Proceedings of the ASME 2021
The International Mechanical Engineering Congress and Exposition
IMECE 2021
November 1-5, 2021, Virtual Conference, Online, USA

arXiv:2107.11510v1 [physics.flu-dyn] 24 Jul 2021

IMECE2021/69657

NEURAL DIFFERENTIAL EQUATIONS FOR INVERSE MODELING IN
MODEL COMBUSTORS

Xingyu Su
Tsinghua University
Beijing 100084, China
su-xy19@mails.tsinghua.edu.cn

Weiqi Ji
Massachusetts Institute of Technology
Cambridge, MA 02139
weiqiji@mit.edu

Long Zhang
Tsinghua University
Beijing 100084, China
long-zha19@mails.tsinghua.edu.cn

Wantong Wu
Tsinghua University
Beijing 100084, China
wwt15@mails.tsinghua.edu.cn

Zhuyin Ren ∗
Tsinghua University
Beijing 100084, China
zhuyinren@mail.tsinghua.edu.cn

Sili Deng †
Massachusetts Institute of Technology
Cambridge, MA 02139
silideng@mit.edu

ABSTRACT
Monitoring the dynamics processes in combustors is crucial for safe and efficient operations. However, in practice, only
limited data can be obtained due to limitations in the measurable quantities, visualization window, and temporal resolution.
This work proposes an approach based on neural differential
equations to approximate the unknown quantities from available
sparse measurements. The approach tackles the challenges of
nonlinearity and the curse of dimensionality in inverse modeling
by representing the dynamic signal using neural network models. In addition, we augment physical models for combustion
with neural differential equations to enable learning from sparse
measurements. We demonstrated the inverse modeling approach
in a model combustor system by simulating the oscillation of an
industrial combustor with a perfectly stirred reactor. Given the
sparse measurements of the temperature inside the combustor,
upstream fluctuations in compositions and/or flow rates can be
inferred. Various types of fluctuations in the upstream, as well as
the responses in the combustor, were synthesized to train and validate the algorithm. The results demonstrated that the approach
can efficiently and accurately infer the dynamics of the unknown

inlet boundary conditions, even without assuming the types of
fluctuations. Those demonstrations shall open a lot of opportunities in utilizing neural differential equations for fault diagnostics
and model-based dynamic control of industrial power systems.
Keywords: Neural Differential Equations, Inverse Problems,
Unsteady Combustion, Auto-differentiation, Machine Learning.

1. INTRODUCTION
Oscillation in combustors can degrade the efficiency and
safety of the combustion system. Unsteadiness in the upstream
is one of the major causes of the oscillation, and thus monitoring
the upstream conditions is important for control and operation.
However, in practice, it is difficult to measure the upstream conditions due to the challenges in diagnostic at harass environment
and the constraints of cost. Therefore, computational tools aiming at the inference of unknown information based on available
measurements have been developed and are often termed as inverse models [1, 2, 3]. Conventional inverse modeling methods
are usually limited to a few unknown parameters and suffering
from the curse of dimensionality since the computational cost
increases significantly as the number of unknown quantities increases.
In the present work, we propose to use neural networks

∗ Address all correspondence to this author.
† Address all correspondence to this author.

