎(𝑖 1 • 𝑤 4 + 𝑖 2 • 𝑤 5 + 𝑖
The "out" value is the first estimated 𝑃 𝑟𝑒𝑑 , that will be noted 𝑃 𝑟𝑒𝑑 ̅̅̅̅̅̅ to avoid confusion with the real value. The feedforward propagation phase is performed for the complete dataset, to obtain a range of estimations of 𝑃 𝑟𝑒𝑑 .

Then the "correctness" of the estimation is examined using a loss function, which in our case consists in the mean square error (mse) evaluation (note that other loss functions such as mean absolute error or custom models may also be used).

𝑚𝑠𝑒 = ∑ (𝑃 𝑟𝑒𝑑 ̅̅̅̅̅̅ -𝑃 𝑟𝑒𝑑 ) 2 𝑎𝑙𝑙 𝑡𝑒𝑠𝑡𝑠 (3)
At last a backpropagation phase is performed to update the weight and biases of the neurons in view of reducing the error. This is the most complex part as in consists in calculating for each weight the partial derivative of the loss function. Details of this calculation may be found in the scientific literature. The important information here is that each weight is updated following a learning rate η, in view of minimizing the final error the 'mse' in our case.

In the field of artificial neural network, a cycle of processing through the complete dataset is referred as an "epoch". Several dozens of epochs may be needed to train the neural network. Note that the initial weights are set randomly in the codes, so that if the user does not alter this initial random distribution, running the same training twice should give close but different results.

Once calibrated with the training data, predictions can be made on the "test data": a set of data that passes only through the feedforward propagation phase, with no updating of the weights. In practice the prediction phase is used on 2 occasions:

1) In our case, to perform estimations of the Pred for given test configurations that are not part of the training dataset. If the experimental measurement is not known, this estimation can be referred as a prediction of the neural network.

2) The second case of use of the prediction method is actually part of the training process. In practice at each epoch, the algorithm selects a certain percentage of the training data (80 %) to perform the learning tasks then runs a prediction phase on the unused data. The resulting error is named "validation error" and is different from the error calculated on the used data, referred as "loss". Thus, in the current work we have at each epoch a validation set of data that is not part of the data used for training nor of the test data (used after the learning phase to make the predictions). Comparison of the evolution of both errors is helpful to examine the convergence of the neural network and eventually stop the training earlier if no progress is made on the validation set. It is also helpful to detect when the model is overfitted. Schematically overfitting could be seen as a model that passes through every point but fails at extracting the general trend, while the underfitting would be the average of the solution vector: for fuel efficiency). The selected neural network to solve this problem comprises two layers of 64 neurons, leading to 4865 trainable parameters. With 398 instances and 1000 epochs, the number of operations is considerable. Even so, the now popular AI frameworks such as TensorFlow and Keras used here, were developed in view of drastically simplifying the implementation and efficiency of such algorithm. The complete coding of such solution in Python is a matter of less than a hundred of lines of code, and the algorithm runs within a few minutes on a standard CPU. Using specific software such as Orange3 [START_REF] Demsar | Orange: Data Mining Toolbox in Python[END_REF] allows to perform the same tasks through visual programming, practically without entering any line of code.

Selected solution

Present work does not aim at looking for the best network architecture but is focused on the use of such tool and the changes it might bring to our field of experience. For this reason, a single MLP is used for the whole study. Starting from the example proposed on TensorFlow website applied to our data, a limited number of tests was performed using various parameters for the network architecture and functions in the early steps of this study to select our best candidate. It relies on 4 densely connected layers of 32 neurons each, for a single output. The chosen activation function is the 'ReLu' function, with the 'Adam' optimizer and the mean square error loss function. These are also the same functions as those used in the example of the official TensorFlow website. A schematic of the MLP structure is presented in Figure 3 (deeper details on the input and output of the MLP are given in the subsequent paragraphs):

Phenomenon studied

The reference case of interest is a vented gas deflagration. The situation is that of an explosible gas mixture ignited in an enclosure equipped with an opening referred as a "vent". The pressure growth in the enclosure eventually leads to the vent opening and discharge of the gases to the surrounding atmosphere with a possible external explosion. Vented deflagrations have been studied since the 1950 th . The physics is not yet totally clear, and the modelling remains challenging. Cooper et al. (1986) published a rather detailed analysis of the explosion dynamics and identified several successive pressure peaks and complex signal shapes, that depend on the explosion conditions.. A typical experimental signal recorded during such event is presented in Fig. 4. The relative importance of the local maxima visible in Fig. 4 or the acoustic perturbations starting after t = 250 ms are common features described by Cooper et al. (1986), they depend on the test parameters, among which the nature and concentration of the combustible gas, the ignition location and the vessel and vent geometries. At the same period of time, no venting model was able to account for all this phenomenology. Here we are only interested in the maximum value of the overpressure measured during the explosion, referred as Pred.

Raw data

Data concerning vented gas explosion was extracted from [START_REF] Bauwens | Vented explosion overpressures from combustion of hydrogen and hydrocarbon mixtures[END_REF]2012a and2014) ; [START_REF] Chao | An analysis of peak overpressures in vented gaseous explosions[END_REF][START_REF] Daubech | Dynamics of vented hydrogen-air deflagrations[END_REF][START_REF] Fakandu | The Effect of Vent Area Distribution in Gas Explosion Venting and Turbulent Length Scale Influence on the External Explosion Overpressure[END_REF] ; [START_REF] Kumar | Vented Explosion of Hydrogen-Air Mixtures in a Large Volume[END_REF][START_REF] Kumar | Vented Combustion Of Hydrogen-Air Mixtures In A Large Rectangular[END_REF][START_REF] Kumar | Vented Turbulent Combustion of Hydrogen-Air Mixtures in A Large Rectangular[END_REF] ; [START_REF] Liang | Scaling effects of vented deflagrations for near lean flammability limit hydrogen-air mixtures in large scale rectangular volumes[END_REF] ; [START_REF] Pasman | Design of Pressure Relief Vents[END_REF] ; [START_REF] Skjold | Vented hydrogen deflagrations in containers: Effect of congestion for homogeneous and inhomogeneous mixtures[END_REF] ; [START_REF] Sun | Effect of vent conditions on internal overpressure timehistory during a vented explosion[END_REF] ; [START_REF] Wang | Effects of hydrogen concentration on the vented deflagration of hydrogen-air mixtures in a 1-m3 vessel[END_REF] and [START_REF] Yao | Explosion venting of low strength equipment and structures[END_REF]. Note that the works from [START_REF] Daubech | Dynamics of vented hydrogen-air deflagrations[END_REF] If the data is grouped by study (author and year), by fuel and by vessel volume, 30 "independent" sets of data can be formed.

Several of these tests were performed in the scope of studying the effect of low Pstat, inertia-less vent panels that often consisted in plastic foils glued to the vessel, attached with duct tape or held with low strength magnets. As such, the Pstat is not a calibrated value and should rather be seen as an estimation based on experimental observation. However, an order of magnitude is enough here, as the purpose of current work is the examination of the neural network, rather than an optimal description of the physics of the explosions.

Data formatting

A first issue arises for the physicist here: which data would be relevant to solve this problem? The authors of the current paper are specialists of industrial explosions. Other profiles may select other data, possibly not relevant to the problem. As this work targets to experts in explosion, we will not expand more on this aspect. It is however important to remind that the works considered here depend on this selection of relevant parameters. For instance, we will not rely on the laminar flame velocity and burnt gas expansion ratio (which could however be an option), but only on the gas nature and its volume fraction -which are related to the former parameters.

Then, there is an issue with the format of the data: how to differentiate in the neural network whether hydrogen or methane is tested, or the difference between central and backward ignition.

To overcome difficulties related to the gas nature, it has been decided to categorize the type of gas and replace the gas name column with 4 columns: "hydrogen", "methane", "propane" and "ethylene", that contains a value of 1 or 0 depending on the gas considered. As published on TensorFlow official website, this appear to be a common practice in neural network related problems.

To account for the dimensions and shape factor of the enclosure, it has been decided to retain the vessel hydraulic diameter, its length and volume and consider a shape variable equal to 0 for cuboids, 1 for cylinders and 2 for spheres.

Then, to model central or backward ignition, we simply use the distance from ignition point to the vent.

