# Prediction of liquid fuel properties using machine learning models with
Gaussian processes and probabilistic conditional generative learning

Rodolfo S. M. Freitas
Dept. Mechanical Engineering
COPPE/ Federal University of Rio de Janeiro
RJ 21941-598, Rio de Janeiro, Brazil
<EMAIL_ADDRESS>
& Ágatha P. F. Lima
Dept. Mechanical Engineering
COPPE/ Federal University of Rio de Janeiro
RJ 21941-598, Rio de Janeiro, Brazil
<EMAIL_ADDRESS>
& Cheng Chen
School of Engineering and Materials Science
Queen Mary University of London
Mile End Road, London E1 4NS, UK
<EMAIL_ADDRESS>
& Fernando A. Rochinha
Dept. Mechanical Engineering
COPPE/ Federal University of Rio de Janeiro
RJ 21941-598, Rio de Janeiro, Brazil
<EMAIL_ADDRESS>
& Daniel Mira
Barcelona Supercomputing Center
Barcelona, Spain
<EMAIL_ADDRESS>
& Xi Jiang
School of Engineering and Materials Science
Queen Mary University of London
Mile End Road, London E1 4NS, UK
<EMAIL_ADDRESS>
The research leading to these results had received funding from the Brazilian
National Agency for Petroleum, Natural Gas and Biofuels (ANP) through Programa
de Recursos Humanos (PRH) under the PRH 8 - Mechanical Engineering for the
Efficient Use of Biofuels, grant agreement numbers F0A5.EDDE.B5C0.3BCB and
2B61.4F5C.A83B.A713.

###### Abstract

Accurate determination of fuel properties of complex mixtures over a wide
range of pressure and temperature conditions is essential to utilizing
alternative fuels. The present work aims to construct cheap-to-compute machine
learning (ML) models to act as closure equations for predicting the physical
properties of alternative fuels. Those models can be trained using the
database from MD simulations and/or experimental measurements in a data-
fusion-fidelity approach. Here, Gaussian Process (GP) and probabilistic
generative models are adopted. GP is a popular non-parametric Bayesian
approach to build surrogate models mainly due to its capacity to handle the
aleatory and epistemic uncertainties. Generative models have shown the ability
of deep neural networks employed with the same intent. In this work, ML
analysis is focused on a particular property, the fuel density, but it can
also be extended to other physicochemical properties. This study explores the
versatility of the ML models to handle multi-fidelity data. The results show
that ML models can predict accurately the fuel properties of a wide range of
pressure and temperature conditions.

_Keywords_ Fuel properties $\cdot$ Molecular dynamics $\cdot$ Deep Learning
$\cdot$ Machine learning models

##  1 Introduction

Fossil fuels have been playing a major role in energy supply and liquid fossil
fuels have dominated the energy use in transport, which will continue to be so
for many decades to come, especially for sectors that are difficult to
decarbonise. With the pressing needs of decarbonisation and sustainable energy
utilisation, renewable fuels and biofuels are becoming increasingly important
Omari et al. (2017); Pélerin et al. (2020). For instance, synthetic fuels like
Oxymethylene Dimethyl Ethers (OMEs) have shown high potential for low-carbon
transport applications due to their capacity to avoid soot formation Pastor et
al. (2020). However, the physicochemical properties of these fuels must be
known for their rapid integration into current infrastructures for storage,
transport and direct injection in combustion engines. This represents a
significant challenge, due to the fact that practical fuels are often composed
by complex mixtures and vary widely in their chemical compositions depending
on the production source and process Omari et al. (2017). For example,
petroleum diesel is a complex mixture involving molecules with carbon chains
that typically contain between 9 and 25 carbon atoms per molecule. To simplify
the complex chemical compositions of these fuels, surrogate models have been
used to represent the chemical composition and combustion characteristics in
practical applications Pitz and Mueller (2011); Lai et al. (2011). In
addition, modern combustion engines have to operate at high pressure
conditions in order to improve the energy conversion efficiency. Fuel
properties at extreme conditions such as high pressure and high temperature
conditions, are very difficult to measure and predict Pastor et al. (2020),
leading to an additional challenge.

Accurate determination of fuel properties of complex mixtures over a wide
range of pressure and temperature conditions is essential to adapt the system
operation to alternative fuels. In recent years, molecular dynamics (MD)
simulations have been used to predict the physicochemical properties of
practical fuels including transport properties at supercritical conditions
Chen and Jiang (2019). By using equilibrium molecular dynamics (EMD) and
nonequilibrium molecular dynamics (NEMD), Yang et al Yang et al. (2021a, b)
predicted the viscosity and thermal conductivity of alkanes (n-decane,
n-undecane and n-dodecane). Kondratyuk et al Kondratyuk et al. (2020a, b);
Kondratyuk and Pisarev (2019) performed a serial of MD simulation to study the
viscosity of hydrocarbons (1-methylnaphthalene, methylcyclohexane and
2,2,4-trimethylhexane) in high pressure conditions up to 1000 MPa. Caleman et
al Caleman et al. (2012) tested the capacity of existing force fields on
prediction of properties (density, enthalpy of vaporization, surface tension
and heat capacity etc) of organic liquids. Although MD simulations provide
molecular details that can be potentially used to accurately predict fuel
properties, they are generally expensive in terms of computational costs. In
addition, MD predictions also need to be validated against experimental
measurements, which can be even more costly especially at extreme conditions.
Accordingly, it is not feasible to establish complete and detailed fuel
property databases consisting of a wide range of pressure and temperature
conditions using either MD simulations or experiments.

