Proceedings of the ASME 2021
Internal Combustion Engine Division Fall Technical Conference
ICEF2021
October 13-15, 2021, Virtual, Online

DATA-DRIVEN APPROACHES TO LEARN HYCHEM MODELS
Weiqi Ji
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139
Email: weiqiji@mit.edu

Julian Zanders
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Cambridge, MA 02139
Email: jzanders@mit.edu

Ji-Woong Park
Argonne National Laboratory
Lemont, IL 60439
Email: ji.park@anl.gov

Sili Deng ∗
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, MA 02139
Email: silideng@mit.edu

ABSTRACT
The HyChem (Hybrid Chemistry) approach has recently
been proposed for modeling high-temperature combustion of
real, multi-component fuels. The approach combines lumped reaction steps for fuel thermal and oxidative pyrolysis with detailed
chemistry for the oxidation of the resulting pyrolysis products.
Determining the pyrolysis submodel requires extensive experimentation on speciation measurements. Recent work has been
directed to learn HyChem from an existing HyChem model for
a similar fuel, which requires less data. However, the approach
usually shows substantial discrepancies with experimental data
within the Negative Temperature Coefficient (NTC) regime, as
the low-temperature chemistry is more fuel-specific than hightemperature chemistry. This paper proposes a machine learning approach to learn the HyChem models that can cover both
high-temperature and low-temperature regimes. Specifically, we
develop a HyChem model using the experimental datasets of ignition delay times covering a wide range of temperatures and
equivalence ratios. The chemical kinetic model is treated as a
neural network model, and we then employ stochastic gradient
descent (SGD), a technique that was developed for deep learn-

ing, for the training. We demonstrate the approach in learning
the HyChem model for F-24, which is a Jet-A derived fuel, and
compare the results with previous work employing genetic algorithms. The results show that the SGD approach can achieve
comparable model performance with genetic algorithms but the
computational cost is reduced by 1000 times. In addition, with
regularization in SGD, the SGD approach changes the kinetic
parameters from their original values much less than genetic algorithm and is thus more likely to retrain mechanistic meanings.
Finally, our approach is built upon open-source packages and
can be applied to the development and optimization of chemical
kinetic models for internal combustion engine simulations.

INTRODUCTION
Predictive chemical kinetic models for real fuels play an important role in modeling the combustion processes in practical
engines. Modeling real fuels is very challenging since real fuels
are comprised of a variety of components with different physical
and chemical properties. Therefore, much effort has been directed to developing compact models for real fuels that are computationally efficient and can accommodate the variations in fuel
compositions. These approaches can be categorized as the ”sur-

∗ Address all correspondence to this author.

This work was authored in part by a U.S. Government employee in the scope of his/her employment.
ASME disclaims all interest in the U.S. Government’s contribution.
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ICEF2021-67925

ing scalability and generalization of the data-driven HyChem
model using techniques developed for training deep neural network models. We propose the stochastic gradient descent (SGD)
optimizer to learn the pyrolysis submodel in HyChem models.
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 [10] 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
[11], which further rationalizes exploiting SGD in training HyChem models. However, the base model is constrained using the
sequential optimization from multiple datasets, and thus changing the kinetic parameters too much has the risk of over-fitting
the experimental dataset. Therefore, we also adopt various regularization techniques used in training deep neural network models to increase the generalization performance of learned models.
Thus, we can achieve a model that has comparable performance
to the conventional GA approach, but with much smaller changes
in the kinetic parameters from the reference values.
However, one of the major challenges in exploiting SGD
for optimizing combustion kinetic models is the lack of software
ecosystems that enable efficient and accurate computation of the
gradient to model parameters. For instance, the finite difference
method (often termed the ”brute-force method”) usually suffers
both computational inefficiency due to the cost increases with the
dimension of the parameters and inaccuracy due to the numerical
truncation error [12]. Conversely, gradient computation methods
based on auto-differentiation (AD) have shown both efficiency
and accuracy in the training of large-scale deep neural network
models. Many open-source AD packages have been developed
in the last decade, such as TensorFlow [13] and PyTorch [14]
in python, ForwardDiff.jl [15] and Zygote.jl [16] in Julia. To
this end, this work employs an open-source simulation package
Arrhenius.jl [17], which enables SGD for combustion model optimization, to the modeling of F-24, which is a variant of Jet-A
with additives.

