Physical quantities reconstruction in reacting flows with deep learning

Nilam Tathawadekar 1

Technical University of Munich Bolztmannstr. 3, 85748 Garching, Germany

Camilo Silva 2

Department of Mechanical Engineering, Technical University of Munich Bolztmannstr. 15, 85747 Garching, Germany

Michael Philip Sitte 3

Siemens Mobility Austria GmbH Leberstr. 34, 1110 Vienna, Austria

Nguyen Anh Khoa Doan 4

Faculty of Aerospace Engineering, Delft University of Technology Kluyverweg 1, 2629HS Delft, Netherlands

ABSTRACT Performing measurements in reacting flows is a challenging task due to the complexity of measuring all quantities of interest simultaneously or limitations in the optical access. To compensate for this, recent advances in deep learning have shown a strong potential in augmenting the information content in datasets composed of partial measurements by reconstructing the quantities that could not be measured. The present work analyses the use of such deep learning tools in two di ff erent cases. First, Convolutional Neural Networks (CNNs) are used to reconstruct the heat release rate (HRR) from velocity measurements in a methane / air premixed flame under harmonic excitation. The CNNs are trained from complete datasets at some specific frequencies and amplitudes of excitation and their ablility to reconstruct the HRR for di ff erent operating conditions with good accuracy is demonstrated. Secondly, an alternate approach based on Physics-Informed Neural Networks that do not require the training data to have all the quantities is explored. It is applied to a pu ffi ng pool fire where the velocity field is reconstructed from observations of pressure, temperature and density with good accuracy.

1 nilam.tathawadekar@tum.de

2 camilo.silva@tum.de

3 philip.sitte@siemens.com

4 n.a.k.doan@tudelft.nl

inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS. ? O? ? GLASGOW

1. INTRODUCTION

Measurements in flow have become an essential feature of fluid mechanical research, allowing to improve our understanding of the flow physics or by providing data that can be used to validate mathematical models and numerical simulations [1, 2]. Among these experimental fluid mechanics techniques, a variety of methods have emerged depending on the quantity to be measured. Particle Image Velocimetry (PIV) or Particle Tracking Velocimetry (PTV) have become standard techniques to measure the velocity in flows [3, 4], starting from enabling the measurements along a single line, to nowadays complex three-dimensional stereoscopic measurements that allow to get the three components of velocity in a box region. Beyond velocity measurements, density measurements have been enabled with schlieren techniques [1], while temperature is typically obtained via a hot-wire measurements or Rayleigh-Raman techniques. In the case of reacting flows, added to the previous methods, Laser-Induced Fluorescence (LIF) techniques have been developed to assess the species distributions in flows enabling a finer understanding of reactions occurring therein. The overview above does not attempt to provide an exhaustive list of flow measurements techniques but highlight the complexity and variety of techniques required depending on the quantity of interest. This complexity often results in the impossibility, in many experiments, of simultaneously measuring all quantities of interest, limiting our ability to analyse the (reacting) flows in detail. To tackle this issue, recent developments in machine learning techniques have shown a great potential [5–13]. For example, methods based on Proper Orthogonal Decomposition (POD) have been used to reconstruct the velocity field from sparse measurements by finding a data-driven based mapping between the measurable quantities (of, for example, sparse sensors) and the velocity field [8] or by using feedforward neural networks to achieve a similar task [9,12]. For reacting flows, an extension of POD, called Gappy POD, was developed with some success to infer the velocity in regions where it is not measured [13]. Further attempts were made at using Convolutional Neural Networks (CNNs) to reconstruct velocity fields from OH-Planar Laser Induced Fluoresence (PLIF) data [5, 6]. Despite their success, the approaches in these works required a database with both the measured quantities and those to be reconstructed, to train the machine learning framework. To circumvent this, a recent framework was proposed by Raissi et al. [11], called Hidden Fluid Mechanics (HFM), where a feedforward neural network is trained with both the governing equations of the system and the measured quantities to perform such a reconstruction task without requiring data of the quantities to reconstruct. The obtained so-called Physics-Informed Neural Network (PINN) was shown able to reconstruct the velocity field in some canonical non-reacting flows. However, this approach has only been applied to non-reacting flows. In this paper, we present the reconstruction capabilities of two di ff erent deep learning techniques applied to reacting flows. In the first one, presented in Section 2, the reconstruction of the heat release rate will be performed in a Bunsen flame under acoustic excitation by using a deep learning framework, called the U-net that relies on a series of CNNs. In the second test case, presented in Section 3, the reconstruction of the velocity field in a pu ffi ng pool fire will be performed using a physics-informed neural network extended to reacting flows. For both test cases, the data is generated from numerical simulations which have been validated against experimental measurements. Only a subset of all the quantities (the "measured" quantities) will be given as input to the deep learning framework. Using numerical data furthermore allows to make a thorough comparison between the quantities reconstructed using the deep learning framework and the actual quantity. A summary of the main results is provided in the final section.