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for approximating the high dimensional unknown quantities and
train the neural networks efficiently with available measurements
by differential programming. In order to achieve good extrapolation capability outside of the training dataset, supervised learning
methods usually require a large number of training datasets that
can cover a wide range of conditions. For the neural differential
equation, physical constraints are imposed during the training
by augmenting the neural network models with the governing
differential equations, such that the training can be viewed as
self-supervised learning to avoid the need for obtaining labeled
training data, in this case, the unmeasurable, which are intrinsic obstacles for modeling industrial combustors with data-driven
approaches.
We demonstrated the inverse modeling approach in a model
combustor system by simulating oscillating combustion in an industrial hot air heater with a perfectly stirred reactor (PSR). Industrial combustion devices such as hot air heaters are widely
used for burning non-standard low-calorific value (NLCV) gases,
such as blast furnace gas and coke oven gas, which are prone
to combustion instability. Zhang et al. [4] studied the oscillating combustion in reactors resulting from periodic fuel burnt out,
fuel replenishment, and ignition governed by the subtle mixingchemical kinetics interaction or due to the inlet flow rate fluctuation under industrial conditions. Zhang et al. utilized forward
modeling to investigate the oscillation of combustion via unsteady PSR simulation. Given the initial conditions and boundary conditions, one can integrate the governing ordinary differential equations (ODEs) to get the temporal evolution profile of
temperature and species concentrations. However, in practical
industrial scenarios, it is often difficult to measure the boundary
conditions with satisfying accuracy, especially for species concentrations. The experimental data that can be effectively obtained are the temperature and flow velocity at the combustor
outlet. In this study, we try to infer the inlet fluctuations in compositions and/or flow rate given the sparse measurements of the
temperature at the combustor outlet. The fluctuation in the inlet
conditions is usually time-dependent, such that one needs to use
a large number of data points to resolve the evolution of the inlet
conditions, and thus the inverse modeling involves high dimensional unknown parameters.
This work aims to tackle the challenges in achieving scalability of the data-driven inverse modeling of unknown measurements in model combustors using techniques developed for training deep neural network models. Specifically, we first parameterize the fluctuations using a neural network model. We then
employ the stochastic gradient descent (SGD) optimizer to learn
the inlet boundary conditions in an unsteady PSR. The SGD optimizer is widely applied in the deep learning communities for
its efficiency as well as scalability in dealing with large datasets,
and its robustness in optimizing high-dimensional non-convex
neural network models. Our recent work [5] has also shown that
a chemical reaction network is equivalent to a neural network

with a single hidden layer. Similarly, solving ordinary differential equations (ODEs) of reaction network models is equivalent to
solving infinite-depth deep residual networks [6], which further
rationalizes exploiting SGD in training ODEs.
However, to exploit neural networks in combustion simulation and optimization, one needs a well-established software
ecosystem that can efficiently and accurately compute the gradient of simulation output to model parameters. For instance,
the finite difference method (often termed as the ”brute-force
method”) usually suffers both the computational inefficiency due
to the cost increases with the number of parameters and the numerical inaccuracy due to the truncation error [7]. Conversely,
gradient evaluation methods based on auto-differentiation (AD)
have shown both efficiency and accuracy in the training of
large-scale deep neural network models in a variety of tasks.
Many open-source AD packages have been developed in the
last decade, including TensorFlow [8], Jax and PyTorch [9] in
Python, ForwardDiff.jl and Zygote.jl [10] in Julia. Thus, this
work utilized a recently developed differentiable combustion
simulation package Arrhenius.jl [11] to enable SGD for combustion model optimizations and applies Arrhenius.jl to infer the
boundary conditions of the model combustor described by PSR.
Specifically, Arrhenius.jl, the differential programming ecosystem in Julia, was employed to encode and solve the governing
ODEs for a perfectly stirred reactor, conduct auto-differentiation,
and train the neural network models. Various types of fluctuations in the upstream, as well as the responses at the combustor’s
outlet, were synthesized to train and validate the algorithm. The
results demonstrated that the approach can efficiently and accurately infer the dynamics of the inlet flow and calorific fluctuations.
This paper is structured as follows: we first introduce the
package of Arrhenius.jl, the governing equations for the combustor, and the inverse modeling using neural differential equations;
we then demonstrate the approach in inverse modeling during
two representative scenarios: fuel switching and inflow fluctuations. Finally, we conclude and discuss the opportunities and
challenges of employing the current approach for the dynamic
control of industrial power systems.