METHODS
Arrhenius.jl
Arrhenius.jl is developed with the programming language of
Julia to take advantages of the rich ecosystems of AD and ordinary differential equation solvers. Arrhenius.jl can do two kinds
of differentiable programming: (I) it can differentiate each elemental computational block. For example, it can differentiate
the chemical reaction source term with respect to the kinetic and
thermodynamic parameters as well as species concentrations; (II)
it differentiates the entire numerical simulation in various ways,
such as forward sensitivity analysis by solving the continuous

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rogate modeling approach” and the ”HyChem approach.”
In the surrogate model approach [1, 2, 3], it is assumed that
the functional groups characterize the combustion properties of a
fuel mixture. More specifically, if a real fuel and a surrogate mixture share similar compositions of functional groups, they should
also share similar combustion behaviors. Besides the functional
groups, the physical properties of the mixture, such as heat capacity and molecular weights, also play important roles. Previous studies [1, 2, 3] have demonstrated that using a handful of
pure hydrocarbon fuels could formulate a surrogate fuel mixture
that effectively reproduces the combustion behaviors of a complex real fuel, including the ignition delay times (IDTs) and laminar flame speeds. However, it is often an art to determine the
choice of surrogate fuel components and the procedures for developing surrogate fuels.
Recently, Wang and his co-workers have proposed an endto-end modeling approach named ”HyChem (Hybrid Chemistry)” [4, 5, 6]. The HyChem approach describes the fuel pyrolysis and the subsequent oxidation of the pyrolysis products in
two submodels. The fuel pyrolysis/oxidative pyrolysis is modeled by several experimentally constrained and lumped reaction
steps. The oxidation of the pyrolysis products is modeled by a
detailed foundational chemistry model. The two submodels are
interconnected. The pyrolysis process provides ”reactants” for
the oxidation process; the oxidation provides heat and radicals to
facilitate the endothermic pyrolysis of the fuel.
The lumped reaction steps are mainly determined via a sequential optimization using species profiles during the hightemperature pyrolysis. One can also use a neural network to
represent the pyrolysis steps [7, 8] and learn the neural network
using species profiles. Compared to the species profiles that require expensive laser diagnostic equipment, ignition delay times
are easier to measures in shock tubes and rapid compression machines. Therefore, recent work [9] has proposed learning the rate
constants of lumped reaction model from an ignition delay times
dataset only by employing the stoichiometric coefficients from
existing models for similar fuels. However, the base model usually shows large discrepancies in the low-temperature regimes,
as the low-temperature chemistry is more fuel-specific compared
to high-temperature chemistry. Thus, one has to optimize the
lumped reaction steps from experimental data such that the optimized model can cover both high- and low-temperature regimes.
Ryu et al. [9] employed genetic algorithms and IDT data to develop a data-driven kinetic model for the Jet fuel of F-24 using
the Jet-A HyChem model as the base model. The core of such
a data-driven modeling approach is the optimization step using
genetic algorithms. However, genetic algorithms (GA) usually
suffer from the curse of dimensionality, and thus they are usually limited to less than one hundred parameters. In addition, it
is usually questionable how well a data-driven model is able to
extrapolate beyond the training dataset.
Therefore, this work aims to tackle the challenges in achiev-

Schematic diagram showing the structure of the Arrhenius.jl [17] package.