2. HEAT RELEASE RATE RECONSTRUCTION WITH CNNS

2.1. Test case: Bunsen flame under excitation The Bunsen flame test case is shown in Fig. 1. It is the laminar multi-slit burner investigated by Kornilov et al. [14], where a premixed methane-air mixture with an equivalence ratio of 0.8 is used and is subjected to single tone velocity perturbation with a loudspeaker. The numerical setup and

Figure 1: Left: Experimental configuration. Right: CFD domain. Figure adapted from [15].

results similar to the one in [16,17] are used: a 2D CFD domain with symmetric boundary condition in the transverse direction and inflow / outflow boundary condition in the streamwise direction is considered. The two-step chemical scheme as detailed in [18] and OpenFOAM [19] are used for the simulation. Additionally, no combustion model is necessary here as all species transport equations are fully resolved. Specifically, the continuity, Navier-Stokes, species mass fraction and enthalpy equations [20] are solved with the CFD solver:

∂ t + ∂ρ u i

∂ρ

∂ x i = 0 (1)

∂ t + ∂ρ u i u j

∂ x i + ∂τ i j

∂ρ u i

∂ x j = − ∂ p

∂ x j + ρ g i (2)

∂ρ Y k

∂ t + ∂ρ u i Y k

∂ x i = ∂

" ρ D ∂ Y k

# + ρ ˙ ω k (3)

∂ x i

∂ x i

∂ρ h

" ρα ∂ h

∂ t + ∂ρ u i h

∂ x i = ∂

# (4)

∂ x i

∂ x i

where ρ is the density, u i denotes the velocity in the i -th direction, p the pressure, Y k is the species mass fraction, ˙ ω k its reaction rate and h is the specific enthalpy of the mixture. τ i j is the molecular stress tensor, D is the molecular di ff usivity and α = λ/ ( ρ C p ) is the thermal di ff usivity, where C p is the specific heat coe ffi cient. The reaction rate and species transport and thermodynamics properties are taken from the two-step chemical mechanism detailed in [18]. It should be noted that for this case of an excited laminar flame, gravity e ff ects are neglected and therefore g i = 0. The heat release rate (HRR) is computed as [20]:

N s X

k = 1 ρ ˙ ω k ∆ h 0 f , k (5)

˙ q = −

where ∆ h 0 f , k is the formation enthalpy of specie k and N s is the number of species.

4 0.35, os —

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x)

0.35 03 0.25 02 015 on 0.05,

Figure 2: Evolution of HRR over one period of excitation for the case with f = 100 Hz and A = 100%. Snapshots are spaced by a time-duration which is one-tenth of the excitation period.

os o7 06

The flame is laminar and the grid is composed of 122 300 cells with a cell size of 0.025 mm in the flame region and area of contractions and cell stretched in the axial direction. This ensures that all flow and reaction lengthscales are fully resolved [15]. At the inflow, a mean inlet velocity of 0.4 m / s and inlet temperature of 293 K are imposed. The plate on which the flame is stabilized is modeled as a no-slip wall with a fixed temperature of 373 K, as measured in the experiment [14]. The CFD simulation is run with an adaptive time-stepping scheme with an average timestep ∆ t = 10 − 6 . Given that this CFD set-up has already been validated in previous studies with respect to experimental data, these validation steps are not repeated here. The interested reader is referred to [16,17] for additional details on the numerical set-up and its validation. This initial set-up is then subjected to acoustic excitation at the inlet, where a normalized harmonic streamwise velocity fluctuation is imposed. The frequencies considered for excitation are f = 100, 150 and 200 Hz with a normalized amplitude of A = 10, 50, 100, 125 and 150% of the mean inlet velocity. Di ff erent combinations of cases will be used to train the deep learning framework and it will then be tested on a di ff erent condition to assess its reconstruction performance. A typical time-sequence of the varying HRR (normalized by its maximum value) over one period is shown in Fig. 2 for the case with f = 100 Hz and A = 100%. It can be seen that the initial flame gets extremely elongated under the large amplitude of excitation before regaining its original shape.

