41
N. Roman, P. dos Santos and P. Konzen,
“ANN-MoC Method for Inverse Transient Transport Problems in One-Dimensional Geometry”,
Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.
ANN-MoC Method
for Inverse Transient
Transport Problems
in One-Dimensional
Geometry
ARTICLE HISTORY
Received 5 March 202
Accepted 19 April 2024
Nelson Garcia Roman
Universidade Federal de Rio Grande do Sul (UFRGS)
Porto Alegre, Brazil
ngroman1992@gmail.com
ORCID: 0009-0006-8794-9500
Pedro Costas dos Santos
Universidade Federal de Rio Grande do Sul (UFRGS)
Porto Alegre, Brazil
pedro.costa4137@gmail.com
ORCID: 0009-0001-9927-2860
Pedro Henrique de Almeida Konzen
Universidade Federal de Rio Grande do Sul (UFRGS)
Porto Alegre, Brazil
pedro.konzen@ufrgs.br
ORCID: 0000-0002-0411-1563
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LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024
ANN-MoC Method for Inverse Transient Transport
Problems in One-Dimensional Geometry
Nelson Garcia Roman
Instituto de Matemática e Estatística
(IME)
Universidade Federal de Rio Grande
do Sul (UFRGS)
Porto Alegre, Brazil
ngroman1992@gmail.com
ORCID: 0009-0006-8794-9500
Pedro Costas dos Santos
Instituto de Matemática e Estatística
(IME)
Universidade Federal de Rio Grande
do Sul (UFRGS)
Porto Alegre, Brazil
pedro.costa4137@gmail.com
ORCID: 0009-0001-9927-2860
Pedro Henrique de Almeida Konzen
Instituto de Matemática e Estatística
(IME)
Universidade Federal de Rio Grande
do Sul (UFRGS)
Porto Alegre, Brazil
pedro.konzen@ufrgs.br
ORCID: 0000-0002-0411-1563
Abstract—Transport problems of neutral particles have
important applications in engineering and medical fields, from
safety and quality protocols to optical medical procedures. In this
paper, the ANN-MoC approach is proposed to solve the inverse
transient transport problem of estimating the absorption coefficient
from scalar flux measurements at the boundaries of the model
domain. The central idea is to fit an Artificial Neural Network
(ANN) using samples generated by direct solutions computed by a
Method of Characteristics (MoC) solver. The direct solver
validation is performed on a manufactured solution problem. Two
inverse problems are then presented for testing the ANN-MoC
method. In the first, a homogeneous medium is assumed, and, in the
second, the medium is heterogeneous with a piecewise constant
absorption coefficient. We show that the method can achieve good
estimates, with accuracy depending on that of the direct solver. We
also include a test of sensibility by studying the propagation of noise
on the input data. The results highlight the potential of the proposed
method to be applied to a broader range of inverse transport
problems.
Keywords—artificial neural network, method of characteristics,
particle neutral transport, inverse problem
I. INTRODUCTION
Neutral particle transport problems have many important
applications in engineering and medical fields. The main
fields of radiative heat transfer and neutron transport share the
fundamental model based on the linear Boltzmann equation
[1], [2]. Applications include engineering at high
temperatures, such as glass and ceramic manufactures [3],
combustion chambers [4], solar energy production [5], nuclear
energy production [6], and optical medicine [7], [8]. Related
inverse problem solutions can enhance the development of
safety protocols, quality control procedures, and technological
innovations.
We consider the time-dependent linear Boltzmann
equation with initial and boundary conditions and with
isotropic scattering
()
()
()
()
denotes the radiation intensity at time
at point
, and in the direction
. The average speed of light in the medium
is denoted by . The total absorption coefficient is
denoted by
, while and
are,
respectively, the absorption and scattering coefficients. The
sources are denoted by
in the domain
and
at boundaries. At ,
initial condition
is assumed. The
average scalar flux is defined as
()
Inverse transport problems have been the subject of
important research for many decades. The books of [9] and
[10] discuss the fundamental methods applied to the solution
of inverse problems. Concerning the problems of parameter
estimation, the main approaches consist of estimating
parameters as solutions to an associated minimization
problem. The problem can then be solved by optimization
methods, which usually require a good initial approximation
of the solution. When this is not known, meta-heuristic
algorithms can be applied to this end (see, for instance [11]).