2. METHODS
2.1 Arrhenius.jl
Arrhenius.jl is built using the programming language of Julia to leverage the rich ecosystems of auto-differentiation and
differential equation solvers. Arrhenius.jl does two types of differentiable programming: (i) it can differentiate elemental computational blocks. For example, it can differentiate the reaction
source term with respect to the kinetic and thermodynamic parameters as well as species concentrations; (ii) it can differentiate
the entire simulator in various ways, such as solving the adjoint
sensitivities [12]. The first type of differentiation plays the basis
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role for the second type of high-level differentiation. Arrhenius.jl
offers the core functionality of chemical reaction simulations in
native Julia programming, which allows users to readily build
the applications on top of Arrhenius.jl and exploit various approaches to do high-level optimizations.
Figure 1 shows a schematic diagram of the structure of the
Arrhenius.jl package. The package reads in the reaction mechanism files in YAML format as the same as those in the Cantera
community, which contains the kinetic models, thermodynamic,
and transport databases. The core functionality of Arrhenius.jl
is to compute the reaction source terms and mixture properties,
such as heat capacities, enthalpies, entropies, Gibbs free energies, etc. Furthermore, Arrhenius.jl makes it possible for users
to define neural network models as submodels and augment them
with existing physical models. For example, one can use a neural
network submodel to represent the unknown chemical reaction
pathways and exploit various scientific machine learning methods to train the neural network models, such as neural ordinary
differential equations [6, 13] and physics-informed neural network models [14, 15]. One can then implement the governing
equations for different applications with these core functionalities and solve the governing equations using classical numerical
methods or neural-network-based solvers. Arrhenius.jl provides
solvers for canonical combustion problems, such as simulating
the auto-ignition in constant volume/pressure reactors and oxidation in perfectly stirred reactors.
Compared to the legacy combustion simulation packages,
Arrhenius.jl can not only provide predictions given the physical models but also the capability of optimizing model parameters or boundary conditions given experimental measurements.
By efficiently and accurately evaluating the gradient of the solution outputs to the model parameters, experimental data can be
incorporated into the simulation pipeline to enable data-driven
modeling with deep learning algorithms.

the volume of the hot air heater. With the assumption of perfect
mixing in the reactor, the dynamics inside the combustor can be
modeled by a non-adiabatic PSR governed by the following set
of ODEs,
Y in −Y
dY
=
+S (Y ),
dt
τres
dT
∑ qi Ri T − Tin Qloss (T − Ta )
= i
−
−
,
dt
ρC p
τres
ρC p

(3)

where Y in and Tin are the species mass fraction and temperature
of the inlet stream, Ta is the ambient temperature, Qloss is the heat
loss coefficient, T is the mixture temperature inside the chamber,
C p is the specific heat of the mixture inside the reactor, Ri is the
i − th reaction rate, qi is the exothermicity of the i − th reaction,
and the vector S (Y ) refers to the chemical reaction source terms
of the compositions. The combustion characteristics of PSR are
determined by the boundary conditions of τres , Y in , Tin , P, Ta and
Qloss and the initial conditions.
This work considers two kinds of NLCV gases, the coke
oven gas and the blast furnace gas. For both two kinds of fuels, the hot air heater is operated under the following conditions:
P = 101325Pa, Tin = 300K, Ta = 873K, Qloss = 4JK −1 m−3 s−1 ,
and τres = 1.0s. The heat loss coefficient Ql oss is estimated by
calculating the equivalent thermal resistance of the heater wall.
Typical compositions of the NCLV gases applied in this study are
listed in Table 1. The initial conditions of the PSR are the adiabatic and isobaric chemical-equilibrium state of the inlet mixture.
Equations (2-3) are numerically integrated with the ODE solver
T RBDF2() in the Julia package DifferentialEquations.jl [12].

TABLE 1. Typical fuel compositions of coke oven gas (COG) and
blast furnace gas (BFG) used (volume fraction) [4]

2.2 Governing Equations for the Model Combustor
Following the work of [4, 16], we employ a non-adiabatic
unsteady PSR to reveal the complex dynamics exhibited in industrial hot air heaters. The chemical kinetic model employed is
DRM22, a reduced version of GRIMECH 1.2 obtained by Kazakov et al. [17], which consists of 22 species and 104 reactions.
The hot air heater in this study operates under constant atmospheric pressure. The heater chamber is cylindrical with a
constant volume of 13.82 m3 as detailed in [4]. The mass flow
rate of the inlet mixture ma is 2.75 kg/s, and the residence time
τres is calculated according to Eq. 1,
ρV
,
τres =
ma

(2)

Fuel composition
Coke oven gas

Blast furnace gas

H2

CO

CH4

N2

COG1

50.0%

9.0%

30.0%

11.0%

COG2

60.0%

9.0%

25.0%

6.0%

BFG

2.5%

22.5%

0.0%

75.0%

2.3 Inverse Modeling
As discussed in the introduction, inverse modeling mainly
involves unknown inlet conditions. To alleviate the high dimensionality of the time-dependent inlets, we use a neural network
model to represent the inlet conditions. The neural network takes

(1)

where ρ is the density of the mixture in the reactor and V is
3

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FIGURE 1.