5

09 os o7 06

2.2. Reconstruction method with CNNs In this section, the reconstruction objective is the inference of the heat release rate (HRR) field from the velocity field. To achieve this, a deep learning architecture based on the U-net will be used [21]. This architecture is presented in Fig. 3. It is composed of a series of CNNs which perform multi- level filtering of the input fields and recombines these into one output field. For this reconstruction problem, it will be considered that the U-net takes as input the full (2-component) velocity fields at 3 di ff erent successive time instants in the past (i.e. u ( x , t ) , u ( x , t − ∆ t ) , u ( x , t − 2 ∆ t )) and the mean HRR field (i.e. ⟨ ˙ q ( x ) ⟩ T , where ⟨·⟩ T represents the time-averaging operation over the period T of the velocity fluctuation) and outputs the HRR fluctuations d ˙ q = ˙ q −⟨ ˙ q ⟩ T . This combination of information allows the U-net to estimate velocity gradients which are necessary to estimate the HRR fluctuation. In this set-up, the U-net architecture requires the data of HRR fluctuations for training which may seem limiting. However, we will show that the U-net is also able to infer the HRR fluctuations for operating conditions not present in the training dataset. This situation would mimic an experimental campaign where measurements of HRR are only available for a few operating conditions while the velocity measurements would be available for all of them. It should be noted that the requirement of having the mean HRR as an input is not excessively constraining given that it can be straightforwardly obtained using luminescent photographs for example.

03 0.25 02 0.15, on 0.05,

0.45, o4 035

0.05,

0.35, 03

0.05,

2D CNN 16

2D CNN 1

skip connection

[256,256,16]

2D CNN 16

2D CNN 16

C

[256,256,1]

2D CNN *

2D CNN 32

skip connection

[256,256,16]

2D CNN 32

2D CNN 32

[256,256,7]

C

[256,256,16]

16

[128,128,32]

[128,128,16]

2D CNN *

2D CNN 64

skip connection

2D CNN 64

2D CNN 64

[128,128,32]

32

[64,64,64]

C

C : Concatenate

[64,64,32]

2D CNN *

2D CNN 128

2D CNN 128

2D CNN 128

skip connection

[32,32,128]

C

MaxPool (2,2) Dropout Upsampling (2,2) Dropout : :

[32,32,64]

[64,64,64]

64

2D CNN *

2D CNN 256

2D CNN 256

[32,32,128]

[16,16,256] 2D CNN: (3,3) ReLU and "same" padding

2D CNN*: Transpose CNN (3,3), ReLU and "same" padding

128

[16,16,128]

Figure 3: Schematic of the U-net architecture

2.3. Results Two di ff erent reconstruction problems are considered. In the first "amplitude interpolation" case, the U-net architecture is trained with the data at specific amplitudes of excitation (10, 50, 100, 150%) and frequencies of 100 and 200 Hz and the U-net is used to reconstruct the HRR fluctuation for the other cases at di ff erent amplitudes of excitation (125%), for the same frequencies. In the second "frequency interpolation" case, the U-net is trained with the case at 2 di ff erent amplitudes of excitation (50 and 100%) for frequencies 100 and 200 Hz and it is tested for a case with frequency of 150 Hz and amplitude 50%. A typical reconstructed velocity field for a representative timestep for the "amplitude" case is shown in Fig. 4a alongside the original data, where the U-net reconstruct the HRR field for a case whose amplitude was not present in the dataset ( A = 125%). It can be seen that the HRR profile is correctly reconstructed from the input (the velocity fields during several past time steps). Especially the formation of the HRR "cusps", which is due to the very large amplitude of the excitation, is recovered. The time evolution of the mean-squared error (MSE) between the predicted HRR field and the exact one is also shown in Fig. 4d where it can be seen that throughout the prediction, the MSE remains small. In addition, an important feature of the flame for thermoacoustic studies is the total HRR fluctuation (the HRR integrated over the entire domain, ˙ Q = R