Machine learning has great potentials to discover the relation between inputs
and outputs in a thermodynamic system directly from the data of complex
systems Freitas et al. (2020). ML can be a powerful tool to predict fuel
properties from chemical compositions of the fuel mixture and/or chemical
structures of the fuel molecules. Several works have been devoted to designing
ML models capable of predicting complex fuels properties from experimental
data. In this regard, ML models obtained accurate predictions of cetane number
(CN) compared to experimental data Ramadhas et al. (2006); Piloto-Rodríguez et
al. (2013); Miraboutalebi et al. (2016). A satisfactory ML approach for
modeling the CN of biodiesel based on four operating conditions given by
iodine volume (IV), carbon number, double bounds, and saponification value was
proposed Faizollahzadeh Ardabili et al. (2019). Moreover, ML models were tuned
with evolutionary algorithms to predict the CN of biodiesel as a function of
its fatty acid methyl ester (FAME) profile Mostafaei (2018); Bemani et al.
(2020). The predictability of the ML approaches also can be improved by using
different optimization algorithms such as teaching-learning based optimization
(TLBO), backpropagation, Quasi-Newton and particle swarm optimization (PSO)
Baghban et al. (2018); Noushabadi et al. (2020); Sánchez-Borroto et al.
(2014). Also, ML models have been used for modeling the kinematic viscosity of
diesel-derived fuels as a function of their FAMEs profiles Yu and Zhao (2019);
Alviso et al. (2020); Meng et al. (2014). In the last years, Multilayer
Perceptron Neural Networks (MLPNNs) have been successfully built to estimate
the physicochemical characteristics of biodiesel Cheenkachorn (2006); Giwa et
al. (2015); Rocabruno-Valdés et al. (2015); de Oliveira et al. (2017)
combining different parameters of model inputs. Furthermore, ML models based
on state variables such as temperature and pressure showed high potential to
obtain physicochemical properties of biodiesel/diesel fuels more accurately
Zhou (2018); Eryılmaz et al. (2015); Eryilmaz et al. (2016). Recently, a ML
approach based on support vector regression (SVR) was proposed by Liu et al.
(2019) for predicting the PVT properties of pure fluids (H2O, CO2, and H2) and
their mixtures, where the training database is provided by the National
Institute of Standards and Technology (NIST) and MD simulations. Also, an ML
approach was proposed to assess the macroscopic ECN Spray-A characteristics
and predictions of fluid properties for the thermodynamic states found in such
conditions Koukouvinis et al. (2021). Yet, from our knowledge, little work has
been dedicated towards exploring the thermodynamic properties of practical
fuels combining MD simulations and ML models. ML can be a powerful tool to
predict fundamental fuel properties directly from the chemical compositions of
the fuel mixture by using databases from MD simulations or available
experimental measurements.

The aim of the study was to demonstrate and validate a ML-MD methodology to
predict fundamental properties of liquid fuels. In this approach, the ML
models are built from data provided by MD simulations, while a combination of
MD and NIST data is used for model assessment and validation. This study is
the first attempt of using ML models with Gaussian process regression
Rasmussen and Williams (2006) and probabilistic conditional generative
learning Yang and Perdikaris (2019a, b) for the property predictions of
single-compound fuels. The ML analysis is focused on fuel density in this
study as one of fundamental properties of liquid fuels, though it can easily
be extended to other physicochemical properties of relevance for practical
applications like viscosity, conductivity or surface tension.

The rest of the paper is organized as follows. Section 2 presents the ML
models and the molecular dynamics simulation methodology. Section 3 describes
the ML results for typical fuel surrogates of diesel. Finally, Section 4
concludes the study with recommendation for further investigations.

##  2 Methodology: Building Machine Learning Models to Describe Physicochemial
Properties

In order to reduce energy consumption and pollutant formation, supercritical
combustion has been increasingly explored in the context of high pressure
internal combustion engines and rocket engines Safarov et al. (2018).
Specifically, in supercritical conditions, the devices operate with pressures
and temperatures higher than the critical values, which implies that
physicochemical properties of fluids are quite different from those at liquid
conditions Pioro et al. (2011). In such scenarios, the design of devices
become more complex, specially due to limitations of replicating flow and
combustion in controlled laboratory environments. In order to cope with these
challenges, computational models can provide adequate tools for obtaining more
accurate predictions of state variables and increase cycle performance in
transcritical conditions.

From a computational fluid dynamics (CFD) perspective, combustion models are
built upon the combination of solid and reliable physico/chemical principles
with closure models, typically describing physicochemical properties of the
fuels and their mixtures using approaches that normally entail uncertainties.
The use of numerical simulations for practical applications encompass a wide
range of conditions, resulting in different fundamental problems depending on
the nozzle geometry, engine architecture or thermodynamic conditions. A good
example is the database from the Engine Combustion Network Pickett et al.
(2010) for which different sprays for diesel- and gasoline-like conditions are
investigated. For instance, pressure can go from sub-atmospheric to 2,000 bar,
and temperatures from cold to highly preheated conditions. In that context,
having accurate values for macroscopic fuel characteristics and properties
over such wide variety of spatial and time scales is one of the main
challenges for physically-driven methods. That is particularly more dramatic
for modern compounds depicting complex chemical compositions, and simplified
surrogate fuels Shen et al. (2021) are employed to estimate the properties of
the original compounds. That allows the systematic use of controlled
experiments and, also, Molecular Dynamics simulations Razi et al. (2018); Xing
et al. (2021). Indeed, here our focus lies on using ML models to leverage such
type of simulations when obtaining liquid fuel physicochemical properties.
Those properties are generally expressed as functions of local thermodynamic
conditions like pressure and temperature, which motivate to refer to closure
models such as the Equations of State (EoS). In general, the EoS is embedded
in complex CFD simulations resulting in divergence or numerical oscillations
when used with traditional methods based on tabular and interpolation schemes
Koukouvinis et al. (2020). It is worth to remark that we are seeking for
models capable of describing physicochemical properties over a wide range of
flow conditions and we expected to observe abrupt changes around critical
conditions.