Schematic diagram showing the structure of the Arrhenius.jl [11] package.

model.

the input of time and outputs the mixture compositions and mass
flow rate of the inlet stream. Recent work of physics-informed
neural network [14] has demonstrated that the neural network
representation of time-dependent state variable not only tackles
the curse of dimensionality but also has the potential to learn
the physics information in the training data and predict the state
variable beyond the training time domain. Therefore, this work
employs a neural network model to infer the unknown inlet conditions, as outlined in Fig. 2. The true inlet conditions uˆin are
essentially unknown, and the optimization process will make the
state variables u fit with the observations û , and thus we get the
NNinlet (t; θ̃ ) as the best estimation of the boundary conditions.
Without loss of generality, we can rewrite Eqs. 2 and 3 as
du
= f (u ,u in ),
dt

du
= f (u , NNinlet (t; θ )).
dt

For a given initial condition u 0 , the ODEs of Eq. 6 can be
solved with a stiff ODE solver from DifferentialEquations.jl and
results in the following solution array:
u (t) = ODESolve( f , NNinlet ,u 0 ,t; θ ).

(7)

We can then compare the predicted evolution of compositions
inside the combustor with available measurements and guide the
optimization of the neural network model. Specifically, with observations û as the training data, one can define the loss function
as the mean square error (MSE) between the solution array and
the training data via

(4)

where f refers to the right hand side of the ODEs. u refers to
the vector of state variables [Y , T ]. We then use a neural network
model NNinlet (t; θ ) to model the inlet mixture, i.e.,
u in = NNinlet (t; θ ),

(6)

Loss = MSE (u , û ) .

(8)

We employ the algorithm of ForwardDiffSensitivity to compute the gradient of the loss function to the neural network
weights. ForwardDiffSensitivity is an implementation of the discrete forward sensitivity analysis through ForwardDiff.jl [10].

(5)

where θ refers to all of the weights and bias of the neural network
4

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constraints on

"!
#$

%
"

unknown
boundary conditions

observation

NN
forward

"
NN
backward

&&#$'() (+; .)
FIGURE 2.

best
estimation
&&#$'() (+; .)

Schematic diagram of the inference of boundary conditions via neural networks.

Note that accurate gradient computation is crucial for training
neural network models as the optimization problem is usually
stiff and it is difficult to reach a good minimum with noisy gradient.
With the computed gradient, we then employ the SGD optimizer to optimize the neural network models. SGD optimizer is
usually very sensitive to the hyper-parameters, such as the learning rate, initialization, etc. We will present the details of the optimizer in the next section. We then treat the neural network that
minimizes the mismatch between the predicted and measured
composition evolution inside the combustor as the best estimation of the inlet conditions.

ally occurs since the switching temperature is usually determined
based on empirical rules and thus it is not readily known and
may not be optimal during the fuel-switching stage. Therefore,
identifying the transition process during fuel switching would be
beneficial for further optimizing the operation and avoiding unexpected extinction.
Here, we simulate the fuel switching from COG1 to COG2
with a linear switching function to synthesize the measurement
dataset. As shown in Fig. 3(a), the volume fraction of COG2 at
the inlet transitions from 0 to 1 to represent the fuel switching
process from COG1 to COG2 in an industrial combustor. As detailed in [4], the burning of COG fuels will show instability from
mixing-chemistry interaction, with an oscillation frequency of
around 20 Hz. Therefore, the simulation time range here is chosen to be [0, 0.5s] with 100 data points in between, and the fuel
switch process occurs between 0.15 s and 0.35 s. Such time interval in the dataset is sufficiently small to resolve the oscillation
inside the combustor.
In practice, it is often the case that we can only measure the
temperature profiles while it is difficult to measure the species
concentrations. Therefore, to simulate this challenging scenario,
we only employ the temperature profiles into the loss functions
for training the neural network model, i.e.,

3. RESULTS
Non-steady inlet boundary conditions are common causes
for the oscillation of temperature and species concentrations inside the combustor. Industrial operations using the NLCV fuels
are mainly concerned with two types of combustion instabilities.
One is due to fuel switching and the other is due to the flow rate
and calorific fluctuations in the inlet flow. In this section, we
demonstrate the present method in the two representative cases
with industry relevant amplitudes and frequencies. We also design different initialization method and different noise levels to
demonstrate the robustness of the optimization process. We will
first demonstrate the identification of fuel switching from the
measurement of temperature history and then demonstrate the
inference of inlet flow rate and calorific fluctuations from the
measurement of temperature and CO profiles.