V ˙ q dV ). It is computed using the HRR reconstruction by the U-net and is shown in Fig. 4d. It can be seen that it closely matches the actual total HRR fluctuation. This indicates that the U-net is able to reconstruct the HRR field from the velocity field, not only in terms of morphology and spatial features, but also in an integral sense. The results for the other test case ( f = 200 Hz and A = 125%) exhibited a similar level of accuracy and are not shown here for brevity.

(a)

(c)

(b) (d)

Figure 4: (a) Actual HRR field, (b) Reconstruction from U-net and (c) di ff erence between the two. (d) Time-evolution of (left axis) the total HRR fluctuation (blue line: U-net, cross: exact) and (right axis) mean squared error for the amplitude interpolation problem (for case f = 100 Hz and A = 125%) .

A similar analysis was performed when considering the frequency interpolation problem, i.e. by requiring the U-net to reconstruct the HRR profile for a case whose frequency of excitation was not present in the training dataset. This is shown in Figs 5a-c where the exact, reconstructed HRR profiles and their di ff erence are shown. Similarly to the other case, one can observe that the reconstructed

04

HRR matches well the actual one. The U-net is also able to reconstruct the HRR accurately in an integral sense (see the time-evolution of the MSE and total HRR in Fig. 5d). Similar accuracy was also found for the other test case in this problem set-up ( f = 150 Hz with A = 100%) and the associated results are not shown for brevity.

(a)

(c)

04 02 san

(b)

(d)

Figure 5: (a) Actual HRR field, (b) Reconstruction from U-net and (c) di ff erence between the two. (d) Time-evolution of (left axis) the total HRR fluctuation (blue line: U-net, cross: exact) and (right axis) mean squared error for the frequency interpolation problem (for case f = 150 Hz, and A = 50%).

3. VELOCITY RECONSTRUCTION USING PINNS

3.1. Test case: Unstable pool fire The second test case considered here is an unstable pool fire at the onset of pu ffi ng. Simulations were carried out using the computational fluid dynamics toolbox OpenFOAM-7 with the solver fireFoam [19]. The governing equations are the continuity equation, Navier-Stokes equations, species mass fraction equations and enthalpy equation. These are formally identical to the ones presented in Sec 2.1 in Eqs 1, 2, 3 and 4 but with a non-zero gravity term. The di ff erence compared to the system solved in Sec. 2.1 is related to the chemical mechanism as detailed hereunder.

04 02 0 san

(a) (b)

san 0.2

Figure 6: (a) Schematic of the numerical domain. Coordinates normalised by pool radius a . (b) Subcritical flames for di ff erent diameters, 2 a = 15 . 9, 17.6 and 19.1 mm. Black line: position of the reaction zone, indicated by the stoichiometric mixture fraction; grey lines: isocontours for T = 600, 900, 1200, 1500 K.

A 2D axisymmetric pool fire with n-heptane fuel and at ambient conditions ( p 0 = 100 000 Pa, T 0 = 300 K, Y O2 = 0 . 233, Y N2 = 0 . 767) was investigated, equivalent to the n-heptane flames with an