As a result, a 268 lines table containing only numbers over 11 columns is obtained. The columns correspond respectively to the type of gas (4 columns), gas volumetric concentration in air, vessel shape coefficient, length, diameter and volume, the ignition location coefficient, the vent area and its static opening pressure, the turbulent flow velocity u' at ignition, and finally the Pred measured in the test.

Note that the Pred is the result vector and all columns before that, correspond to data that are directly part of the test configuration and no specific modelling such has flame velocity calculation is considered.

The principle here is to train the ANN on this test data in the so-called training phase. Then in the prediction phase, the knowledge of Pred is required. In other words, once the training is terminated, the knowledge of the test parameters is enough with no further calculation of any intermediate variable, to estimate a Pred.

Data splitting

The goal of the present study is not the development of a code able to predict the vented vessel overpressure in case of a gas explosion, but to examine the capabilities of the neural network to do so. Therefore, several tests are designed in view of examining the performances and robustness of the neural network. They consist in using a part of the data to train the network, then validate it with the other part, that has not been used for the training. It is the same work as that performed during the training, with the difference that the data used later for validation is chosen by the user rather than randomly selected by the code. Two types of data splitting are investigated: the general random separation and the laboratory or group separation.

Basic configuration: general random data separation

In this case, 10 % of the tests are randomly selected as test data in the original 268 test table . During a same study, the tests configurations can be assumed to be relatively similar because the vessel and fuel are the same, in some occurrences only the fuel concentration is varied. Removing a few samples for the database and trying to test the missing configurations should yield satisfying results. The subjacent idea is that if knowledge of the test results in similar conditions is available, the network should perform well.

Laboratory/Group data separation

Here, the separation is between our (INERIS) measurements and the other published. The question concerns the generalisation capabilities of the model: can we, from our tests only, predict all the results published by other scientists on this topic? and the opposite: would it possible to find the INERIS measurements from the data published by researcher other than those from INERIS? This approach differs from the earlier as, in this case, the test configuration may differ significantly between the test and the training data. For instance, INERIS has neither 63.7 nor 120 m 3 vessels.

Then other tests are performed selecting randomly 3 groups of studies as test data. The 30 groups available in total correspond to tests from different studies/publications also with different fuels and different vessels.

Data augmentation

Each time a training data set is prepared from the earlier separation, it is expanded through a data augmentation method. Neural networks are particularly efficient when large amounts of data are available, while they tend to lack of stability with smaller data samples. However real scale explosion tests involve significant costs, and their number is limited. Given the fact that we need to reduce our database to perform the validation tasks, the number of tests becomes insufficient. Based on the idea that experimental measurements may not be exact, the dataset is expanded 100 times supposing for each relevant feature (typically not the gas nature) a random error factor between 0 and a maximum. This maximum is 5 % for the metric dimensions: those of the vessel, the vent, the ignition location. An error up to 5 % is also assumed on the vent Pstat and the measured Pred, while a maximal error of 1 % is considered on the fuel volume fraction.

To put it simply, for a test with 10 % of methane gas, the algorithm generates 100 neighbours with a random methane concentration between 9.9 and 10.1 %. Different random factors, within the ranges described earlier, will also be applied simultaneously to the other variables of the same test.

Data and code publication

In paragraph 2.3, it has been asserted that the data manipulation and development and use of MLP based method with libraries such as TensorFlow in Python is relatively accessible, without significant knowledge in computer sciences. To illustrate this point, the code used to prepare the current article is published on a Zenodo repository [START_REF] Grégoire | Vented gas explosion overpressure calculation based on a multi-layered neural network -Code and data[END_REF], https://doi.org/10.5281/zenodo.5497942). It consists in 2 csv files containing the test database and the rules used to perform the data augmentation, as well as 3 python files dedicated to the data augmentation algorithm, the construction, training and use of the MLP. The example given in the repository concerns more specifically the works discussed in paragraph 5.1: the random selection of 10% of the tests in the database as independent test data, the 90 % other being the data firstly augmented then used to train the network. However, enough material is given to be able to reproduce all of the MLP results shown in the paper, as the user will only need to tweak either the selection in the original database (most cases) or the creation (for physical tests) of the test data.

"Traditional" modelling approach

In view of comforting the analysis, a few of the ANN predictions are also compared to those made with various kinds of empirical or phenomenological model. However, the purpose of this section is not to document the model performances but rather to highlight the complexity of the phenomenon described and give a focus on the consequent research work still needed.

4 With Av the vent area, KG the gas explosion index, Pstat the vent static opening pressure and V the enclosure volume.

This formula is generally admitted being overly conservative except in the case of turbulent deflagrations of highly reactive gases (such as hydrogen).

Alternatively, the National Fire Protection Association's (NFPA) standard on Explosion protection by deflagration venting, NFPA 68, provides venting models for gaseous mixtures explosions. This standard is regularly updated, and the model can change drastically from one release to the next. Here the example of the 2013 version is presented. The expressions are presented in such a way that the vent area Av can be calculated as function of the reduced internal overpressure Pred.

If the reduced overpressure Pred is lower than 0.5 bar :

𝐴 𝑣,0 = 𝑆 𝑢 𝜌 𝑢 2 𝐺 𝑢 𝜆 𝐶 𝑑 [( 𝑃 𝑚𝑎𝑥 +1 𝑃 0 +1 ) 1 𝛾 𝑏 -1] (𝑃 0 + 1) 1 2• 𝐴 𝑠 𝑃 𝑟𝑒𝑑 1 2 ⁄ (7)
When the reduced overpressure P red is over 0.5 bar :

𝐴 𝑣 = 𝐴 𝑠 [1-( 𝑃 𝑟𝑒𝑑 +1 𝑃 𝑚𝑎𝑥 +1 ) 1 𝛾 𝑏 ] [ ( 𝑃 𝑟𝑒𝑑 +1 𝑃 𝑚𝑎𝑥 +1 ) 1 𝛾 𝑏 - (( 𝑃 𝑠𝑡𝑎𝑡 +1 𝑃 0 +1 ) 1 𝛾 𝑏 -1) (( 𝑃 𝑚𝑎𝑥 +1 𝑃 0 +1 ) 1 𝛾 𝑏 -1) ] 𝑆 𝑢 𝜌 𝑢 2 𝐺 𝑢 𝜆 𝐶 𝑑 (8) Where -
As is the internal area of the enclosure -Pmax is the maximum pressure developed in a closed deflagration by ignition of the same flammable mixture (in bar) -P0 is the enclosure pressure before the ignition (in bar) -Su is the laminar burning velocity of the flammable mixture ρu is the density if the unburned flammable mixture -Gu is the unburned flammable mixture sonic flow mass flux (𝐺 𝑢 = 230.1 𝑘𝑔 𝑚 2 ⁄ . 𝑠𝑒𝑐) λ is the flame wrinkling factor due to the turbulence and flame instabilities γb is the ratio of the specific heats for the burned gases (between 1.1 and 1.2 depending on the gas mixture) -Cd is the vent flow discharge (𝐶 𝑑 = 0.8) -Pstat is the nominal vent deployment or the static activation pressure (in bar)

The flame wrinkling factor λ calculation is not trivial, it depends on various parameters such as the presence of obstacles and the Reynolds number at the vent. [START_REF] Rodgers | NFPA 68 new gas venting equations[END_REF] showed that NFPA68:2013 model gave conservative estimations of the vent are in a vast majority of industrial cases. Fakandu et al ( 2013) compared both NFPA68:2013 and EN14994:2007 models and demonstrated a better fit on experimental data with the former, despite it seem insufficient in cases with turbulent hydrogen, or when many small obstacles are present near the vent.

4.2

Ulster University models [START_REF] Molkov | Modeling of vented hydrogen-air deflagrations and correlations for vent sizing[END_REF][START_REF] Molkov | Passive ventilation of a sustained gaseous release in an enclosure with one vent[END_REF]) [START_REF] Molkov | Modeling of vented hydrogen-air deflagrations and correlations for vent sizing[END_REF] proposed a model considering the flame instabilities and of the interaction between flow created by the vent (turbulence) and the internal explosion. These two aspects are covered using the laminar and turbulent "Bradley" numbers (Br and Brt : Bradley and Mitcheson;1978). The turbulent Bradley number depends on the Degree Of Interaction (DOI) between the unburnt gas flow through the vent, creating turbulence, and the internal explosion. A few fitting parameters remain, tuned on existing experimental data.