Loss = MSE T, Tb .

(9)

Since the neural differential equations involve stiff combustion chemistry, differentiating a stiff neural ODE is usually
much more computationally expensive than a non-stiff problem,
as pointed out in [13]. In addition, the computational cost scales
with the number of neural network parameters. To reduce the
burden of automatic differentiation, the neural network model
should be as concise as possible. Here, a simple neural network
model with a single hidden layer with five hidden nodes is employed. The activation functions used are tanh(x) for the hidden
layer and exp(x) for the output layer.
The popular first-order optimizer of Adam [18] is adopted
for the optimization. The optimization proceeds step-by-step
with a decaying learning rate. Specifically, we first employ a

3.1 Inverse Modeling for Fuel Switching
In combustion devices such as hot air heaters, coke oven
gas can be directly ignited at a normal temperature, for it has
half of the calorific value of natural gas, about 4.75 kW h/m3 .
On the contrary, blast furnace gas with one-tenth the calorific
value of natural gas, about 0.95 kW h/m3 , needs to be burnt at a
higher temperature. During the fuel switch, the distinct combustion characteristic between the two gases can induce oscillating
combustion, and thus cause the vibration of the heater body and
the instability of the heated air. In addition, extinction occasion5

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(a) inlet conditions

(b) state variables

FIGURE 3. Initial states of the fuel switching case. (a) The ground truth and the initialized inlet conditions of COG2 ratio. (b) The ground truth and
predicted species and temperature profiles based on the inlet conditions in (a).

(a) inlet conditions

(b) state variables

FIGURE 4. Trained results of the fuel switching case. (a) The ground truth and the inferred inlet conditions of COG2. (b) The ground truth and
predicted species and temperature profiles based on the inlet conditions in (a).

learning rate of 1e-1 to capture the most important information
with 100 training epochs. We then employ 1e-2 for another 100
epochs and 5e-3 for extra 200 epochs to fine-tune the model. Figure 5 presents the history of the loss function. As can be seen,
the loss decreases stably, which indicates that the learning rate is
properly chosen. The loss function reaches a plateau after 300
epochs, and thus 400 epochs are sufficient for the training.
The initialization of the neural network models is achieved
by assuming a steady inlet with only COG1. Even though the
inlet condition is steady, oscillation is observed in the combustor due to intrinsic instability, as indicated by the dashed line
shown in Fig. 3(b). Similar oscillation is found in the evolution of the state variables corresponding to the ground truth inlet
condition, where fuel switches from COG1 to COG2. Shown as
the symbols, 1% of noise has been added to represent the mea-

FIGURE 5.

6

The loss history of the fuel switching case.

Copyright © 2021 by ASME

surement uncertainty. Such synthesized measurements are also
shown in Fig. 3(b) and utilized to train neural network models
to infer the inlet conditions. Therefore, the lines in Figs. 3(a)
and 3(b) demonstrate the inferred inlet conditions and predicted
state variables, respectively. As can be seen, the inferences and
predictions agree very well with the ground truth. In addition,
the predicted species profiles for methane and carbon monoxide
also agree well with the ground truth, although they are not included in the loss function. Such capability of revealing hidden
information demonstrates the good generalization capability of
the approach. The training takes about 2 hours on a Linux server
with a single CPU core, and the training can be potentially accelerated by improving the weight initialization strategy and parallel
computing [13].

in the hidden layer and the output layer. ai , bi are the weights and
biases of the mapping from the input layer to the hidden layer.
Ai j and Bi j are the weights and biases of the mapping from the
hidden layer to the output layer.
In practice, the fluctuation frequency and amplitude in the
temperature profile can provide insights for the initialization of
the neural network. For example, Fig. 6(a) shows the NNinlet initialized with similar frequencies and amplitudes as the temperature profiles, and the corresponding state variables are shown in
Fig. 6(b) with the dashed red lines.
However, due to the complexity of the coupling between
flow and calorific fluctuations, it is difficult to identify the boundary conditions with only temperatures as the observed data.
Therefore, temperature along with CO mass fractions are used
to constrain the neural networks via the loss function