sensitivity equations [18] as did in CHEMKIN [19] and Cantera [20], and the adjoint methods [21, 22]. The first type of
differentiation usually serves as the basis of the second type of
high-level differentiation. Arrhenius.jl provides the core functionality of reacting flow simulations in native Julia programming, such that users can readily develop applications on top of
Arrhenius.jl’s basic modules and exploit various approaches to
do high-level differentiation.
Figure 1 shows a schematic diagram of the Arrhenius.jl
package. The package reads in the chemical mechanism files
in YAML format from the Cantera community. The YAML file
contains the kinetic, thermodynamic, and transport databases.
The core functionality of Arrhenius.jl is computing the reaction
source terms and mixture properties, such as heat capacities, enthalpies, Gibbs free energies, etc. Furthermore, Arrhenius.jl provides flexible interfaces for users to define neural network models as submodels and augment them into the governing equations.
For example, one can employ a neural network submodel to represent the unknown chemical reaction pathways and exploit Scientific Machine Learning (SciML) approches to train the neural network models, such as neural ordinary differential equations [11,23] and physics-informed neural network [24,25]. One
can then implement the governing equations for various applications with these core modules and solve the equations using classical numerical methods or neural-network-based solvers, such

as physics-informed neural networks [24, 25]. Arrhenius.jl ships
with applications for canonical combustion problems, such as
simulating the auto-ignition in constant pressure/volume reactors
and oxidation/pyrolysis in jet stirred reactors.
Compared to other open-source combustion simulation
packages, Arrhenius.jl can not only perform deterministic simulations given the physical models, but also can optimize model
parameters based on experimental measurements. By efficiently
and accurately computing the gradient of the numerical solutions
to the model parameters, experimental data can be incorporated
into the modeling pipeline to enable data-driven modeling with
deep learning inspired algorithms.
Dataset and Base Model
Following the work of [9], we employ the IDT dataset compiled in [9], including the IDTs of the lean, stoichiometric, and
rich F-24/air mixtures measured at 20 bars in a rapid compression machine (RCM) and a shock tube (ST) over the temperature
range of 625 K to 1250 K. The RCM is driven pneumatically
with a hydraulic stop apparatus, and the initial chamber temperature and pressure are achieved by adjusting the compression ratio [26]. Low-temperature (up to 700 K) IDTs were measured
in the RCM. The shock tube uses a polycarbonate diaphragm to
separate the driver and driven sections, the lengths of which are
2.744 m and 4.227 m, respectively. A tailored mixture of He and

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FIGURE 1.

Gradient Evaluation
As the only available dataset is the IDT, the key step in utilizing SGD for the optimization of the HyChem model is to compute the gradient of the IDT with respect to the kinetic parameters. Recent work [27, 18, 28, 29] have substantially advance
the algorithms for computing the gradient of IDT to model parameters, this work employs the sensBVP method [27]. The approach converts the initial value problem (IVP) to a boundary
value problem (BVP) by treating the temperature at ignition as a
boundary condition and the IDT as a free variable to solve. The
algorithm is implemented with Arrhenius.jl and further enhanced
with the auto-differentiation, multi-threading, and sparse matrix
techniques. In addition, the Arrhenius.jl enables gradient computation w.r.t. all of three Arrhenius parameters, which is crucial
for the optimization, while the existing package of SensBVP in
C++ [27] can only compute the gradient w.r.t. the pre-factor A.
For the size of the skeletal HyChem model considered in this
work, the computational cost of the sensitivity of IDT is negligible comparing to computing the IDT.
It is worthy to note that the gradient of laminar flame speed
to kinetic parameters is also usually computed using the boundary value problem approach as in the sensBVP approach for
IDTs. Thus, the computational cost for the gradient computation of flame speed is also negligible comparing to computing
the flame speed. Therefore, the gradient-based optimization approach proposed in this work can be readily extended to the experimental dataset of flame speed as well.
Optimization Techniques
Following the HyChem approach, the 16 steps of the pyrolysis submodel, listed in Table 1, are subject to optimization, and
the rest of the core model is fixed. Although in the work of [9],
only the pre-factor A and activation energy Ea are optimized, the
SGD optimizer can easily scale up to high-dimension, such that
all three parameters, including the non-exponential temperature
dependence factor b, are optimized. Consequently, the optimization involves 48 parameters.
As discussed in the introduction, we would like to regularize the model to achieve an optimized model that can fit the experimental datasets well but only moderately change the kinetic
parameters from the base model. To facilitate regularization,

the model parameters are scaled and centered at zero, such that
the relative changes of Arrhenius parameters is to be optimized.
Specifically, we define the scaled parameters as
p = [ln(A/A0 ), b − b0 , Ea − Ea0 ],