Since the number of data increased over time, various versions of this model were proposed (1999,2001,2008). The correlations and fitted parameters are given in the following tables.

These formulas rely on the definition of Degree of Interaction factor (DOI) which coefficients were initially determined empirically: Where E is the combustion products expansion coefficient; γ the reactants specific heat ratio; Av the vent area (m 2 ); V, the volume of enclosure (m 3 ); Su, the initial laminar burning velocity (m/s); cu the speed of sound (m/s); V# the dimensionless volume (numerically equal to enclosure volume in cubic meters); Pv, the dimensionless static activation pressure (𝑃 𝑣 = (𝑃 𝑠𝑡𝑎𝑡 + 𝑃 𝑖 )/𝑃 𝑖 ); Pi#, the dimensionless initial pressure (numerically equal to initial pressure in absolute atmospheres); Pi the initial pressure (Pa) and Pstat the static activation pressure (Pa).

Later, [START_REF] Molkov | Passive ventilation of a sustained gaseous release in an enclosure with one vent[END_REF] proposed a new way to estimate DOI based on the findings and methods used derived computational fluid dynamics (CFD) calculations. The DOI number was since then defined a product of flame wrinkling factors:

𝐷𝑂𝐼 = 𝛯 𝐾 * 𝛯 𝐿𝑃 * 𝛯 𝐹𝑅 * 𝛯 𝑢 ′ * 𝛯 𝐴𝑅 * 𝛯 𝑂 (15)
• ΞK is the wrinkling factor due the Landau-Darrieus flame instabilities.

• ΞLP is the wrinkling factor due to "leading point" mechanism, related to flame acceleration toward the vent. • ΞFR is the wrinkling factor due to fractal increase of flame surface area linked to the development of the Rayleigh-Taylor instabilities. • ΞAR a wrinkling factor to account for aspect ratio of the enclosure • ΞO is the wrinkling factor to account for the presence of obstacles. Because of limited data, Molkov admitted that Ξ 𝑂 = 1 without obstacles and Ξ 𝑂 = 3.5 when obstacles are located between the ignition point and the vent. • Ξu' is the wrinkling factor due to the initial turbulence defined on the basis of the disturbed laminar burning velocity :

Along with this updated version of the DOI, the following new correlation estimate Pred was given:

𝑃 𝑟𝑒𝑑 = 0.91 * 𝐵𝑟 𝑡 -1.06 (16)
Comparisons between these different models were made by [START_REF] Rocourt | Vented hydrogen-air deflagrations in a small enclosed volume[END_REF] and [START_REF] Jallais | An inter-comparison exercise on engineering models capabilities to simulate hydrogen vented explosions[END_REF]. It was concluded that the model of 1999 generally gave the best results as compared to that of 2001 and 2008. The latter version of the model ( 2014) was not available at that time. (Bauwens, 2012) Bauwens's model ( 2012) proposes an estimation of the major pressure peaks P identified in experiments (those described by Cooper et al., 1986). A single expression is used for all of those:

FM Global Model

𝑃 - 𝑃 0 = 𝑃 𝑒 𝑃 0 (1 - ( 𝛾+1 2 ) 𝛾 𝛾-1 (𝐴 𝑣 * ) 2 ) -1 (17) with: 𝐴 𝑣 * = 𝐶 𝐷 ( 𝑅 𝑇 𝑣 𝛾(𝛾+1) 2 𝑀 𝑣 ) 1 2 • 𝑎 𝑐𝑑 * 𝐴 𝑣 𝑆 𝑢 𝐴 𝑓 (𝜎-1) (18)
Where P_, Av, and Av* are pressure (for the different peaks), vent area, the "vent parameter" and the subscripts 0, e, f, and v corresponding to ambient, external, flame and vent conditions. In addition, E, γ, cd, R, Tv, and Mv are the expansion ratio, the specific heat ratio, a discharge coefficient (equal to 0.6), the universal gas constant (8.314 J/kg/mol), and the temperature and molar mass of the vented gas, respectively.

In case of an ignition on the wall opposed to the vent, the different values of P are estimated by choosing appropriate values for Af and Su; as shown in the table thereafter:

P4 𝐴 𝑓 (𝑃2) = 0.9 (𝐴 𝑐𝑤 -𝐴 𝑐𝑜𝑛𝑡𝑎𝑐𝑡 )

With 𝐴 𝑐𝑜𝑛𝑡𝑎𝑐𝑡 = 0.9 (𝐴 𝐵𝑊 -𝐴 𝑣 ) 𝑆 𝑢 ~ Ξ 𝐴 𝑆 𝐿

Where p = 1.6075, L, is twice the length of the enclosure; w the width and h height. Acw is the internal area of the enclosure, α=0.63, BR is the average area blockage and N is the average number of layers of obstacles in the flame path. SL is the laminar burning velocity and Le the Lewis number. ΞA is an empirically determined constant flame-wrinkling factor. Note that if the ignition does not occur on the face opposed to the vent, different formulas are proposed by Bauwens et al. (2012).

In terms of physics, this model is more accessible than that of Molkov. However, [START_REF] Jallais | An inter-comparison exercise on engineering models capabilities to simulate hydrogen vented explosions[END_REF] showed that, in presence of obstacles, the results of the experiments of Daubech et al. (2013) are not correctly estimated (overestimation by at least a factor 2). Also, for small enclosures, the maximum internal overpressure of a vented deflagration tends to the maximum adiabatic overpressure (10 b) which is unrealistic.

INERIS's SECEX model

To model confined gas explosion phenomena and keep a satisfying understanding on the course of the events, INERIS developed the SECEX code, a phenomenological software consisting of interlinked models each dedicated to a single aspect such as flame propagation in a volume, turbulence characteristics prediction, combustion rates, mechanical resistance, pressure effects… Deeper details on this model may be found in recent works of Proust and Leprette (2010), [START_REF] Daubech | Propagation of a confined explosion to an external cloud[END_REF] and [START_REF] Duclos | Développement de modèles phénoménologiques et de maitrise des risques d'explosion pour la filière émergente hydrogène-énergie[END_REF]. Most of the physics is derived from fundamental research. Each model is qualified separately, and the overall consistency can be compared to realistic full-scale experimentation or actual accidents. This tool has been designed in a context of process safety; this implies that it was built to provide conservative estimations in most situations. To model the secondary explosion phenomenon the model partly relies on empirical observations. In the vessel, the flame is modelled as an ellipsoid surface directed towards the vent. In front of the vent, [START_REF] Lannoy | Analyse des explosions air-hydrocarbure en milieu libre : Etudes déterministe et probabiliste du scénario d'accident. Prévision des effets de suppression[END_REF] model is used to estimate the overpressure effects of the secondary explosions:

Δ𝑃 𝑒𝑥𝑡 = 3 2 . 𝜌. 𝑉 𝑒𝑥𝑝 2 (22)
with Vexp, the flame velocity in front of the vent. Vexp was measured in INERIS tests and an empirical correlation [START_REF] Proust | The dynamics of vented gas explosions[END_REF] was extracted from these tests to determine a value of Vexp for the other tests of the database (Figure 5).

Fig. 5. Proust et Leprette (2010) correlation on the outer cloud expansion velocity during vented gas explosions

The correlation takes the form:

𝑉 𝑒𝑥𝑝 = 10. 𝑉 𝑣𝑒𝑛𝑡 0,5 (23)
The secondary explosion contribution to the internal pressure effects is considered under 2 angles: a part of the energy (proportional to the vent area over enclosure cross section ratio) contributes to the vessel compression while the overpressure outside also impedes the gas discharge through the vent. In terms of complexity this model is relatively simple as it does need for instance, the knowledge of a flame wrinkling factor but it still relies on an iterative calculation and does not account for specific features such as obstacles.

The SECEX model yields the following results when applied to a part of the dataset [START_REF] Duclos | Développement de modèles phénoménologiques et de maitrise des risques d'explosion pour la filière émergente hydrogène-énergie[END_REF]:

Fig. 6. Comparison of the SECEX tool estimations with experimental data.