2a = 15.9 mm 10 17.6 mm 19.1 mm -l 0 1 ; 1800 + 1500 + 1200 900 600 300

xt t 50! Pressure outlet } Pressure outlet Isothermal wall ofl 23 4 5 Fuel inlet

isothermal brass base plate studied experimentally and numerically by Moreno-Boza et al. [22]. A schematic of the numerical domain is shown in Fig. 6a. The domain size was 50 a in axial and 5 a in radial direction, where a is the fuel pool radius. A 2D structured grid was used with 600 × 130 grid points. In the region of interest the grid resolution was O (0 . 1 mm) to ensure that all flow and reaction lengthscales are fully resolved. Numerical schemes were of second order in space and first order in time with a fixed time step of 10 − 5 s, so that CFL < 0 . 1, and using a fractional step scheme for the chemical source term. In the present simulation, radiation was neglected and turbulence modelling was not necessary since the flame was in the laminar regime at the onset of pu ffi ng. Combustion chemistry was modelled as the irreversible single-step reaction (C 7 H 16 + 11 O 2 → 7 CO 2 + 8 H 2 O), whose rate constant is given by the Arrhenius law, K = BT β exp( − T A / T ) with the model constants β = 0, T A = 12 000 K and B = 5 . 5 × 10 7 m 3 / (mol s) , following the reasoning of Fernandez-Tarrazo et al. [23] but without correcting heat release rate (HRR) and T A with equivalence ratio ϕ . This modelling of the chemical reaction was chosen for the sake of simplicity. It is su ffi cient for the present case of a laminar di ff usion flame characterised by high Damköhler number and controlled by mixi ng [22]. Density is given by the ideal gas law. Viscosity is computed from Sutherland law, µ = A s √

T / (1 + T s / T ), independent of species composition with A s = 1 . 672 × 10 − 6 and T s = 170 . 67 in the appropriate SI-units. Molecular and thermal di ff usivity are computed based on the assumptions of unity Lewis number, Le = α/ D = 1, and constant Prandtl number, Pr = µ/ ( ρα ) = 0 . 7. Specific heat and enthalpy are obtained from NASA polynomials but modifying the formation enthalpy of n- heptane to reduce the flame temperature to approximately 1750 K to account for the e ff ect of radiative heat loss on the flame, following previous work [22]. The liquid surface is modelled as a boundary condition following Moreno-Boza et al. [22]. The liquid surface is assumed to be at the boiling temperature of n-heptane, T B = 371 . 5 K. Walls are assumed isothermal at the ambient temperature T 0 . The fuel mass flow rate at the liquid surface is determined by the evaporation rate relating the conductive heat flux to the liquid with the surface normal velocity u n as in [22]. The CFD simulations are validated against the experimental and numerical results for the n-heptane flame with isothermal walls from Moreno-Boza et al. [22], notably the reproduction of the flame length, the critical point and the pu ffi ng frequency. Figure 6b shows temperature contours for the subcritical flames of diameters 2 a = 15 . 9, 17.6 and 19.1 mm with corresponding flame lengths of 7 . 0 a , 7 . 7 a and 8 . 4 a . The flame length was determined based on the downstream tip of the stoichiometric mixture fraction isocontour. These values are in good agreement with the experiments, where the flame lengths ranged from 6 . 7 a to 8 . 0 a . In the present simulations, the critical point – defined as the characteristic diameter where pu ffi ng first occurs – is found at 2 a = 20 . 1 mm, in line with experimental findings for the isothermal base plate. At this condition, the pu ffi ng establishes as a periodic process with a constant frequency of approximately 12.0 Hz, very close to the experimentally determined value of 12.8 Hz. Figure 7 shows one cycle of this pu ffi ng behaviour and the temporal variation of HRR during the pu ffi ng cycle. In this sequence, we can see the dynamics of the pu ffi ng flame, where a "cusp" is formed due to buoyancy-driven vorticity generation. These numerical results closely resembles the time sequence of the flame recorded in the experiments. Therefore, the present simulations have been shown to accurately reproduce the behaviour of the n-heptane pool fire in the vicinity of the critical point, studied by Moreno-Boza et al. [22]. For the reconstruction problem discussed in the next section, the pu ffi ng flame with 2 a = 20 . 1 mm is considered. The dataset used to train the PINN consisted of 450 snapshots recorded at 1 000 frames per second, corresponding to more than 5 periods with about 83 snapshots per period.