(1)

where the subscript 0 refers to the base model and A, b, Ea
are the three Arrhenius parameters in a Arrhenius equation
a
AT b exp( −E
RT ). The dimension of Ea is specified as kcal/mol
rather than cal/mol as a change of 2 kcal/mol in Ea is roughly
equivalent to changes of e times in A, such that the changes in
A and Ea will be balanced and avoid stiffness in the parameter
space.
The loss function is defined as the mean square error (MSE)
between the predicted IDTs in the logarithmic scale using the
optimized model and the measured IDTs, as shown in Eq. 2:
Loss = MSE (log(IDT op ), log(IDT exp ))

(2)

The choice of logarithmic scale is similar to the choice of relative error between measured and predicted IDT as in [9]. The
Adam [30] optimizer, a popular variant of SGD, with the default
learning rate of 0.001 is adopted. Weight decaying and early
stopping [31] are employed to regularize the parameters, such
that the optimization will prefer kinetic parameters that are close
to their original values.

RESULTS
In this section, we first present an overview of the learning
results and then present the details of the learning algorithms.
The code implementations can be found in the Github repository
for Arrhenius.jl [17].
Overview of Learning Results
Figure 2 show the comparisons between the experimentally measured IDTs and the predictions using the base HyChem
model, the optimized model using genetic algorithms (GA) by
[9], and the current model. Overall, both of the optimized models agree with experimental data very well, with an MSE of 0.008
for SGD and 0.044 for GA on the entire dataset. As pointed out
by [9], the base model substantially underpredicts the IDTs in the
NTC regimes for all three equivalence ratios, while it performs
reasonably well in the high-temperature regimes. Conversely,
both of the optimized models substantially improve the performance in the NTC regime.
Since the slope of the response curve is indicative of the
global activation energy, it is interesting to note that the current
model shows a smaller global activation energy at lower temperature regimes, where the temperature is below 700 K, than the

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N2 was used as the driver gas, and premixed F-24/air mixtures
were introduced to the driven section.
For the base model, we adopt a HyChem version of the JetA skeletal mechanism [5] that covers the NTC regime. It has
48 species and 254 reactions. In addition, details of the formulations and stoichiometric coefficients in the F-24 model are the
same as in [9], and they are listed in Table 1. For simplicity and
consistency, all of the simulations are carried out under constant
volume conditions using Arrhenius.jl.

TABLE 1.

List of reaction pathways in the pyrolysis submodel presented in the skeletal Jet-A HyChem model [5]

1. The current model indeed underpredicts the activation energy due to large uncertainties in the training data, since the
IDT measurements in the NTC regime are usually associated with large uncertainties. In addition, the simulations
for training and predictions are carried out under constant
volume assumptions for simplicity and consistency, which
might not be appropriate for two-stage ignition [34]. These
uncertainties in the training dataset and modeling may result in a biased model. In such case, the data re-balance
procedure in [9] can be employed to increase the weights of
low-temperature data on the model training.
2. The reported measurements are subject to heat loss, and
therefore overestimate the global activation energy. The heat
loss for experiments with long testing time in RCM usually
has a substantial effect on the IDTs. The reported experimental data using the temperature at the end of compression
gives higher global activation energy than the actual one.
In such case, the effective temperature [35, 36] is usually a
better representative temperature than the temperature at the
end of compression.
On the other hand, the SGD optimization tends not to over-fit the