The average absolute error is 245 mbar and the standard deviation 250 mbar on this set. Here we purposely showed a limited part of the dataset, consisting in tests performed in vessels of parallelepipedal shapes with vents of low Pstat, with is the reference configuration for which this model has been developed. The database mostly consists in test in this configuration but also includes cylindrical and spherical vessels and some tests with vent panels having a Pstat up to 300 mbar. Thus, on the full dataset the average error is bigger, on the order of 400 mbar.

Synthesis on the "traditional" modelling approach

Different limits are exposed for these models that have been design with different goals (pure vent dimensioning on a process, accurate estimation of the pressure peaks, description of secondary explosions, etc..). The accuracy of the prediction put aside, one can notice a certain complexity of each model which rely on highly non-linear equations, and in some cases: input parameters that may be difficult to access such as the wrinkling factor, the DOI or a specific flame surface calculation In contrast the neural network implementation does not require the user to solve equations and the input data here is chosen to be fairly easily accessible as it mostly consists in geometric dimensions of the experiment and information on the vent Pstat and the test conditions (gas nature, concentration, ignition location and initial turbulence).

Results of the ANN predictions

Results are presented for each of the cases described in the paragraph 3.

Basic configurations: general random data separation

In a first attempt, 10 % of the lines in the database are randomly selected and isolated as the test data (Table 6). The rest of the data -268 -26 = 242 test is augmented 100 times to build a 24442 lines matrix (242+242*100). It is used to train the neural network. Thus, the samples from the test database are different but should remain generally close to the training dataset. The typical results obtained in this configuration are shown in Fig. 7. The absolute error is 55 mbar (the average error is 6.2 mbar) and the standard deviation is also 55 mbar. If compared to the phenomenological model results, absolute error attained with the ANN is the lowest in 22 of the 26 cases, the latter showing an average error above 250 mbar. However, the ANN only leads to a conservative estimation in 16 cases versus 22 for the phenomenological model.

. Among those randomly selected tests, only half ( 13) are sufficiently documented to compute the Pred with all the other models listed in section 4. In Fig. 8 (numerical data in table 7), the predictions obtained with the various models are presented. Those consist in tests with hydrogen, in the following test configurations:

Table 6 List of randomly selected tests in the database to verify the functioning of the neural network and for which sufficient data is available to estimate the overpressure with the models of NFPA68:2013, Bauwens et al. (2012) and [START_REF] Molkov | Modeling of vented hydrogen-air deflagrations and correlations for vent sizing[END_REF][START_REF] Molkov | Passive ventilation of a sustained gaseous release in an enclosure with one vent[END_REF]."I" refers to the ignition point: it is 0.5 if the ignition occurs in the center of the enclosure and 1 if it is carried out at the bottom opposite the vent. S.I units These results point out an overall better efficiency of the neural network. Worse, if only the mean absolute error is retained (which is certainly not a good practice if industrial safety is considered), this model performs at least about twice better than any other, on this specific sample. Of course, a statistical analysis on such small sample is irrelevant. Furthermore, it has been chosen to highlight the best result at each line of Table 7, not considering that in some cases other models also lead to very satisfying estimations, which is disputable, considering that a 20 mbar difference is negligible on such measurements. The discrimination between lower and stronger (above 100 mbar) underestimates remains also subjective. In comparison, one can put on the line that the 100 mbar values correspond to the static opening pressure of most explosion vents and the lower limit for the Pstat parameter in the European standard vent dimensioning formulas. The objective here is to give a focus on the excellent performance of the neural network, on these tests that were chosen randomly in the database, given the fact that barely any time was spent by the authors studying the physics or configuring the neural network, as we basically kept the example proposed on the official website of TensorFlow. Furthermore, the computation cost is negligible at this scale, as a few minutes are needed on a standard CPU to train the neural network while the prediction phase is practically instantaneous.

However, if the ANN model can be seen as some sort of polynomial fit over the dataset, one can expect that close configurations will yield results close to those used for the regression as effectively shown here. The real problem is in the generalisation capabilities of the model.

Laboratory/Group data separation

The data is split in 2 sets: one contains all the tests performed at INERIS (105 tests), the other contains the measurements published by other researchers (163 tests). In one case INERIS data is used to train the ANN and the other set is the test data. In the second case it is the opposite. This illustrative path yields the following results: It seems that INERIS dataset is slightly better to train the ANN. In some occurrences unphysical results such as negative overpressures may be found. However, in both cases large discrepancies appear, confirming that the missing data has an adverse effect on the neural network performance.

Another test was run selecting randomly 3 groups of data as test data (and removed from the initial database used to train the neural network). As a reminder, we define as a "group of data", the tests performed by a same author, during a same study with a single combustible gas and a single volume. The operation has been repeated 3 times (i.e. 3 random selections of 3 groups), leading to a total of 89 tests, to limit the bias introduced by the random selection of the samples. An histogram classifying the relative differences between the predictions / estimations (respectively of the ANN and the SECEX tool) and the experimental measurements are presented in Fig. 10.: So, this appear that despite it needs to rely on significant amount of data, the ANN approach yields consistent results and can often provide estimations that are on average closer to the experimental measurement than those given by the physics-based code. However, the errors are evenly distributed around zero, while with the SECEX model, most of the estimations yield conservative results. Such result was expected as the ANN was designed to fit data while the SECEX model is set to provide conservative estimations, so that it could be use in a context of industrial safety. It is however conceivable to tweak a neural network to reproduce such behaviour, for example by purposely increasing the targeted pressures values by a few percent in the training phase or applying safety margins on the network output.

Physical relevancy

In this section, the interest is focused on the behaviour of the ANN model. A sort of "sensitivity analysis" is performed to evaluate the physical relevancy of the model proposed by the neural network. In practice, the ANN is firstly trained with the whole dataset. Then 2 representative test configurations are selected in the database for which the largest amount of "resembling" data is available. The objective is to have the largest confidence in the capabilities of the neural network in the vicinity of those 2 configurations (Table 8). From these 2 test configurations, test data is produced by varying a single input parameter (for instance the gas concentration) and keeping the other parameters constants. In the current examples, the varied parameters are:

• the volume % of the combustible gas in air, which is varied from 0 to 80 %,

• the vent area, which is varied from 1 to 25 m 2 ,

• the vent Pstat, which is varied between 5 and 200 mbar.

First, for the volume concentration, the code leads to the results presented in figure 11. Clearly meaningless results are produced with negative overpressures.

Fig. 11. ANN predictions as function of fuel fraction in air

Note that for methane, the tests were performed in the 8-13 % range with a large majority of tests around 10 %, while those with hydrogen in the 5-34 % range, with a large majority of tests around 20 %. As already known, the ANN entirely ignores the physics, but it is especially critical here for the hydrogen case. Indeed, the model even fails to yield physical result in the when the gas concentration exceeds 28 %. Note that one could have expected a better performance as the database includes tests with volume fraction up to 34 % (yet not on this 63.7 m 3 vessel). In the methane case, negative overpressures are reached after 20% but it is not as critical as in the hydrogen case: the higher explosive limit of methane being located around 15%, the mixture in not explosive and this result is less prone to dramatic misinterpretations. A slight improvement of the model could be made assuming that the overpressure is null as soon as the gas mixture is out of the explosivity range of the gases (about 5-15% for methane in air and 4-75% for hydrogen).

Those curves are globally showing false trends, if the whole picture is examined. However, they should be also presenting physically representative trends in a local window more representative of the training dataset.

Unfortunately, the discrimination between both behaviours is difficult. A better mastering of the neural network, and in particular a way to constrain it to respect specific physical laws such as the "Pred is higher than Pstat" may be helpful to verify trends or derive new empirical models from these predictions. Such approach seems possible working on the ANN loss function and output conditions [START_REF] Liu | Multi-Fidelity Physics-Constrained Neural Network and Its Application in Materials Modeling[END_REF], but it involves significant effort and a deeper understanding in neural network processing.

Because the initialisation of the network is based on initial random numbers and the training consists in minimizing the error of a group of functions, the neural network, despite leading to similar results within its functioning range, is never the same after a training phase. Thus, a solution to the problem of the definition of the range of validity of the ANN could be to train a second network (that may keep the same architecture, to save possible computational development costs) on the exact same training dataset. Doing so, when the same test data is introduced in this a second version of our ANN, named v2 we obtain the curves of Fig. 12. The prediction may not be more relevant but now the regions where V1 and V2 diverge are likely to point out the ranges where the ANN method fails.