Here, we build two different ML models, namely Gaussian Processes (GP)
Rasmussen and Williams (2006) and a probabilistic conditional generative
approach Yang and Perdikaris (2019b). We train both in a supervised learning
fashion using data produced with expensive MD simulations. Therefore, we rely
on their ability to learn from a small amount of data and their capacity of
extrapolation. Moreover, we also want to take into consideration the
unavoidable uncertainties arising from limited information (epistemic) and
from noisy data (aleatoric).

GPs have become popular due to its success on being a proxy for high-fidelity
models in different applications Rasmussen and Williams (2006); Perdikaris et
al. (2017); Su et al. (2017); Chen et al. (2007); Yuan et al. (2008); Guerra
et al. (2019). Another well proved ML approach are the so called generative
models that explore existing low-dimensional structures capable of explaining
high-dimensional data introducing probabilistic latent variables.

In the remainder of this chapter, we present a brief description of both ML
models for a generic property $\gamma(p,T)$ function of pressure and
temperature, along with the corresponding training algorithms. For the
training of the models, we assume the availability of, potentially expensive,
dataset comprising input/output pairs $\\{(p,T)_{i},\mathbf{\gamma}_{i}\ \ \
i=1,...,n\\}$ generated by an implicit mapping $g$ characterizing the
macroscopic thermodynamic relation between the property and the state
variables:

${\gamma}=g({p,T};\boldsymbol{\xi}).$ (1)

The role of $g$ here is played by upscaling MD simulations or, to a less
extent, by experimental available data. The vector $\boldsymbol{\xi}$ denotes
potential noisy and is often considered a random. In order to keep a compact
notation, we refer to the above dataset as
$\mathcal{D}=(\mathbf{x},\mathbf{y})$, with $\mathbf{x}\in\mathbb{R}^{2n}$ and
$\mathbf{y}\in\mathbb{R}^{n}$ vectors containing inputs and outputs. We
intentionally do not use the word surrogate to designate any of the two ML
models to avoid misleadings. In the combustion technical literature, it is
employed to refer to compounds with simpler compositions to replace complex
fuels in experimental or numerical analysis.

###  2.1 Gaussian process regression

A GP is an infinite collection of random variables, in which any finite number
of such variables depict a joint Gaussian distribution Rasmussen and Williams
(2006). In line with Bayesian estimation, to approximate $g$ we assign a GP
zero mean prior $f(\mathbf{x})$ , i.e., $f\sim
GP(f|\mathbf{0},k(\mathbf{x},\mathbf{x}^{{}^{\prime}};\boldsymbol{\theta}))$,
where $k$ is a kernel parametrized by a vector of hyper-parameters
$\boldsymbol{\theta}$ to be learned from $\mathcal{D}$ and engenders a
symmetric positive-definite $n\times n$ covariance matrix
$K_{ij}=k(x_{i},x_{j};\boldsymbol{\theta})$. Instead of choosing the squared
exponential form of the kernel as usual Rasmussen and Williams (2006), here,
we test some forms of covariance matrix belonging to the Matern family. More
specifically, we employ the Mayern 3/2 covariance matrix given as

$k(\mathbf{r})=\sigma^{2}\left(1+\sqrt{6}\frac{|r|}{l}\right)\text{exp}\left(-\sqrt{6}\frac{|r|}{l}\right)$
(2)

with $\mathbf{r}=\mathbf{x}-\mathbf{x}^{{}^{\prime}}$ denoting the distance
between different inputs. The hyper-parameters are the standard deviation
$\sigma$, and the correlation lengths
$\mathbf{l}=\\{l_{1},l_{2},\dots,l_{n_{k}}\\}$, and $n_{k}$ denotes the
dimension of input $\mathbf{r}$. Hence, the hyper-parameters vector reduces to
$\boldsymbol{\theta}=\\{\mathbf{l},\boldsymbol{\sigma}\\}$.

We do not follow a fully Bayesian approach, and obtain the vector of hyper-
parameters $\boldsymbol{\theta}$ by maximizing the marginal log-likelihood of
the model, i.e.

$\text{log}p(\gamma|\mathbf{x},\boldsymbol{\theta})=-\frac{1}{2}\text{log}|\mathbf{K}|-\frac{1}{2}\gamma^{T}\mathbf{K}^{-1}\gamma-\frac{n}{2}\text{log}2\pi.$
(3)

using a conjugate gradient descend method.

The final goal of the regression is obtaining a predictive model for $\gamma$,
which means to compute its value for an untested state $\mathbf{x}_{*}$
Perdikaris et al. (2017)

$\mu_{*}(\mathbf{x}_{*})=k_{*n}\mathbf{K}^{-1}\mathbf{y}$ (4)

and

$\sigma_{*}^{2}(\mathbf{x}_{*})=k_{**}-k_{*n}\mathbf{K}^{-1}k_{*n}^{T}$ (5)

where
$k_{*n}=[k(\mathbf{x}_{*},\mathbf{x}_{1}),\dots,k(\mathbf{x}_{*},\mathbf{x}_{n})]$
and $k_{**}=k(\mathbf{x}_{*},\mathbf{x}_{*})$. The predictions are computed
using the posterior mean $\mu_{*}$, and the uncertainty associated with that
predictions is quantified through the posterior variance $\sigma_{*}^{2}$. It
is worth to mention that in absence of noisy in the training data, the later
represents epistemic uncertainty due to lack of data.