data at low temperatures compared to the GA optimization, since
multiple regularization techniques are employed to prevent overfitting. From this point of view, the experimental data are likely
giving an overly large global activation energy than the actual
one. Nevertheless, good data-driven modeling not only can help
us develop a predictive model from experimental data but also
help reveal potential experimental uncertainties and guide further
experimental designs to advance our physical understandings.
To assess the generalization capability of the learn model,
we further evaluate the performance of the optimized models in
predicting laminar flame speeds, although the model is not optimized against flame speed data. To the best of our knowledge,
there are no experimental data of flame speed for F-24 in the
open literature. Thus, we apply the model to predict flame speed
for Jet-A as F-24 is derived from Jet-A and might share similar
flame propagation behaviors. The predicted and measured flame
speeds for Jet-A are presented in Fig. 3.
While it is difficult to assess which model performs better given the uncertainties in the experimental data, we can use
the base model as a reference, for the base model is calibrated
for high-temperature pyrolysis. The base model is expected to
perform well for flame propagation that is dominated by hightemperature chemistry. Overall, both SGD and GA models predict the similar flame speed with the base model, with the differences within 2 cm/s. On the other hand, the GA model systematically predicts slower laminar flame speeds than the base model.
The predictions by the current model agree with those by the base
model at fuel-lean conditions and show similar trends as those by
the GA model under fuel-rich conditions. This could be poten-

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one suggested by the experimental data and the GA model. Such
global activation energy is related to the key reactions that govern the low-temperature auto-ignition [32, 33], and thus it is important to accurately predict the global activation energy to gain
physical insight of the ignition behavior of F-24. The discrepancies in the predicted global activation energy can be attributed to
several plausible reasons:

FIGURE 2. Ignition delay times from F-24 RCM/shock tube measurements (Exp), and predicted by the skeletal Jet-A HyChem model
(Base), the optimized HyChem model using SGD in current work
(SGD), and optimized HyChem model using genetic algorithms in [9]
(GA), at 20 bars for lean (φ = 0.5), stoichiometric, and rich (φ = 1.3)
fuel/air mixtures.

tially attributed to the regularization in the SGD optimization.
As shown in Tables 2 and 3, the kinetic parameters in the current
model are closer to the base model compared to those in the GA
model. For the GA model, the activation energy for the direct
decomposition reaction of R1 is changed a lot, and such changes
may lead to systematic changes in flame speed under both fuellean and rich conditions. For the SGD model, it is possible that
the flame speed is only sensitive to the changes of kinetic parameters under fuel-rich conditions, as the fuel-lean and rich conditions correspond to the different radical pools. Future work
could exploit advanced pathway analysis tools, such as global
pathway analysis [37], to further investigate the different trends
under lean and rich conditions. Nevertheless, the SGD model
tends to less affect the model parameters for high-temperature
chemistry while improving the low-temperature chemistry from
experimental data. Thus, comparing to the GA model, the SGD
model is more likely to be able to extrapolate to conditions beyond the range of the training dataset.
While the experimental dataset of laminar flame speed can
be readily incorporated in the SGD approach as discussed in the
methodology section, the results in Fig. 3 suggest that learning
the HyChem model from the ignition delay time dataset might
be sufficient to develop a model that works well for both ignition
and flame propagation with the help of regularization. Such generalization capability is especially useful recognizing that experimental measurements of laminar flame speed are usually limited
to near-atmospheric pressures due to the onset of turbulence at
elevated pressures [38, 39]. Nevertheless, the SGD approach can

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FIGURE 3. Laminar flame speeds from Jet-A measurements [5]
(Exp), and predicted by the skeletal Jet-A HyChem model (Base), the
optimized HyChem model using SGD in current work (SGD), and optimized HyChem model using genetic algorithms in [9] (GA), at one atm
and 423 K.