Fig. 12. ANN predictions as function of fuel fraction in air. Comparison of 2 versions of the ANN

Alternatively, a good superposition enhances but does not guarantees good predictions. Because of the low computational costs involved in such approach, it could be extended to larger numbers of iterations. Note that for the methane case the 2 MLP lead to similar result over most of the flammability range (8-13% to compare to 5-15%), indicating that on this specific configuration (cylindrical vessel of volume 63.7 m 3 with a 5.4 m 2 vent and a Pstat as low as 5 mbar) the amount of training data could be dense enough to perform reliable estimations. Further tests remain needed to validate or extend this hypothesis.

In the following, two versions of the ANN are systematically employed. Note that this method may be insufficient to trace out of all the validity difficulties. Again, the objective of current paper is to provide information on the practice rather than an efficient model.

The effect of vent area and vent bursting pressure is shown on Fig. 13: a) b)

Fig. 13. ANN predictions as function of vent area (a) or vent Pstat (b)

Concerning the vent area, for methane the ANN clearly indicates a strong influence on Pred decreasing for vent sizes increasing from 1 to 6 m 2 . This is an expected physical behaviour. In further details it appears that in this area the vent size is linearly dependent on the square root of the overpressure, with is in line with the "traditional" models for vent dimensioning shown in paragraph 4. Thus, using neural networks in such way could also be useful to develop new physical models. The agreement is not as good for the case of hydrogen and vent areas around 3-4 m 2 but the overall trend is still respected.

However, the agreement is generally poor on the vents Pstat. On the initial training dataset, most of the tests are performed with vents of Pstat lower than 20 mbar (200 over 268 tests), often assumed to be close to 5 mbar, in the absence of any better measurement (as said earlier, these vents often consisted in plastic foils glued or taped on the vessels). In any case, this underlines the strong dependency of the network estimations on consistency of the dataset used to train it.

Robustness

Other tests of this kind, on the vessel volume or turbulence intensity could be made, but it is not the object of current paper. The authors wish to point out the specific difficulty that may be encountered when dealing with these algorithms and linked to their extreme robustness. The exercise performed in section 5.1 in which 10% of the dataset is selected as test data is conducted a second time with the exact same tests (no new random selection), but with the introduction of an error in the training dataset. It consists in shifting by 1 cell all the overpressures in the initial table (i.e. the last column of Table 2), the first value of the table is replaced arbitrarily by the last one. Note that this accidental shifting of the ANN output column could be due a manipulation error from the user such as a coding error in the table indexes. Then the data is augmented (see paragraph 3.5) and the ANN is trained with this corrupted dataset. The code yields the following results: Only limited differences appear between this case and that shown in Fig. 5. The absolute error is 125 mbar (versus 55 mbar before) and the standard deviation is 190 mbar (versus 55 mbar in Fig. 5.). In practice, there is no way to detect the error in the dataset: the ANN was trained on corrupted data on which it managed to perform an excellent fit. The output result is however subject to more randomness despite it is barely visible in the current case. Thus, because of their extreme ability to fit data, those algorithms, when they are trained with corrupted data, produce false models without the user being informed.

Discussion

On a general basis, in this work, a remarkable efficiency and robustness of the ANN based tool is shown, and in particular its extreme capability to fit data. As they stand, this kind of tools produces results based on a purely "inductive" method (Escande et al., 2013). Despite they consist in purely mathematic tools, with no physics behind, it is possible to extract general physical trends from the results. Note that in the current work, a single neural network architecture was selected and not modified for the whole study. It is based on the default pre-programmed parameters and online recommendations of the official TensorFlow's tutorials. This underlines the simplicity of the task and the consequent work achieved by the developers of the concerned libraries. Placing more attention on the specific aspect of the network architecture could lead to an even increased performance of the tool described in the current paper. However, it is also show here that, despite its simplicity of implementation, this type of calculation is far from trivial and the results of simulations made with this type of model must be examined with the greatest care. If the code is intended to be empirical, the critical analysis of the results may nonetheless be very complex because of:

• the data: the multiplicity of hypotheses that can be made on the training data is a first complexity if for example, as here, we rely on data from very different sources, more or less documented. Note that the selection of the meaning parameters covering the phenomenology (ex: if nature of the gas is lacking a relevant parameter would lack) to include in the data is a critical part of the physicist job in this problemthe other part being the examination of the results consistency in regard to physics law's (for instance obtaining positive pressures at the outputs of the ANN). Having a significant amount of data is critical to build the model, which may pose a difficulty as real scale explosion tests are expensive. A numerical data augmentation method based on the measurement's accuracy was proposed and successfully tested, but it can be seen as a way to weight certain selected tests and to orientate the ANN. This may not be satisfactory. • the structure of the network: there is a great diversity of possible neural architectures: we did not focus on this point because it was not the object of the study, but the construction of the network is based on choices on the number of neurons, of layers of neurons, of minimization criteria, of convergence, a certain number of iterations or of output parameters. An interesting alternative solution would consist in the adding physical constraints. In addition of releasing the limitation of the breadth of the database, this solution may help to avoid the well-known intrinsic limitations of "inductive" method. "Physically" guided neural network exist and have been the object of publications in various fields (for example refer to [START_REF] Liu | Multi-Fidelity Physics-Constrained Neural Network and Its Application in Materials Modeling[END_REF]). In the present example, it could consist for instance in specifying that in the absence of the vent, the Pred obtained is the gas Pmax or that at too lean or to rich concentrations, the Pred is 0. • the choice of the outputs: one chose the Pred here but one could have asked as well the 4 peaks of pressure of Cooper's model ( 1986), a signal of pressure or a physical parameter like a term of speed of flame to be used then in other models. In the meantime, it is suggested to perform comparison of several models trained on the same datasets, to verify locally the overall consistency of the model predictions. Two models trained on the same data should lead to similar predictions. When it is not the case, it may mean that the ANN perform an extrapolation too far from the initial training dataset.

Finally, it is envisioned that once enough confidence on the ANN is reached, a parametric study on its input data is a computationally cheap way to extract or verify empirical observations as well as developing new physical models. It may for instance be useful to identify specific behaviours, unknown sensitive variables or unknown unknowns in a problem.

Conclusions

The problem of overpressure generated during a gas explosion in a vented enclosure has been examined under the angle of a regression problem with the use of a multilayer perceptron, a specific type of artificial neural network. The use of such algorithms drastically increased over the past 5 year and is particularly supported by the publication from a few sector leaders of open source tools such as TensorFlow, among others. These tools were specifically designed in view of simplifying and democratizing the access to neural networks and the associated methods. In the current work, a remarkable efficiency and robustness of such tools is shown, and in particular their extreme capability to fit data. However, it is also demonstrated that despite its simplicity of implementation, this type of calculation is far from trivial and the results of simulations made with this type of model must be examined with the greatest care. Recommendations are made, mostly on the training data and the necessary post-processing of the results in view of improving the interpretability of the results obtained with such method.

Fig. 2 .

 2 Fig. 2. Illustration of model underfitting or overfitting

Fig. 3 .

 3 Fig. 3. Schematic of the MLP structure used in this study

Fig. 4 .

 4 Fig. 4. Example of experimental signal of the overpressure recorded during a vented gas deflagration (16.5 % H2-air ignited at the center of a 4 m 3 vessel with a 0.5 m 2 vent)

  Fig. 7. a) typical convergence curves for the MLP training. b) comparison of ANN predictions and measurements on 26 randomly selected tests (not present in the training data)

Fig. 9 .

 9 Fig. 9. ANN Predictions vs. measurements for datasets arranged by laboratory: a) the train data is that of INERIS while the test data is the other published test and b) opposite case: the train data is the published test from other laboratories while INERIS data is the test data.

Fig. 10 .

 10 Fig. 10. Histogram of the relative error (prediction/estimation minus experimental measurement) calculated with the ANN and the SECEX models when randomly selecting as test data 3 studies in the initial dataset (again those are not used in the training phase and the graphic is compilation of the results of 3 trained neural networks).