3.2. Reconstruction method with PINNs To reconstruct the unmeasured quantity, the approach called Hidden Fluid Mechanics proposed by Raissi et al. [11] will be used. This approach relies on a Physics-Informed Neural Network [24], illustrated in Fig. 8, and uses a neural network to infer the measured and unmeasured quantities in a given flow which is governed by the (reacting) Navier-Stokes equations. To achieve the inference of

(a) (b)

Figure 7: (a) Time sequence of the pu ffi ng flame with diameter 2 a = 20 . 1 mm. (b) Corresponding time evolution of HRR, normalised by period τ p and max(HRR); instances of snapshots are indicated as ◦ .

unmeasured quantities, the PINN relies on two sources of information: (i) the data which is related to the measurable quantities and (ii) the governing equations of the flow. A PINN is a conventional feedforward neural network (blue box in Fig. 8) which is trained with a specific loss function that accounts for the governing physical equations ( ϵ p , red part) and the data error ( ϵ m , green part). Feedforward neural networks map the input to the output, as shown in Fig. 8 where an input, two hidden and an output layers are represented. The network is termed feedforward as the output of a given layer is not fed back into the input or preceding layers. Each hidden layer consists of neurons which are fully connected, meaning that each neuron in a given layer l − 1 is connected to all neurons in the following layer l through a weight matrix W l . Therefore, the intermediate output of the hidden layer l can be written as Z l = W T l X l − 1 + b l where X l − 1 is the output of the layer l − 1 and b l is the bias in layer l . Following this, nonlinearities are introduced through the element-wise activation function g , so that X l = g ( Z l ). For what follows, the activation function used for all hidden layers will be the swish function [25] with a linear activation in the final output layer. Other choices of activation functions are possible, such as the sigmoid or ReLU activation, but swish was shown to provide su ffi cient accuracy for the presented case as will be discussed in Sec. 3.3.

(Grow TL t/t,

ρ p

t x

T

v u

r

Figure 8: Schematic of the HFM framework.

In the HFM architecture, the PINN takes as input a space-time location ( x , t ) (here, in 2D axisymmetric coordinates, x = ( x , r )) and outputs the flow state vector at that location, i.e. its output is e Φ = [ ρ, p , T , u , Y ], where e · indicates a prediction from the PINN, u = ( u , v ) is the velocity vector, with axial component u and radial component v , and Y is the chemical composition vector. For simplicity, in this first attempt at using the HFM, the species mass fraction will not be considered in the reconstruction problem, and therefore, the considered flow state is just e Φ = [ ρ, p , T , u , v ]. In the specific reconstruction problem considered here, it should be stressed that, for the training of the PINN, only a subset of these outputs have associated target data (i.e. the measured quantities). Such measured states will subsequently be noted ϕ m for the target data and e ϕ m for the PINN prediction, with their associated space-time locations noted as ( x m , t m ). In what follows, it will be considered that the measurable quantities are ϕ m = [ ρ, p , T ]. To enable the PINN to reconstruct the other unmeasured quantities (in this case, the velocity field u ), the loss function used to train the

PINN includes the residual of the reacting Navier-Stokes equations, noted R , which is estimated at collocation points (noted ( x c , t c )) spread over the time-space domain covered by the simulation. Therefore, the loss function to train the PINN is:

N m X

N c X

L = 1

+ 1

n = 1 | ϕ m ( x n m , t n m ) − ˜ ϕ m ( x n m , t n m ) | 2

i = 1 |R ( e Φ ( x i c , t i c ) | 2

(6)

N m

N c

| {z } ϵ m

| {z } ϵ p

In the equation above, the first term, ϵ m (green part in Fig. 8), corresponds to a standard mean- squared-error between the target data and the prediction of the PINN for the N m space-time locations  ( x n m , t n m ) where data is available. It should be again emphasised that only a subset of all the flow states are considered measurable and therefore ϕ m does not account for the full flow state. The second term ( ϵ p , red part in Fig. 8) represents a physics-based loss which is the residual of the reacting Navier-Stokes equations computed using the prediction of the PINN at arbitrary space-time collocation points n ( x i c , t i c ) o . This second term enables the network to identify suitable predictions for the unmeasured quantities that satisfy the governing equations. This residual is computed using automatic di ff erentiation [26] as in past work on PINNs [11,27]. For simplicity, in the present work, the collocation points are taken to be the same space-time locations as where target data is available, i.e. N c = N m = 78 000 and n ( x i c , t i c ) o n ( x i m , t i m ) o . In this work, similarly to an earlier work on HFM [11], a feedforward neural network of 20 hidden layers with 300 neurons each will be used. The training is performed in two stages: (i) the network is pre-trained using only the available data, i.e., the loss function only contains the first term in Equation 6 which allows for a rapid partial weight optimisation as it corresponds to a traditional supervised training process with a mean-squared error (MSE) loss; (ii) all the network weights are optimised using the full loss function, as in Equation 6 enabling the reconstruction of the unmeasured quantities. All the training processes are performed with the ADAM optimiser [28] using a learning rate of 0.001 with a batch size of 10 000. The training is run until the loss function reaches a plateau indicating a fully trained network.