3.2 Inverse Modeling for Inflow Fluctuations
Another important mechanism for oscillating combustion
behaviors is the uncertain inlet conditions with flow and calorific
fluctuations during the stable burning of the BFG gas. According to Zhang et al., [16], typical inflow fluctuations of hot air
heater consist of 10% of fluctuation in inlet flow rates and 3% of
fluctuation in inlet calorific value, with frequencies around 1 Hz
and 0.05 Hz, respectively. In this study, we employ a sine function with 20% of noise added to synthesize the ”ground truth”
inlet boundary conditions, as shown in Fig. 6(a). The inlet flow
rate fluctuation is characterized by the fluctuations in the resident
time τres = ρV /ma , which has a nominal value of τ̄res = 1.0s and
frequency of 1 Hz. For simplicity, we assume that the fluctuation
of inlet calorific value is due to the composition variation in the
fuel blend of COG1 and BFG. Thus, the volume fraction of BFG
can well characterize the change in the inlet calorific value. To
reduce the computational cost, the frequency of calorific fluctuation is set to be 0.2 Hz and the time range for the ODE integration is [0, 5 s] for demonstration. In addition, to demonstrate the
robustness and capability of the feature learning of the current
method, 10% of noise is added to the state variables obtained
at the presumed inlet boundary conditions to be utilized as the
training dataset for the neural network.
The structure of the neural network NNinlet is similar to that
mentioned in the fuel switching case, but has two output neurons corresponding to the two inlet variables in this case, i.e., the
volume fraction of BFG and the residence time τres . Besides, to
capture the periodic inlet boundary conditions, we employ sin(x)
as the activation function for the hidden layer. This means that
the neural network inlet NNinlet is a linear combination of sine
functions



Loss = MSE [T,YCO ], [Tb, YbCO ] .

Similar to the case of fuel switching, we gradually decaying
the learning rate. We first employ a learning rate of 5e-3 to capture the most important information with 100 training epochs.
Then a learning rate of 2e-3 for another 100 epochs and 1e-3 for
extra 200 epochs are employed to fine-tune the results. The training results are shown in Fig. 7, which takes about 30 minutes to
train with a single thread. The losses of the training process are
shown in Fig. 8.
As can be seen from Figs. 7(a) and 7(b), the inferred inlet conditions and the predicted temperature evolution with the
inferred inlet conditions agree with the ground truth very well.
In addition, the predicted species profiles for methane and other
species also agree well with the ground truth. Such capability of
revealing coupled hidden inlet conditions further shows the potential of the approach in fault diagnostics for sourcing the causes
of combustion oscillation.

4. CONCLUSION
This work presents a machine learning approach for inverse
modeling of inlet boundary conditions of a model combustor.
The approach is demonstrated to recover the inlet conditions under two common modes of unsteady inlets: fuel switching and
inlet flow rate/calorific fluctuations. Using synthesized data, the
results show that the approach can accurately infer the temporal evolution of the inlet boundary conditions from limited measurements of temperature and/or CO profiles. In addition, the
temporal profiles of methane inside the combustor can also be
accurately inferred although methane is not exposed to the training process. Therefore, the approach is shown to efficiently and
accurately reveal the hidden dynamics of the upstream compositions and flow rate.

!
NNinlet (t) j = exp

∑ (Ai j sin(ai x + bi ) + Bi j ) ,

(11)

(10)

i

where i = 1, 2, ..., 5, j = 1, 2, correspond to the indices of neurons
7

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(a) inlet conditions

(b) state variables

FIGURE 6. Initial states of the inflow fluctuating case. (a) The ground truth and the initialized inlet conditions of the volume fraction of BFG (black)
and the flow rate represented by τres (red). (b) The ground truth and predicted species and temperature profiles based on the inlet conditions in (a).

(a) inlet conditions

(b) state variables

FIGURE 7.
Trained results of the inflow fluctuating case. (a) The ground truth and the inferred inlet conditions of the volume fraction of BFG
(black) and the flow rate represented by τres (red). (b) The ground truth and predicted species and temperature profiles based on the inlet conditions in
(a).

The major limitations of this approach are that the algorithm
relies on an accurate combustion kinetic mode, and there could
be multiple possible solutions to the inference due to the uncertainties in the experimental data. In future works, we will incorporate Bayesian neural networks [19] to quantify impact of the
uncertainties in the model and the experimental data to the inferences solution. The estimated uncertainty will greatly facilitate the development of dynamic control of industrial combustion
systems.