###  2.2 Probabilistic conditional generative model

Now, we explore a probabilistic conditional generative approach Yang and
Perdikaris (2019a, b), that integrates variational auto-encoders (VAE) Kingma
and Welling (2014) and generative adversarial networks (GANs) Goodfellow et
al. (2014). Moreover, it employs a probabilistic perspective that enables to
take into consideration noisy and limited data from the beginning. It is also
capable of dealing with high-dimensionality of inputs and outputs, what is not
explored here due to the specific aspects of our needs.

The final goal is to build probabilistic neural networks that follow a
conditional probability density function $p(\gamma|(p,T),\mathcal{D})$ learnt
from the data. So, the surrogate model can deploy accurate values for the
property $\gamma$ by estimating the expectation
$\mathbb{E}(\gamma|(p,T),\mathcal{D})$, and also, to quantify the uncertainty
associated with that prediction in CFD calculations.

The main ingredient for this approach is the introduction of a vector of
latent random variables aiming at seeking for a hidden low dimensional
structure for explaining the data structure. In a formal abstract perspective,
such latent variables allow us to express the conditional probability
associate to the data $\mathcal{D}$, not included in the expression to keep
the notion clear, $p(\gamma|p,T)$, as an infinite mixture model through

$p(\gamma|p,T)=\int p(\gamma,\mathbf{z}|p,T)\ d\mathbf{z}=\int
p(\gamma|p,T,\mathbf{z})\ p(\mathbf{z}|p,T)\ d\mathbf{z}$ (6)

where $p(\mathbf{z}|p,T)$ is a prior distribution on the latent variables. The
above hierarchical mathematical ansatz, despite being very elegant and
rigorous, has to be approximated Yang and Perdikaris (2019b), where a
regularized adversarial inference framework is proposed and detailed. The
final result is a generator model $\gamma=f_{\phi}(p,T,\mathbf{z})$
parametrized by vector $\phi$, like trained deep neural networks. In
conjunction with $p(\mathbf{z})$, the statistics of $\gamma$ can be
characterized. More specifically, we can compute its low order statistics via
Monte Carlo sampling. It is important to remark that the predictions with the
identified probabilistic generator, that, in present context, plays the role
of a proxy for obtaining macroscopic thermodynamic properties of mixtures for
pressures and temperatures not contained in $\mathcal{D}$, is negligible when
compared to MD simulations. The mean and variance of the predictive
distribution at a new point $(p^{*},T^{*})$ are computed as

$\mu_{\gamma}(p^{*},T^{*})=\mathbb{E}[\gamma|p^{*},T^{*}\mathbf{z}]\approx\frac{1}{N_{s}}\sum_{i=1}^{N_{s}}\left[f_{\phi}(p^{*},T^{*},\mathbf{z}_{i})\right]$
(7)
$\sigma_{\gamma}^{2}(p^{*},T^{*})=\mathbb{V}\text{ar}[\gamma|p^{*},T^{*},\mathbf{z}]\approx\frac{1}{N_{s}}\sum_{i=1}^{N_{s}}\left[f_{\phi}(p^{*},T^{*},\mathbf{z}_{i})-\mu_{\mathbf{y}}(p^{*},T^{*})\right]^{2},$
(8)

where $\mathbf{z}_{i}\sim p(\mathbf{z})$, $i=1,\dots,N_{s}$, and $N_{s}$
corresponds to the total number of samples.

At this point, it is important to clarify that the predictive uncertainty
encoded in $\mathbf{z}$ is due to noise in the Molecular Dynamics computations
originated by numerical approximations and to the potential small amount of
data employed in the training process. Therefore, it encapsulates aleatoric
and epistemic uncertainties.

Later, we explore the versatility of the probabilistic ML model employing the
fusion of data produced by MD with experimental data obtained for
supercritical behavior of the mixture.

###  2.3 Density prediction in EMD simulation

In this study, all MD simulations are performed in Gromacs package Van Der
Spoel et al. (2005) with Transferable Potentials for Phase Equilibria (TraPPE)
force field Martin and Siepmann (1998). United-atom molecular description is
used in order to reduce the computational cost. Before simulation, 500
molecules are distributed in a box with relatively large edge length of 14 nm
to avoid atom’s overlap. After energy minimisation, a 500 ps simulation is
followed in isobaric-isothermal ensemble. The temperature is controlled by
velocity rescale, and the pressure is maintained by using the Parrinello-
Rahman approach Parrinello and Rahman (1981). The time step is set to be 1 fs.
The fixed bond length in TraPPE force field is achieved by using LINCS
algorithm Hess et al. (1997). The density is averaged over the last 300 ps
simulation.

##  3 Results and discussion

Here, we demonstrate the performance of the proposed methodology. Despite
alternative fuels can be very complex mixtures consisting of hundreds of
compounds, we consider single-component alkanes CnH2n+2, so reliable data for
model assessment and validation can be used. In general, realistic fuels are
usually described by surrogate models Chen and Jiang (2019) because of
availability of validated chemical mechanisms and experimental measurements.
The data to train our ML models consist of properties of a family of alkanes,
ranging from normal to supercritical conditions. More specifically, we
construct ML models to characterize density dependency on some operational
conditions in which data is not available. As mentioned before, in order to
take into consideration unavoidable uncertainties, we approximate the
conditional probability $p(\gamma|\mathbf{x},\boldsymbol{\theta})$, with
$\mathbf{x}$ being the input vector with components pressure $p$, temperature
$T$ and chemical composition. Moreover, it is worth mentioning here that we
consider as the input that characterizes the chemical compositions the number
of atoms of carbon $C$ in the fuel molecule, a categorical variable. Also, for
the GP learning model, the hyper-parameters vector reduces to
$\boldsymbol{\theta}=\\{\mathbf{l},\boldsymbol{\sigma}\\}$, and for the
generative model $\boldsymbol{\theta}$ represents a set of latent variables
$z\sim p(z)$. Here, we employ a one-dimensional latent space with a standard
normal prior, $p(z)\sim\mathcal{N}(0,1)$.