Details of Learning Algorithms
We then present the details of the learning algorithms that
produce the SGD-optimized model. Figure 4 shows the loss history in the training and validation of the model. 76 out of 96
samples were randomly chosen as the training dataset, and the
rest were for the validation. Within each epoch, the training
dataset was randomly shuffled to introduce randomness to the
training process to prevent over-fitting. Mini-batch optimization
with a batch size of one was adopted such that the model parameters were updated for each sample and 76 times for each epoch.
Cross-validation and early-stopping were exploited to prevent
over-fitting. For instance, the training is stopped when the loss in
the validation reaches a plateau, although the loss in the training
is still decreasing. The validation loss reached a plateau at around
60 epochs, although we let the training continues to 110 epochs
to confirm the plateau. We thus select the model at 58 epochs,
which is the closest checkpoint near 60 epochs, and the checkpoint is saved every time the validation loss reaches a new minimal. In summary, we adopted multiple techniques, including
mini-batching, cross-validation and early-stopping, to regularize
the model and prevent over-fitting. The regularization in SGD in
general leads to better generalization performance, which is one
of the most important advantages comparing to GA optimization,
although both SGD and GA can give a similar fitness in a small
dataset with less than one hundred parameters.
As far as the computational cost is concerned, SGD optimization is also much more efficient than genetic algorithms. To
estimate the computational costs for both algorithms, we denote
the computation of the IDT for all 96 samples as one computational unit. Since the computational cost for gradient evaluation is negligible, the current SGD optimization for F-24 costs
about 100 units. For comparison, the GA optimization for the
same datasets and base model costs about 105 units, with 32
particles and 300 generations [9]. Such direct comparison reveals 1000 times of acceleration, which also results in a reduced
need for computational power. For example, the current training
was performed on a desktop workstation for hours, while the GA
optimization in [9] was performed on a high-performance cluster. Note that one can accelerate the GA optimization by parallel
evaluation of the particles in each generation. In addition, the
acceleration factor scales with the size of the dataset and the dimension of the model parameter. Therefore, such computational
saving is expected to be more significant when the kinetic system
is more complex and more data are available.

FIGURE 4. The history of the loss function in the training (with 76
samples) and validation (with 20 samples). Within each epoch, the training dataset was randomly shuffled and the optimizer updated the model
parameters for each sample and 76 times for each epoch.

CONCLUSION
This work presents a machine learning approach for learning
and optimizing HyChem models. By treating the kinetic model
as a neural network and using the stochastic gradient descent for
optimization, HyChem models can be learned from ignition delay time measurements. Such an approach was demonstrated in
training a HyChem model for the jet fuel of F-24. Compared to
the commonly utilized genetic algorithms, the current approach
has 1000 times less computational cost and better generalization
performance with minimal degradation to the high-temperature
chemistry while significantly improving the performance at low
temperatures. In addition, further computational cost-saving is
expected for larger datasets and more complex kinetic systems.
Finally, the code for the current approach is built upon opensource packages and is available online.
While the current work focuses on developing HyChem
models, such optimization techniques can be readily applied to
other computational tasks. For example, optimizing overly reduced chemical models is a technique that has been pursued in
practical engine simulations [40, 41]. In such tasks, high dimensional kinetic systems with hundreds of model parameters are
usually present. Therefore, the current SGD-based approach will
be superior to genetic algorithms.

ACKNOWLEDGMENT
WJ and SD would like to acknowledge the funding support
from Weichai Power Co., Ltd. WJ would like to acknowledge
the help from Dr. Vyaas Gururajan on the implementation of the
sensBVP method and the help from Dr. Je Ir Ryu on simulating

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potentially enable us to develop a data-driven kinetic model by
providing only the high-pressure ignition delay time dataset but
can work well for predicting both ignition and flame propagation
at engine relevant elevated pressures.

TABLE 2. The kinetic parameters of the pyrolysis submodels in the skeletal Jet-A HyChem model [5] (Base), the optimized model using SGD in
current work (SGD), and the optimized model using genetic algorithm in [9] (GA).

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TABLE 3. The changes of the kinetic parameters in the optimized models in current work (SGD) and in [9] (GA) with respect to the base model of a
skeletal Jet-A HyChem model [5].

F-24 ignition with the GA-optimized HyChem model.

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