Alternatively, Deep Learning [12] techniques are also applied
[13], [14]. A well-known approach is to fit an Artificial Neural
Network (ANN, [15]) with samples built from solutions to the
associated direct problem.
In this context, we introduce the ANN-MoC approach to
the inverse transport problem of the absorption coefficient
estimation from the scalar flux measured at the boundaries of
the model domain. The core concept is to fit an ANN using
data derived from direct solutions of Eq. (1) computed by a
solver based on the Method of Characteristics (MoC) [16].
The designed methodology is here presented together with
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10.5281/zenodo.12191947
N. Roman, P. dos Santos and P. Konzen,
“ANN-MoC Method for Inverse Transient Transport Problems in One-Dimensional Geometry”,
Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.
selected test cases. After testing the direct solver, two inverse
problems are considered. The first is a transport problem in a
homogeneous medium. In the second, the medium has two
regions with different absorption coefficients.
In the following, the methodology of the MoC direct
solver and the ANN model are presented. Numerical
experiments with the proposed approach are then presented.
They include the selection of ANN architectures, data
preprocessing, and model sensibility tests. Conclusions are
then presented.
II. T
HE ANN-MOC METHOD
The ANN-MoC approach consists of solving the inverse
transport problem by an Artificial Neural Network (ANN)
trained from samples generated by directly solving a set of
transport problems by the Method of Characteristics (MoC).
A. MoC direct solver
The MoC direct solver computes an approximation of Eq.
(1) built with the Discrete Ordinates Method (DOM) [1]
followed by an implicit Euler time discretization [17]. The
raised system of ordinary differential equations is decoupled
by a Source Iteration (SI, [1]) scheme and then, solved with
the Method of Characteristics (MoC, [15]).
Discrete ordinates formulation. The following DOM form
of Eq. (1) is obtained by assuming the Gauss-Legendre
quadrature
, with even
,
()
()
()
()
where the notation
(analogous to the
others) is assumed with
. The scalar flux is
approximated by
()
Time discretization. For the time discretization, it is assumed
that
(see Fig. 1).
The implicit Euler formulation of Eq. (3) gives an iterative
procedure with initialization
()
, and the following steps
()
()
()
where the notation
(analogous to the
others) is assumed with
and
. For the sake of simplicity, in the following the
index will be suppressed, with
denoting
and
(analogous to the others).
Source iteration. The decoupling of system Eq. (6) is
performed with the Source Iteration (SI) technique. From a
given initial scalar flux
, successive approximations
are iteratively computed from
()
()
()
where
()
for
until some given stop
criteria are fulfilled.
Fig. 1. Scheme of the space-time discretization. Points
and intervals
(lines and sets) for directions (blue) and (red)
Method of characteristics. At each time step and each source
iteration, Eq. (7) is solved by the Method of Characteristics
(MoC). First, it is observed that Eq. (7a) can be rewritten as
()
and . Again, for the sake of
simplicity, the index j is suppressed in the following.
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The MoC form of Eq. (9) is obtained by assuming
from where Eq. (9) is rewritten as
()
. This linear first-order differential equation can
now be solved using an integration factor, which gives the
solution from
()
where
and
()
.
One observes that choosing
, Eq. (11) gives the particle
intensity
at each domain
for a given
direction . Analogously, by choosing
, one
obtains the particle intensity point for a given direction .
Direct solver algorithm. Assuming a spatial mesh with
nodes
, and mesh size
, see Fig. 1, the direct solver algorithm can be
summarized as follows:
1. Set time, mesh and quadrature parameters
2. From initial condition, set
()
()
3. (Time loop). For
a. (SI loop) For
a.1. For
and
For
()
a.2. For
and
For
()
a.3. Compute new scalar flux
()
a.4. SI stop criterion
B. ANN inverse model
The inverse problem is solved by fitting a Multilayer
Perceptron network (MLP, [14]) from a data set
generated from computed solutions
of the direct problem for several values of the absorption
coefficient. The MLP of
layers is written as
()
where, in the l-th network layer with
neuron units,
denotes the triple of the activation function,
the bias
-vector, and the weights
-matrix. By
denoting the input
of detector measurements, its
forward propagation through the network layers
is given by
()
and the output is the estimated absorption coefficient
(see Fig. 2).