The fuels considered here correspond to n-octane, n-nonane, n-decane,
n-dodecane, and n-hexadecane, operating within conditions ranging from high-
pressure nozzle condition to supercritical chamber environment. The dataset
used to build the ML models consists of 1200 density values. Specifically, the
dataset is composed of 240 densities values of each diesel surrogate fuel at a
regular temperature grid $T\in[320,900]$ K, varying by 20K, and at specific
pressures: $p=[3,4,6,8,10,20,100,150]$ MPa. Also, it is worth remarking that
in this dataset we included density values for supercritical regions, more
specifically values above the critical temperature ($T_{c}$) of the
compositions, being the critical values for n-octane ($T_{c}=569.32K$),
n-nonane ($T_{c}=594.55K$), n-decane ($T_{c}=617.7K$), n-dodecane
($T_{c}=658.1K$), and n-hexadecane ($T_{c}=722K$), which replicate engine-like
conditions.

In the learning process, 80% of the data points are selected randomly to
training the ML models. The remaining 20% are used to testing them. Moreover,
the training data set is organized in three subsets with 10%, 50%, and 100% of
data available to train the models. The aim here is to evaluate the
convergence and impacts of constructing the ML models in a small data regime.
Accuracy is measured using the distance between the expected values predicted
with the ML models and the predictions computed with the MD simulations. We
check this accuracy computing the $L_{2}$ mean relative error ($L_{2-MRE}$)

$L_{2-MRE}=\frac{1}{N}\sum_{i=1}^{N}\left(\frac{\rho_{i}-\hat{\rho}_{i}}{\rho_{i}}\right)^{2}$
(9)

where $\rho_{i}$ is the density computed with MD simulations, $\hat{\rho}_{i}$
is the expected ML output and $N$ is the number of test samples. Also, we
compute the coefficient of determination ($R^{2}$-scire) metric Weisberg
(2005)

$R^{2}=1-\frac{\sum_{i=1}^{N}\parallel\rho_{i}-\hat{\rho}_{i}\parallel_{2}^{2}}{\sum_{i=1}^{N}\parallel\rho_{i}-\overline{\rho}\parallel_{2}^{2}}$
(10)

where $\overline{\rho}=\frac{1}{N}\sum_{i=1}^{N}\rho_{i}$ is the mean density
of test samples. The $R^{2}$-score metric represents the normalized error,
allowing the comparison between ML models trained by different data sets, with
values close to 1 corresponding to the ML models best accuracy.

We obtain the GP regression model of Eq. (1) via maximizing the marginal log-
likelihood of Eq. (3) using the Mattern 3/2 kernel function, as that shown in
Eq. (2). Also, we have used the gradient descend optimizer L-BFGS Liu and
Nocedal (1989) using randomized restarts to ensure convergence to a global
optimum. The GP learning model was implemented in GPy: Gaussian Process (GP)
framework written in python GPy (since 2012).

On the other hand, to construct the generative learning model, we departed
from the architecture proposed and validated by Yang and Perdikaris Yang and
Perdikaris (2019b). More specifically, the conditional generative model is
constructed using fully connected feed-forward architectures for the encoder
and generator networks with 4 hidden layers and 100 neurons per layer, while
the discriminator architecture has 2 hidden layers with 100 neurons per layer.
The neural networks are constructed by combining try-and-error and Hyperopt
algorithm Bergstra et al. (2013) to search for the hyperparameters that give
the lowest $L_{2-MRE}$. All activation uses a hyperbolic tangent non-
linearity. The models are trained for 50,000 stochastic gradient descent steps
using the Adam optimizer Kingma and Ba (2017) with a learning rate of
$10^{-4}$, while fixing a two-to-one ratio for the discriminator versus
generator updates. Furthermore, we have also fixed the entropy regularization
parameter to $\lambda=1.5$ and $\beta=0.5$, respectively. The proposed model
was implemented in TensorFlow v2.1.0 Abadi et al. (2015), and computations
were performed in single precision arithmetic on a single NVIDIA GeForce RTX
2060 GPU card.

We also explore some alternatives versions of the above described ML models by
proposing fusion with experimental data and the use of multi-fidelity
formulations.

###  3.1 ML results for typical fuel surrogates

Tables 1 and 2 show the coefficient of determination (R2-score) and $L_{2}$
mean relative error, respectively, for GP and the probabilistic conditional
generative models. The accuracy metrics are computed with the test samples. We
observe that they are not satisfactory in the small training data scenario,
with 10% of training data. R2-scores for the GP and conditional generative
models in this specific training scenario are 0.8538 and 0.9359, respectively.
For a data richer situation, with 50% of training data, we observe that the
models return good predictions with R2-score higher than $0.99$. Also, we
observe that the conditional generative model returns better predictions than
the GP model in a small data scenario, with an accuracy of
$L_{2-MRE}=2.8989\times 10^{-3}$ while the GP accuracy is
$L_{2-MRE}=4.7428\times 10^{-2}$. Finally, with 100% of the training data, we
can see that the surrogate models return excellent predictions with R2-score
very near $1.0$ and mean relative errors lower than 0.03%.