3.3. Results In this section, we demonstrate the ability of the PINN to reconstruct the velocity field of the pu ffi ng flame from measurements of temperature, density and pressure. To do this, the PINN is trained as discussed in Sec. 3.2 using the dataset obtained from the simulations. It should be emphasised that, in the training dataset, only the temperature, density and pressure fields are provided to the network and that it never receives any velocity information. Therefore, the training dataset is incomplete and the role of the (trained) network is to be able to infer the missing states in that specific (training) dataset (and there is therefore no "testing" dataset as in other traditional supervised learning problem). Specifically, here, the network must be able to infer the two components of the velocity fields using the residual of the reacting Navier-Stokes equations and the incomplete information provided by the temperature, density and pressure fields. The reconstructed velocity field for a representative snapshot is shown in Figure 9a for u and Figure 9b for v . As can be seen in those figures, the two components of the velocity field are accurately reconstructed by the PINN and only minor di ff erences can be observed at the base of the flame where strong gradients are present. This shows that the PINN manages to infer the dynamics of the buoyant plume from the residual of the Navier-Stokes equations without any observation of the velocity field. Additionally, the overall L 2 -error remains small over the majority of the domain and most features of the velocity fields are accurately reconstructed. While this is shown for a specific time instant, the reconstruction accuracy was similar for most snapshots and the mean squared error (averaged over the computational domain) in function of time is shown in Fig. 9c. It can be seen that the error is overall low except at the initial time ( t = 0 s). This higher error for that initial time instant is

related to the relative lack of measurements data (in time) that prevents an appropriate estimation of the time-derivative in the residuals [11].

v (a) (b)

Diff. Abs. Diff. (c)

Exact PINN

u Abs. Exact PINN

Figure 9: Comparison between actual and reconstructed (a) u and (b) v velocity for a representative snapshot. Abs. Di ff . stands for absolute di ff erence between the prediction and actual velocity fields. Superscript + indicates coordinates normalised by a . (c) Evolution of MSE normalized by averaged squared velocity of the reconstruction of u and v .

4. CONCLUSIONS

This work presented two di ff erent deep learning based methods for the reconstruction of unmeasured quantities in reacting flows. The first one is based on CNNs in a U-net architecture and was aimed at reconstructing the HRR fluctuation in an acoustically excited flame from past velocity information. The U-net was trained with the HRR dataset for some specific frequencies and amplitudes of excitation and could accurately reconstruct the HRR for other frequencies and amplitudes not in that training dataset. The second method did not require any prior training data of the quantities to be reconstructed and is based on the physics-informed neural network architecture. It was shown able to reconstruct accurately the velocity field from the pressure, density and temperature field in an intrinsically unsteady pu ffi ng flame. These two techniques should be viewed as complementary of each other as in the first, no physical considerations are necessary while in the latter no information on the quantities to be reconstructed is required. Indeed, there can be cases where obtaining / implementing the relevant physical information (such as the complex chemical reactions) may be di ffi cult, making the second PINN-based approach more di ffi cult while other reconstruction tasks may not have any data on the quantities to be reconstructed making the first U-net approach unusable. Nonetheless, these results demonstrate the potential of deep learning techniques in supplementing information when only partial data is available. In future work, these techniques are extended to three-dimensional turbulent flames and will be tested using experimental data.

ACKNOWLEDGEMENTS

N. Tathawadekar acknowledges the financial support of the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 766264 and the ERC Consolidator Grant SpaTe (CoG-2019-863850).

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