  Fig. 14. a) Convergence curves for the ANN training with corrupted data. b) comparison of ANN predictions and measurements on the same 26 tests as in Fig.5 with the wrongly trained ANN.

  3

  • 𝑤 6 + 𝑏 1 ) ℎ 21 = 𝜎(ℎ 11 • 𝑤 7 + ℎ 12 • 𝑤 8 + 𝑏 2 ) ℎ 22 = 𝜎(ℎ 11 • 𝑤 9 + ℎ 12 • 𝑤 10 + 𝑏 2 ) 𝑜𝑢𝑡 = 𝜎(ℎ 21 • 𝑤 11 + ℎ 22 • 𝑤 12 )

Table 2 :

 2 consists in tests performed by the authors of the current paper. Randomly selected lines in the database, right side columns for the same tests as in Table1 -The shape column refers to the vessel volume, shape, the ignition is performed either at the amrous of the vessel or at its back (BW stands for backwards) far from the vent u' is the turbulent velocity at ignition time

	In total we found 202 published tests. Other still unpublished data from INERIS is used hereafter, which consist
	in 90 tests. Because printing all those tests in the current paper would be unpractical (268 lines over 17
	columns) and is not necessary for the current analysis, only a sample is presented as an illustration in tables
	1 and 2 thereafter. Geometric data on the test vessel (shape, volume, vent area, vent bursting pressure),
	information on the fuel (nature, concentration, laminar burning velocity, expansion ratio) are listed with the
	maximum recorded overpressure.
	Table 1: Randomly selected lines in the database, left side columns. L, W and H respectively stand for the
	length, width and height of the rectangular enclosures (for cylinders, L is for the length, but W and H are both
		the vessel diameter; for spheres, only the diameter is displayed)
	Test #	Author	Year	Fuel	%vol	L (m)	W (m)	H (m)
	2	Bauwens	2011	Hydrogen	16.5	4.6	4.6	3
	49	Chao	2010	Hydrogen	18	4.6	4.6	3
	69	Daubech	2011	Hydrogen	14	5.5	1.6	1.6
	102	Kumar	1989	Hydrogen	10	2.36	2.36	2.36
	118	Kumar	2009	Hydrogen	5.9	10	4	3
	155	Liang	2017	Hydrogen	6.2	4.5	4	3
	171	Skjold	2018	Hydrogen	15	5.867	2.352	2.385
	177	Sun	2018	Ethylene	7	2	1.2	0.6
	197	Wang	2018	Hydrogen	34	1.8	1	0.55
	218	Ineris	2019	Hydrogen	10	2	1	2
	266	Ineris	2009	Methane	11.1	1.83	0.834	0.834

All information listed in this table, except Pred is known before the test. We intend to use this information in a neural network to predict Pred. In total, the database contains test results: • from 22 different studies from 12 authors between 1974 and 2019, • of tests performed in 18 different volumes: o ranging from 10 l to 120 m 3 , o of parallelepipedal, cylindrical or spherical shapes, o with vents of static opening pressure Pstat up to 300 mbar but with a vast majority (259 over 268) under 100 mbar, including 200 tests with Pstat under 20 mbar. • performed mostly with hydrogen (180) and methane (60), but also including propane (9) and ethylene tests (19), • mostly in fluids at rest as 185 tests over 168 were performed with a turbulent velocity u' lower than 0.1 m/s • with Pred recorded up to 2 bar but almost 60 % of the tests in the 0-200 mbar range (and 90% of tests below 1 bar).

Table 3

 3 Correlations of 1999; 2001 and 2008 Molkov models

	Bradley number	𝐵𝑟 =	𝐴 𝑣 𝑉 2 3 ⁄ •	𝑐 𝑢 𝑆 𝑢 (𝐸 -1)	(9)
	Degree Of Interaction	𝐷𝑂𝐼 = 𝛼 [	(1 + 𝑒𝑉 # 𝑔 )(1 + 0.5𝐵𝑟 𝛽 ) 1 + 𝑃 𝑣 ]	𝛿	𝑃 𝑖# 𝜔	(10)
	Turbulent Bradley number		𝐵𝑟 𝑡 =	√𝐸 𝛾 ⁄ √36 𝜋 3	𝐵𝑟 𝐷𝑂𝐼	(11)
	Reduced overpressure (1999)	𝑃 𝑟𝑒𝑑 = 𝐵𝑟 𝑡 -2.4 𝑃 𝑟𝑒𝑑 = 7 -6 * 𝐵𝑟 𝑡 0.5 (𝑖𝑓 𝐵𝑟 𝑡 < 1; 𝑃 𝑟𝑒𝑑 > 1) (𝑖𝑓 𝐵𝑟 𝑡 > 1; 𝑃 𝑟𝑒𝑑 < 1)	(12)
	Reduced overpressure (2001)	𝜋 𝑟𝑒𝑑 𝜋 𝑣 2.5 = 5.65 * 𝐵𝑟 𝑡 -2.5 (𝐼𝑓 𝐵𝑟 𝑡 > 2;	𝜋 𝑟𝑒𝑑 𝜋 𝑣 2.5 < 1)	(13)
	Reduced overpressure (2008)	𝜋 𝑟𝑒𝑑 𝜋 𝑣 2.5 = 7.9 -5.8 * 𝐵𝑟 𝑡 0.25 (𝐼𝑓 𝐵𝑟 𝑡 < 2;	𝜋 𝑟𝑒𝑑 𝜋 𝑣 2.5 > 1)

Table 4

 4 Empirical coefficients for theDOI number correlation of 1999; 2001 and 2008 models

	Model	α	B	δ	E	g	ω
	1999	0.9	1	0.37	10	0.33	0
	2001	1	0.8	0.4	10	0.33	0.6
	2008	1	0.8	0.4	2	0.94	0

Table 5

 5 Af and Su for the different values of P

	Pressure peak (see Cooper 1986, Fig. 4)				Af				Su	Equations
	P2	𝐴 𝑓 (𝑃1-𝐵𝑊) ~2𝜋 [ 𝐿 𝑝 ℎ 𝑝 + 𝐿 𝑝 𝑤 𝑝 + 𝑤 𝑝 ℎ 𝑝 3	] 𝑝 1	𝑆 𝑢 ~	0.9 𝐿𝑒	𝑆 𝐿	(19)
	P3	𝐴 𝑓 𝑜𝑏𝑠𝑡 𝐴 𝑓	= (1 +	4 3	𝜎 1-𝛼 (𝐵𝑅)	1 2 𝑁 𝛼 ) 2		𝑆 𝑢 ~	0.9 𝐿𝑒	𝑆 𝐿

Table 6 Table 7 Experimental

 67

	Test	Author	Year	%vol	I	L	D	V	Av	u'	Su	E
	A	Bauwens	2011	14.9	0.5	4.6	3,63	63.7	5.4	0.1	0.83	4.6
	B	Bauwens	2011	18.1	0.5	4.6	3,63	63.7	5.4	0.1	1.3	5.2
	C	Bauwens	2011	19	0.5	4.6	3,63	63.7	5.4	0.1	1.4	5.4
	D	Bauwens	2011	18.3	1	4.6	3,63	63.7	5.4	0.1	1.32	5.25
	E	Bauwens	2011	19	1	4.6	3,63	63.7	5.4	0.1	1.4	5.4
	F	Bauwens	2011	15.1	1	4.6	3,63	63.7	2.7	0.1	0.85	4.6
	G	Bauwens	2011	18.3	0.5	4.6	3,63	63.7	5.4	0.1	1.3	5.2
	H	Bauwens	2011	18.5	0.5	4.6	3,63	63.7	5.4	0.1	1.35	5.3
	I	Daubech	2011	27	1	1.66	0.94	1	0.13	0	2.5	6.6
	J	Kumar	2006	11	1	10	3.43	120	0.55	0	0.36	3.65
	K	Kumar	2009	10.2	0.5	10	3.43	120	0.55	1	0.27	3.42
	L	Kumar	2009	8.8	0.5	10	3.43	120	1.1	1	0.19	3.1
	M	Duclos	2019	16	1	2	1.33	4	0.49	0	1	4.8