Table 1: Gaussian Process training accuracy. Train data | $L_{2-MRE}$ | R2-score
---|---|---
10 % | $6.2805\times 10^{-2}$ | $0.8538$
50 % | $4.7438\times 10^{-2}$ | $0.9976$
100 % | $2.7272\times 10^{-2}$ | $0.9991$
Table 2: Generative model training accuracy. Train data | $L_{2-MRE}$ | R2-score
---|---|---
10 % | $4.9316\times 10^{-2}$ | $0.9359$
50 % | $2.8989\times 10^{-3}$ | $0.9983$
100 % | $2.1409\times 10^{-3}$ | $0.9990$

As a further illustration of the performance of such approaches to predict the
density, we plot its values for n-octane, n-dodecane, and n-hexadecane
densities with respect to temperature for the ML models trained with 50% of
the dataset, since this training scenario returns the best relation between
accuracy and computational cost. Figure 1 shows the n-octane density
predictions at the pressures equal to 3, 10, and 100 MPa. We can observe that
at 3MPa the GP model fails to deliver good results around the transcritical
region, while the generative model provides robust predictions with
uncertainties bounds that capture the data. The predictive uncertainty of the
proposed approaches reflects limited data for training the models, the
epistemic uncertainty. We can also note that both models perform well at 10
and 100 MPa, wherein the density dependency on the temperature has a smooth
behavior.

(a) 3 MPa

(b) 10 MPa

(c) 100 MPa

Figure 1: n-Octane predictions with the GP (top) and probabilistic conditional
generative models (bottom) at the pressures 3, 10, and 100 MPa.

Also, the n-dodecane and n-hexadecane densities are depicted along with
temperature in Figs. 2 and 3. We observe that the ML models return robust
predictions at three different pressures. Besides, it is noted that the GP
model returns larger uncertainty bounds at high pressures, specifically at
density points not used in the training process.

(a) 3 MPa

(b) 10 MPa

(c) 100 MPa

Figure 2: n-Dodecane predictions with the GP machine learning model (top) and
conditional generative machine learning model (bottom) at the pressures 3, 10,
and 100 MPa.

(a) 3 MPa

(b) 10 MPa

(c) 100 MPa

Figure 3: n-Hexadecane predictions with the GP machine learning model (top)
and conditional generative machine learning model (bottom) at the pressures 3,
10, and 100 MPa.

We also test how the proposed ML technology perform in an extrapolation
scenario. We test them for the n-heptane, a fuel not used for building the
models. In order to do that, instead of employing data provided by ML
computations, we use an experimental database furnished by the National
Institute of Standards and Technology (NIST). Figure 4 shows that at 3 MPa and
liquid condition the ML model returns good predictions of the n-heptane
density behavior, with small uncertainties. However, at supercritical
conditions ($T_{c}=540.13K$), the GP model returns density predictions far
from satisfactory. Also, we note that the generative model has uncertainty
bounds able to capture the thermophysical property. The $L_{2}$ mean relative
error between the NIST dataset and the expected values predicted by the GP and
conditional generative models are $7.1697\times 10^{-2}$ and $2.0838\times
10^{-2}$, respectively. We can also note that at higher pressure where the
density behavior is smooth, the models present better predictions, with the GP
model showing larger uncertainties bounds and the generative model returns
smaller uncertainty bounds. Moreover, the $L_{2}$ mean relative errors of the
GP model at 10 and 100 MPa are respectively $1.8152\times 10^{-4}$ and
$6.3072\times 10^{-4}$, and for the conditional generative model the $L_{2}$
mean relative errors at the same pressures are $8.4484\times 10^{-5}$ and
$2.0322\times 10^{-4}$.

(a) 3 MPa

(b) 10 MPa

(c) 100 MPa

Figure 4: n-Heptane predictions with the GP machine learning model (top) and
conditional generative machine learning model (bottom) at the pressures 3, 10,
and 100 MPa.

Furthermore, we use a coefficient of variation to measure the degree of
uncertainty of the density predictions. It is defined as the ratio between the
standard deviation $\sigma_{\rho}$ and the mean $\mu_{\rho}$ of the prediction

$\text{cv}({p,T})=\frac{\sigma_{\rho}({p,T})}{\mu_{\rho}({p,T})}$ (11)

Figure 5 gives an overall picture by displaying a mapping between the
operating conditions and the uncertainty on n-octane density predictions. We
present an explicit quantification of the epistemic uncertainty resulting from
the lack of data, which helps to understand limits of the ML models. More
specifically, to make more accessible the visualization of the results, we
plot this mapping for $\text{log}_{10}p\in[0.5,2.5]$ MPa and $T\in[320,900]$ K
with regular intervals of 20K, allowing us to make explicit the strong
dependence of the epistemic uncertainties regarding different regions of
operating conditions. A critical aspect to be remarked is the higher values of
cv in particular regions of the operating conditions space, especially at
transcritical conditions displaying higher gradients of the property. We can
note that the GP model returns a degree of uncertainty slightly large in this
region. That can be mitigated by providing more training data for this
specific region. Also, it is noted that variability of density provided by the
conditional generative model is less pronounced at liquid regions and for
high-pressure supercritical regions, which is due to the smooth density
behavior resulting in a low degree of uncertainty in the predictions at these
regions.

(a) GP model

(b) Conditional generative model

Figure 5: n-Octane density variability for a range of temperatures and
pressures.

###  3.2 Data-fusion Machine Learning models

Although MD simulation is considered to be a robust tool to predict
thermodynamic properties, it returns incorrect values at critical
points/transcritical regions. It was shown Chen and Jiang (2019) that the
transport properties predictions of diesel surrogate fuels are far from
satisfactory near such critical points. That is also the case with n-dodecane
in that study. Density predictions with MD simulations and NIST data at
transcritical regions present considerable discrepancies, as shown in Fig. 6.
More specifically, in operating conditions near the critical point of
n-dodecane, critical pressure ($P_{c}=1.8170$ MPa) and critical temperature
($T_{c}=658.1$ K), our ML models based on the MD data fail to accurately
predict the density. Figures 7 (a) and 8 (a) show the density predictions at 2
MPa for GP and conditional generative model, respectively. Moreover, Figures 7
(b) and 8 (b) also show that the main discrepancies between the expected
values of ML models against the NIST database are into the transcritical
regions.