FIGURE 8.

ACKNOWLEDGMENT
ZR and XS would like to acknowledge the support from National Natural Science Foundation of China No. 52025062.

The loss history of the inflow fluctuating case.

8

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REFERENCES
[1] Cui, M., Mei, J., Zhang, B.-W., Xu, B.-B., Zhou, L., and
Zhang, Y., 2018. “Inverse identification of boundary conditions in a scramjet combustor with a regenerative cooling
system”. Applied Thermal Engineering, 134, pp. 555–563.
[2] Perakis, N., Strauß, J., and Haidn, O. J., 2019. “Heat flux
evaluation in a multi-element ch4/o2 rocket combustor using an inverse heat transfer method”. International Journal
of Heat and Mass Transfer, 142, p. 118425.
[3] Lee, K. H., and Baek, S. W., 2011. “Inverse estimation of
inlet parameters in an axisymmetric cylindrical combustor
with thermal radiation effect”. International journal of heat
and mass transfer, 54(25-26), pp. 5386–5394.
[4] Zhang, L., Xie, W., and Ren, Z., 2020. “Combustion stability analysis for non-standard low-calorific gases: Blast
furnace gas and coke oven gas”. Fuel, 278, p. 118216.
[5] Ji, W., and Deng, S., 2021. “Autonomous discovery of unknown reaction pathways from data by chemical reaction
neural network”. The Journal of Physical Chemistry A,
125(4), pp. 1082–1092.
[6] Chen, R. T., Rubanova, Y., Bettencourt, J., and Duvenaud,
D., 2018. “Neural ordinary differential equations”. arXiv
preprint arXiv:1806.07366.
[7] Baydin, A. G., Pearlmutter, B. A., Radul, A. A., and
Siskind, J. M., 2018. “Automatic differentiation in machine
learning: a survey”. Journal of machine learning research,
18.
[8] Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A.,
Dean, J., Devin, M., Ghemawat, S., Irving, G., Isard, M.,
et al., 2016. “Tensorflow: A system for large-scale machine learning”. In 12th {USENIX} symposium on operating systems design and implementation ({OSDI} 16),
pp. 265–283.
[9] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury,
J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N.,
Antiga, L., et al., 2019. “Pytorch: An imperative style,
high-performance deep learning library”. arXiv preprint
arXiv:1912.01703.
[10] Revels, J., Lubin, M., and Papamarkou, T., 2016. “Forwardmode automatic differentiation in julia”. arXiv preprint
arXiv:1607.07892.
[11] Ji, W., and Deng, S., 2021. Arrhenius.jl: A differentiable
combustion simulation package. https://github.
com/DENG-MIT/Arrhenius.jl.
[12] Rackauckas, C., and Nie, Q., 2017. “Differentialequations.
jl–a performant and feature-rich ecosystem for solving differential equations in julia”. Journal of Open Research Software, 5(1).
[13] Kim, S., Ji, W., Deng, S., and Rackauckas, C., 2021. “Stiff
neural ordinary differential equations”. arXiv preprint
arXiv:2103.15341.
[14] Raissi, M., Perdikaris, P., and Karniadakis, G. E., 2019.

“Physics-informed neural networks: A deep learning
framework for solving forward and inverse problems involving nonlinear partial differential equations”. Journal
of Computational Physics, 378, pp. 686–707.
[15] Ji, W., Qiu, W., Shi, Z., Pan, S., and Deng, S., 2020. “Stiffpinn: Physics-informed neural network for stiff chemical
kinetics”. arXiv preprint arXiv:2011.04520.
[16] Zhang, L., Xue, Y., Xie, Q., and Ren, Z., 2021. “Analysis
and neural network prediction of combustion stability for
industrial gases”. Fuel, 287, p. 119507.
[17] Kazakov, A., and Frenklach, M., 1994. http://www.
me.berkeley.edu/drm/.
[18] Kingma, D. P., and Ba, J., 2014. “Adam: A method for
stochastic optimization”. arXiv preprint arXiv:1412.6980.
[19] Gal, Y., and Ghahramani, Z., 2016. “Dropout as a bayesian
approximation: Representing model uncertainty in deep
learning”. In international conference on machine learning,
PMLR, pp. 1050–1059.

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