Fig. 8. Experimental measurement and results of predictions or calculations for the tests listed in measurement for the tests listed in Table 6 and results of predictions using various models. Overpressures in mbar; ANN = Artificial Neural Network. The conservative best estimate is in italic

and highlighted in green. Underestimates of effects are in bold, highlighted in red and dark red and underlined for strong underestimates of effects (greater than 100 mbar difference 1 ).M.A.E. stands for the mean absolute error (with respect to the experimental value, for each column here)

1 see details in the article for explanations on this threshold

Table 8

 8 Reference configurations selected for the physics tests

	Comb. %Vol Length (m) Diameter (m) Volume (m 3 ) Ignition Vent area (m 2 ) Pstat (bar) u'
								(m/s)
	H2	19	4.6	3.63	63.7	Central 5.4	0.005	0.1
	CH4	10	4.6	3.63	63.7	Central 5.4	0.005	0.1


Valentin Penaud--Polge
email: valentin.penaudpolge@etu.emse.fr

Yann Gavet
email: gavet@emse.fr

Robustness Study of Edge Segmentation and Completion by CNN Based Method for Tessellation Images

Keywords: Tessellation, Cell Segmentation, Edge Completion

In this paper, the robustness of tessellation images segmentation using Convolutional Neural Network (CNN) is presented. Particularly, this paper aims to study the effect of the quality and number of the images used for training on the robustness of the edge segmentation and completion. Three kinds of image deterioration are considered : the absence of a part of the tessellation, the discontinuity of the edges constituting the tessellation and the presence of a background noise. Five CNNs are trained with several kinds of deteriorated images and the trained models are then compared according to the results obtained with four classes of images.

I. INTRODUCTION

Segmentation of tessellation or mosaic images can often arise in biological or industrial problems, such as the segmentation of corneal endothelium images [START_REF] Gavet | Visual perception based automatic recognition of cell mosaics in human corneal endothelium microscopy images[END_REF]. Many articles have been proposed to deal with this issue: using watershed [START_REF] Vincent | Morphological image processing and network analysis of cornea endothelial cell images[END_REF], active contours [START_REF] Charłampowicz | Automatic segmentation of corneal endothelial cells using active contours[END_REF] or directional filters [START_REF] Habrat | Cell detection in corneal endothelial images using directional filters[END_REF]. With the recent breakthrough of deep learning methods, several solutions have been studied using neural networks and CNN [START_REF] Fabijańska | Automatic segmentation of corneal endothelial cells from microscopy images[END_REF] [6] [START_REF] Fabijańska | Segmentation of corneal endothelium images using a u-net-based convolutional neural network[END_REF], even though, a common problem that occurs using CNN is the lack of a sufficient number of images and their relatively poor quality. The present paper constitutes a study of the robustness of the segmentation, based on CNN, of tessellation images depending on the quality and the number of images used to train a deep learning model. In order to control the quantity and the diversity (number of cells, deterioration) of images used, the tessellations are synthetic and computed with a Poisson-Voronoi model (Poisson Point Process (PPP) coupled with a Voronoi Diagram). By being spatially completely random, the PPP offers the most diversified Voronoi Diagrams possible.

The present paper is organised as follows. Sect. II presents the tessellation model used to generate the images. Then, Sect. III introduces three different ways of deteriorating the images. Sect. IV offers details about the experimentation procedures. Sect. V presents the results obtained. And finally Sect. VI concludes this study and offers new perspectives.

II. TESSELLATION MODEL

A tessellation is a partition of the plane, defined by a set of points (the seeds) and a distance (generally the Euclidean distance). The set of points is randomly generated by the realization of a PPP, and the partition is defined as a Voronoi diagram on these seeds.

A. Poisson Point Process

The PPP is a fast and simple way to generate seeds for a Voronoi Diagram. The PPP uniformly generates n λS random points {p i } in the spatial domain of the subset D ⊂ R 2 . n λS comes from a Poisson law P with parameter λS, where S represents the Lebesgue measure of D in R 2 (area) and λ is a density parameter.

n λS ∼ P (λS) (1)
For D = [a, b] × [c, d] with a, b, c, b ∈ R: ∀i ∈ [[1; n λS ]] : p ix ∼ U ([a, b]) and p iy ∼ U ([c, d]) (2)
Fig. 1 shows on the left a realization of the PPP with λ = 0.0005 and S = 224 pixel 2 .

B. Voronoi Diagram

A Voronoi Diagram of a set of points {p i } in a subset D ⊂ R 2 with regard to a distance d corresponds to the partitioning of D into cells {C i }, defined as followed :

C i = {x ∈ D | d (x, p i ) ≤ d (x, p j ) ∀j = i} (3)
The Voronoi Diagram, as the one in Fig 1, is the intersection of all the cells.

V or ({p

i }) = i,j ; i =j C i ∩ C j (4)
For more details about Voronoi Diagrams, the reader can refer to [START_REF] Aurenhammer | Voronoi diagrams[END_REF].

III. DETERIORATION MODELS

The following section describes the three alterations applied on the images used in order to train the models. We group these three alterations into two kinds of deterioration: a deterioration concerning the entire image, which is presented in Sect. III-A, to study the robustness with regard to an additionnal noise and deteriorations concerning only the edges of the cells, described in Sect. III-B and III-C, in order to study the robustness concerning the lack of edges information and the capacity of the trained models to close edges [START_REF] Gavet | Visual perception based automatic recognition of cell mosaics in human corneal endothelium microscopy images[END_REF].

A. Gaussian Random Field Background

A Gaussian Random Field (GRF) is a specific Random Field [START_REF] Adler | Random fields and geometry[END_REF] defined by the following covariance: ∀x, y ∈ R 2 C (x, y) = C (τ ) = e -τ 2 2σ 2 ; τ = ||x -y|| R 2 (5) This defines a continuous and stationary Random Field. A noisy background based on GRF can be applied to a given tessellation image T V or({pi}) built from a Voronoi Digram V or ({p i }):

T V or({pi}) : D ⊂ R 2 -→ R x -→ 1 if x ∈ V or ({p i }) 0 otherwise (6)
The resulting image T GRF V or({pi}) is defined as:

T GRF V or({pi}) = F -1 F (C 0 ) F (S) S = W + snr × 1 -T V or({pi}) (7)
With C 0 : x ∈ R 2 → C (0 2 , x). F corresponds to the Fourier Transform, σ represents the standard deviation, 0 2 is the neutral element of R 2 for the usual addition, W is a white noise and snr is the signal-to-noise ratio.

For all images used with a GRF background deterioration σ follows a Poisson law of parameter 4 and snr follows a Poisson law with parameter 10: σ ∼ P (4) and snr ∼ P [START_REF] Serra | The boolean model and random sets[END_REF].

The aim of adding a GRF background is to simulate a potential texture of the cells. Fig. 2 represents the result (bottom left) of a tessellation (top left) immersed into a GRF (top right). An example of a real corneal endothelium image is given as a comparison (bottom right).

B. Boolean Model Holes

This degradation consists in removing the cell edges which intersect a Boolean Model [START_REF] Serra | The boolean model and random sets[END_REF] realization whose germs are computed by a PPP with parameter λ BM S and whose grains are disks. The radius of each disk is uniformly chosen between two extremum values R min and R max . Given a Voronoi Diagram V or ({p i }) , and a boolean model BM , the result V or BM ({p i }) is defined as follows:

V or BM ({p i }) = V or ({p i }) ∩ BM c (8)
. c represents the complementary. An example of a resulting image is showed in Fig. 3 (bottom images), before and after being embedded in a GRF respectively.

C. Density Reduction

Contrary to the holes created by using the Boolean Model Holes (BMH) alteration which concentrates the loss of information in few areas, the Density Reduction (DR) aims to remove uniformly the edges information. Each point of the edges is removed or kept depending on the realization of a Bernoulli law B of parameter q. We define as V or q ({p i }) the resulting set. Five CNNs have been trained with different datasets descibed below. The architecture is the same for each CNN and is the U-net [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF] based architecture inspired from [START_REF] Vigueras-Guillén | Fully convolutional architecture vs sliding-window cnn for corneal endothelium cell segmentation[END_REF], see Fig. 5.