Aiming at improving the predictability of our ML models at transcritical
regions, we adopt two strategies, exploring the fusion of MD simulations with
experimental data. The aim here is not to compare these different strategies
but to evaluate their potential. Both are formulated with the same idea,
promoting the fusion of data from MD simulations and experiments datasets. In
the first one, we propose a data-fusion strategy in which density points of
the transcritical region provided by the NIST database are simply concatenated
into the training dataset. The second differs as we propose a multi-fidelity
arrangement of the data. A detailed description of both strategies is given
further ahead.

Figure 6: Comparison between n-dodecane density predictions along with
temperature at 2 MPa between MD simulations against NIST dataset.

(a) GP model

(b) $L_{2}$ relative error

Figure 7: n-Dodecane density predictions GP model: (a) n-Dodecane density
along with temperature at 2 MPa. (b) $L_{2}$ error between the expected value
predicted by the ML model against NIST.

(a) Conditional generative model

(b) $L_{2}$ relative error

Figure 8: n-Dodecane density predictions conditional generative model: (a)
n-Dodecane density along with temperature at 2 MPa. (b) $L_{2}$ error between
the expected value predicted by the ML model against NIST.

In the data-fusion approach, we add three density values from NIST to the
original training dataset, as depicts in Fig 9 (a). Note that the fusion
improves considerably the predictions of the conditional generative model with
relative errors lower than 5%, while the GP model still returns relative
errors not satisfactory. Further details about this data-fusion approach can
be found in the Appendix A.

(a) 2 MPa

(b) L2 relative error

Figure 9: n-Dodecane density predictions with the GP model (top) and
conditional generative model (bottom) using the data-fusion approach with
three density points from the NIST database.

As discussed above, generating reliable data with MD simulations to be used in
supervised learning might require a great computational effort. To tackle such
a drawback, numerical formulations combining models displaying different
levels of fidelity are frequently employed. Those multi-fidelity simulators
employ, for instance, coarse grid discretizations, models based on simplified
physics, or simplified iterative methods. Here, again we merge experimental
data with MD simulations, restricting our approaches to two levels of
fidelity.

In this new context, we propose extensions of the previous introduced ML
models. We start by obtaining high-fidelity $\\{\mathbf{x}_{H},\gamma_{H}\\}$
and low-fidelity $\\{\mathbf{x}_{L},\gamma_{L}\\}$ input-output samples.
Typically, the number of samples in the first case tends to be much smaller
due to the related costs. We assign the high-fidelity score to the
experimental data, according to the considerations above about the potential
inaccuracy of the MD obtained computed properties for transcritical regions.

We start with our first, in this multi-fidelity context, ML model
approximating the conditional probability
$p(\gamma_{H}|\mathbf{x}_{H},\gamma_{L},z)$, using the generative model
$\gamma_{H}=f_{\phi}(\mathbf{x}_{H},\gamma_{L},z)$, $z\sim p(z)$. In another
words, the ML model is supposed to capture the correlation between the two
level of fidelity data. Once this is achieved, we have a predictive model
computing outputs for a new point $\mathbf{x}^{*}$:
$\mathbf{y_{H}}^{*}=f_{\phi}(\mathbf{x}^{*},f_{L}(\mathbf{x}^{*}),z)$. At this
point, it is worth remarking that one of the inputs is the output of the low-
fidelity model, leading to a recursive scheme to obtain the predictions of the
multifidelity model. In fact, here the considered low fidelity data is
produced with expensive MD simulations. Therefore, in order to achieve a
feasible scheme, we need to build an auxiliary, cheap to compute and accurate,
proxy for the low fidelity model using the available data.

As a second approach, the one based on GPs, we employ the nonlinear
autoregressive multi-fidelity GP (NARGP) regression model Perdikaris et al.
(2017). The main idea of the NARGP model is to extend GP modeling to capture
nonlinear correlations from data generated by sources of different fidelity
Gratiet and Garnier (2014); Kennedy and O’Hagan (2000). It enables the
construction of probabilistic models prone to encapsulate uncertainties, built
upon the recursive relation $y_{H}=g(x_{H},f_{L}(x_{H}))$ involving low and
high fidelity data, in which $f_{L}$ is a GP model for the former. Moreover,
we put a GP prior on $g$. After the training, we obtain the predictive model,
which turns to be also a GP, $y_{H}=g(x^{*},f_{L}(x^{*}))$.

To assess the above multi-fidelity ML approaches, we use an illustrative
example involving data from "low-fidelity" MD simulations and "high-fidelity"
NIST experimental values. For both approaches, the training dataset consists
of 7 density values of n-dodecane $\rho_{H}(p,T_{H})$ and $\rho_{L}(p,T_{L})$,
at the pressure of $2$ MPa and a set of temperatures given by
$T_{H}=T_{L}=\\{320,440,500,620,660,680,700\\}$ K. Note that we prioritize
points located in the transcritical part, since this region presents larger
discrepancies between the values predicted by MD simulations and the NIST
database.

The conditional generative model is constructed using fully connected feed-
forward architecture for the encoder and generator networks with 4 hidden
layers and 100 neurons per layer, while the discriminator architecture has 2
hidden layers with 100 neurons per layer. All activation uses a hyperbolic
tangent non-linearity. The models are trained for 20,000 stochastic gradient
descent steps using the Adam optimizer Kingma and Ba (2017) with a learning
rate of $10^{-4}$, while fixing a one-to-five ratio for the discriminator
versus generator updates. Furthermore, we have fixed the entropy
regularization parameter to $\lambda=1.5$, and we also employed a one-
dimensional latent space with a standard normal prior,
$p(z)\sim\mathcal{N}(0,1)$.

We train the NARGP model via maximizing the marginal log-likelihood using the
Mattern 3/2 kernel function. The gradient descend optimizer L-BFGS is used
considering randomized restarts to ensure convergence to a global optimum.
Once the high-fidelity recursive GP is trained, we can compute the predictive
posterior mean and variance at a given untested point $\mathbf{x}^{*}$ by
sampling the probabilistic predictive model.

The main results are summarized in Fig 10. More specifically, the results
indicate that the NARGP model was able to satisfactorily reconstruct the high-
fidelity data. To make this comparison quantitative, we compute the mean
$L_{2}$ relative error between the expected values predicted by the generative
model and the NIST data. It shows predictions with accuracy of
$L_{2-MRE}=1.4524\times 10^{-2}$. Moreover, it returns good uncertainty bounds
able to capture the high-fidelity response at the transcritical region. Also,
we note a perfect agreement between the expected value provided by the
probabilistic conditional generative model and the high-fidelity data,
resulting in an accuracy of $L_{2-MRE}=4.4782\times 10^{-5}$. Finally, we
observe that the multi-fidelity model returns small uncertainty bounds despite
the small amount of data employed in the training process.

(a) NARGP model

(b) Conditional generative model

Figure 10: Multi-fidelity modeling of n-dodecane diesel surrogate fuel
density.

##  4 Conclusions

In this work, we propose a computational methodology based on the use of ML
with Molecular Dynamics simulations to compute physicochemical properties of
single compound fuels at engine-relevant conditions. The ML models have
revealed to be a powerful tool to accurately predict the fuel properties from
the chemical composition for a wide range of alkanes. Moreover, this study
explores the versatility of the ML models to handle data from different
sources, which can then be integrated efficiently in the context of UQ
workflows with many-query tasks.

We place our contribution in the emerging area of physics-aware ML, where the
final model, in many different ways, blends two main components: availability
of experimental data and/or often expensive computational models relying on
first principles and phenomenological closure equations, and deep learning
data-driven models. Such combination allows describing physicochemical
properties over a wide range of flow conditions at relatively low cost, and
also offers a broad spectrum of opportunities to enhance CFD codes.

This study has shown a successful prediction of physical quantities, in this
case density, that can also be extended to more complex fuel molecules or
multicomponent mixtures like dimethyl ethers or oxymethylene dimethyl ethers.
The generation of reliable physicochemical properties of renewable fuels is an
important step forward towards the generation of digital tools that can assist
on the decarbonization by the use of renewable fuels.

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##  Appendix A Data-fusion studies

In order to enhance the predictability of the ML models at transcritical
regions, here we propose a data-fusion approach. Specifically, we concatenate
density points of the transcritical region provided by the NIST database into
the training dataset. The aim here is to improve the density predictability of
our ML models, by supplying reliable information about this state variable in
the specific region where MD data is scarce. Following this purpose, the first
attempt is to add one density point from the NIST database. Here, we
concatenate the n-dodecane density at pressure 2 MPa and temperature 660 K to
the training data. By adding this point to the training set, it is verified
that the ML models can recover the density at 660 K, as shown in Fig. 11.
However, the $L_{2}$ relative errors between the expected values predicted by
ML models and the NIST data are still considerable in transcritical regions.
Also, we can note that the conditional generative model has larger uncertainty
bounds at the transcritical region trying to recover density behavior due to
the lack of data in this region. Furthermore, Figure 12 shows that adding
density points from NIST into the training data does not change the degree of
uncertainty at other operating conditions.

(a) 2 MPa

(b) $L_{2}$ relative error

Figure 11: n-Dodecane density predictions with the GP model (top) and
conditional generative model (bottom) using the data-fusion-fidelity approach
with one density point from the NIST database.

(a) 10 MPa

(b) 100 MPa

Figure 12: n-Dodecane density predictions with the GP model (top) and
conditional generative model (bottom) using the data-fusion-fidelity approach
with one density point from the NIST database at the pressures 10 and 100 MPa.

As a further attempt to enhance the density predictions at the transcritical
region, we now concatenate one more density point from the NIST database. More
specifically, in addition to concatenating the n-dodecane density at pressure
2 MPa and temperature 660 K to the training data, we also add the n-dodecane
density at 680 K. Figure 13 shows that adding two density points from NIST
data in the transcritical region slightly improves the predictions of the GP
model, while the relative error remains considerable. However, we can verify
that the generative model returns satisfactory predictions with $L_{2}$
relative error lower than 10% in the transcritical region. This shows the
capability of the conditional generative model to enhance the predictability
of the density when some pieces of information about the correct behavior of
the transport property are given to the model.

(a) 2 MPa

(b) L2 relative error

Figure 13: n-Dodecane density predictions with the GP model (top) and
conditional generative model (bottom) using the data-fusion-fidelity approach
with two density points from the NIST database.

Finally, to further increase the predictability of our ML models, a third
attempt is proposed based on adding three density points from NIST to the
training data, those being the n-dodecane densities at 2 MPa and temperatures
660, 680, and 700K. Figure 9 depicts that in this training scenario the
density predictions of the GP model have some improvements, but the $L_{2}$
relative error is still considerable. Furthermore, we can verify that the
conditional generative model returns accurate predictions, with relative
errors lower than 5% in transcritical regions. Finally, we note that the
generative model has uncertainty bounds able to recover the density
predictions near the critical point.