V or q ({p i }) = {x ∈ V or ({p i }) | u x = 1 ; u x ∼ B (q)} u ∼ B (q) ⇔ P (u = k) = q k (1 -q) 1-k ; k ∈ {0, 1} (9)
All datasets are generated with 3 different values of λ : 0.0002, 0.0008, 0.0016 equally split during the image generation. A morphological dilation was applied on each ground truth as better results were observed when using thicker ground truth edges. B. Evaluation 1) Images: In order to evaluate the robustness of the five trained CNN, four classes of images are used. Two of them are computed in the same way as the images used to train the CNN models, but were not used for the training process. The two others are used to determine the ability of the models to close edges.

• EV-GRF-DR: This class of images represents a tessellation immersed in a GRF background, iteratively deteriorated by a DR process. The corresponding values of q go from 0.1 to 0.9 with a step of 0.01 between each image. The background is the same for the 81 images. Fig. 6 shows an example of tessellation for q = 0.3, q = 0. The holes are disks with radius going from 0 to 15 pixels, with a step of 1 between each image. The length of a side of one square of the grid in the initial situation (before any holes) is 32 pixels. Fig. 8 shows the grid with disk holes having respectively 3 and 13 as radius and the initial grid used as the ground truth.

• BIN-KAN: This last class of images represents binary

Kanizsa triangle patterns [START_REF] Kanizsa | Subjective contours[END_REF] as it is a reference in Fig. 6: Left: Tessellation for q = 0.3. Center: Tessellation for q = 0.7. Right: Tessellation ground truth.

2) The

Dissimilarity Criterion: The criterion used to measure the segmentation accuracy is the criterion presented in [START_REF] Gavet | A geometric dissimilarity criterion between jordan spatial mosaics. theoretical aspects and application to segmentation evaluation[END_REF] [14], its definition is the following. A simple definition of this criterion would be that it corresponds to a symetric difference with a tolerance introduced be the Minkowski addition of a ball of radius ρ.

ρ G (X) = #{(X\G ⊕ B (0, ρ)) ∪ (G\X ⊕ B (0, ρ))} #{G ⊕ B (0, ρ)} (10)
With #, ⊕, B (0, ρ), G and X representing respectively the cardinal sign, the Minkowski addition, the disk of center 0 and radius ρ (the tolerance) as a structural element for the Minkowski addition, the ground truth and the predicted segmentation.

As all the ground truths in the training datasets were dilated, the ground truths used in Sect. V were morphologically dilated before computing the values of . The value of the tolerance ρ is fixed to 1.5 for the rest of the paper. corresponds to the model trained on the BIN-BMH (respectively GRF, GRF-BMH-3k, GRF-BMH-9k and GRF-BMH-DR) dataset .

• EV-GRF-DR (Fig. 10-a . Finally, as it can be observed in Fig. 11, even though the GRF-BMH-9k properly extends the grid, it obtains worst results than GRF-BMH-3k as it adds false positive pixels. • BIN-KAN (Fig. 10-d): For this class of images too, the BIN-BMH model gives good results (see Fig. 11), for the same reasons as the ones described for the BIN-GI images. But by adding barbules, the BIN-BMH model obtains scores close to the GRF model which only segments the existing edges (Fig. 11).

Here again, by the important number of false positive pixels, the GRF-BMH-3k, the GRF-BMH-9k and the GRF-BMH-DR models obtain worst results than the two other models. well build, as it is the case for the BIN-BMH model for the BIN-GI class of images, but without being able to generalize in unknown situations (in the presence of a GRF background). The overall better results of the GRF-BMH-DR model over the GRF-BMH-9k model (except for EV-GRF-BMH) show that the diversity of images is an important factor. Moreover, the completion process tends to penalize the models by adding unwanted edges or isolated pixels. Finally, as the main aspects underlined are the presence of a GRF background and the lack of constraints for the completion process, a new perspective to this work would be to use numerical twin images adapted to a specific application domain to study the capacities of a CNN on real life tessellation images. A first approach could be to use specific shapes of mosaics for a particular application to constrain the completion process to a certain range of angles or situations. Indeed, the Voronoi Diagrams resulting from the realisations of the PPP used in Sect. II, present a high cells shape diversity, due to the lack of spatial interaction in this process.

Images \ Models BIN- BMH GRF GRF- BMH- 3k GRF- BMH- 9k GRF- BMH- DR EV-GRF-DR EV-GRF-BMH ∼ BIN-GI ∼ BIN-KAN ∼ ∼ ∼

Fig. 1 :

 1 Fig. 1: Left : Result of a PPP. Right : Voronoi Diagram associated

Fig. 2 :

 2 Fig. 2: Top Left: Tessellation. Top Right: Gaussian Random Field. Bottom Left: Tessellation with the Gaussian Random Field background. Bottom Right: Endothelium Corneal Image

Fig. 3 :

 3 Fig. 3: Top Left: Tessellation. Top Right: Boolean Model. Bottom Left: AND(Tessellation, NO(Boolean Model)). Bottom Right: Resulting Image

Fig. 5 :

 5 Fig. 5: U-net architecture.

•

  BIN-BMH: The BIN-BMH dataset is composed of 3000 images. The only deterioration used is the BMH process with two similarly divided sets of parameters: λ BM S = 2, R min = 20, R max = 30 and λ BM S = 15, R min = 10, R max = 15. As there is no background alteration, the images of this dataset are binary images.• GRF: The GRF dataset is composed of 3000 images only deteriorated with a GRF background. • GRF-BMH-3k: This dataset contains 3000 images deteriorated with a GRF background and the BMH using the same two sets described in the BIN-BMH dataset. • GRF-BMH-9k: This dataset contains the same kind of images used in the GRF-BMH-3k dataset, the difference is that it is composed of 9000 images. • GRF-BMH-DR: All kinds of deterioration have been used to compute this last dataset. All the images have a GRF background. 3000 images are only altered by a background, 3000 others are images corresponding to the same kind of images as in the GRF-BMH-3K dataset and the last 3000 other images are deteriorated with a GRF background and a DR process : the following values of q have been evenly used : 0,25 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7.

  7 and the associated ground truth. Results presented in Sect. V represent the mean values calculated over 100 iterations of this class. • EV-GRF-BMH: The images of this class correspond to a tessellation iteratively altered by an BMH deterioration with disks of radius r = 10. Each image corresponds to a remaining proportion of edges information. The proportions go from 0 to 0.95 with a step of 0.05 between each image. The background is the same for the 20 images. Fig. 7 represents the image with 90% of the initial edges information, the image with 15% of the initial edges information and the corresponding ground truth. Results presented in Sect. V represent the mean values calculated over 100 iterations of this class. • BIN-GI: The binary grid images class represents a grid with iteratively growing holes at the intersection points.

Fig. 7 :

 7 Fig. 7: Left: Tessellation with 90% of edges information. Center: Tessellation with 15% of edges information. Right: Tessellation ground truth.

Fig. 10 and

 10 Fig. 10 and Tab. I summarize the results obtained. In the rest of the paper, each trained model holds the name of its training dataset.The BIN-BMH (respectively GRF, GRF-BMH-3k, GRF-BMH-9k and GRF-BMH-DR) model

Fig. 9 :

 9 Fig. 9: Top Left: Kanizsa motif with a gap of 40 pixels. Top Right: Kanizsa motif with a gap of 80 pixels. Bottom Left: Ground truth of the Top Left images. Bottom Right: Ground truth of the Top Right image.

Fig. 10 :

 10 Fig. 10: (a): Results of the EV-GRF-DR images. y-axis: , xaxis: q. (b): Results of the EV-GRF-BMH images. y-axis: , x-axis: Proportion of remaining edges information. (c): Results of the BIN-GI images. y-axis: , x-axis: Radius of the disk holes. (d): Results of the BIN-KAN images. y-label: , x-label: Size of the gap in number of pixels

Fig. 11 :

 11 Fig. 11: From top to bottom and from left to rigth: Image 1: Input image from the grid images. Image 2: Output of image 1 by the BIN-BMH model. Image 3: Output of image 1 by the GRF model. Image 4: Output of image 1 by the GRF-BMH-9k model. Image 5: Input image from the Kanizsa images. Image 6: Output of image 5 by the BIN-BMH model. Image 7: Output of image 5 by the GRF model. Image 8: Output of image 5 by the GRF-BMH-DR model.

TABLE I

 I

	: Summarized Results
	VI. CONCLUSION
	Results showed that a CNN can either learn accurately a
	segmentation or a completion task if the training dataset is