Fig. 2. Architecture of a MLP neural network with
neurons on each
hidden layer
Basic training algorithm. The basic training algorithm can
be summarized as follows:
1. Set the MLP architecture.
Sets
,
,
, and initial
,
and a global learning
rate
.
2. Loop over epochs
2.a. Forward the training set.
()
2.b. Compute the loss function.
()
2.c. Backward the loss function to compute the gradients
()
2.d. Perform an optimizer gradient based step.
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LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024
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N. Roman, P. dos Santos and P. Konzen,
“ANN-MoC Method for Inverse Transient Transport Problems in One-Dimensional Geometry”,
Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.
(
()
where
, and
is a given learning rate.
The MLPs reported in this paper have been implemented
with the help of the machine learning package PyTorch [18],
and trained with the Adam method [19]. The learning rate has
been set to
.
ANN model test. The test of the trained neural network model
consists of verifying its performance for a new data set
which has not been used for training. The
test data set has also been computed by solving the direct
problem for several values of the absorption coefficient. The
accuracy of the network estimated values
can be
measured by the squared error
and the coefficient of
determination.
C. Data preprocessing
Data preprocessing for deep learning may reduce
generalization errors and reduce the size of the model needed
to fit the training set [12]. There are many available
techniques [20], and we have chosen to work with the
preprocessing now as Standard Scaler. This function
transforms the features to have zero mean and unit standard
deviation. The general formula for the transformation is:
()
where is the original value of the feature, is the
mean and the standard deviation over the data set .
It ensures that features have comparable scales, which is
known to enhance training gradient-based methods.
III. R
ESULTS
Numerical experiments with the proposed ANN-MoC
approach are presented. First, the direct solver validation is
presented on a manufactured solution problem. Two inverse
problems are then discussed. In the first, a homogeneous
medium is assumed, and, in the second, it is considered a two-
region heterogeneous medium.
A. Direct solver test
In order to test the direct solver, we have considered the
manufactured solution
()
By substituting Eq. (24) into Eq. (1.1), the source is found to
be
()
and from the definition of the scalar flux Eq. (2), one has
.
After numerical tests, we have chosen the solver
parameters
, and
as the absolute
-norm tolerance for the SI stopping
criterion. Table I shows a comparison between the direct
solver approximations and the exact scalar flux solutions at
for different absorption coefficients. The relative
-
error is denoted by
and indicates that the chosen
parameters were enough for the direct solver to produce an
accurate solution with
.
TABLE I. COMPARISON BETWEEN THE DIRECT SOLVER APPROXIMATIONS
AND THE EXACT SOLUTION AT
Exac
t
--x--
B. Inverse problem 1 – homogeneous medium
In the inverse problem 1, we assume a homogeneous
medium with a constant absorption coefficient. The problem
consists of estimating from detectors
measurements of the scalar fluxes at
,
and at
time
. Boundary conditions are taken as
, for all , and
, for all . The source
is considered null, and the initial condition is
, and
for all .
The ANN inverse model has the detectors’ measurements
as inputs and outputs the
estimated absorption coefficient . For its training, we have
used the direct solver to build a training set
of
samples (patterns)
with
. The test set
has been generated with
with
uniformly distributed random choices
(see
Fig. 3).
Fig. 3. Inverse problem 1. Training (circles) and test (stars) samples
We have performed several numerical tests to choose an
adequate MLP architecture. Here, we tried architectures
( inputs,
neurons on each
hidden layer,
and output). Training has been stopped when the loss
function
. Due to the stochasticity of the training
method, each test has been repeated three times. Table II
presents the results with the hyperbolic tangent and
the identity as activation functions in the hidden and in the
output layers, respectively. The demanded averaged total
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number of epochs
and computational time
are
tabulated. Table III presents results for similar numerical test,
but with the as activation function in hidden layers.
We observe that, if MLP, with have demanded last
resources to train with small architectures, the in an
MLP was even better.
TABLE II. INVERSE PROBLEM 1. TRAINING TESTS FOR MLP
ARCHITECTURES WITH TANH AS ACTIVATION FUNCTION
To enhance the training, we have then performed trials
with data preprocessing. Inputs of the training samples have
been scaled with the Standard Scaler. Setting the as
activation function in hidden layers, several MLP
architectures have been tested, and the results can be found
in Table IV. The enhancement with preprocessing is notable,
with the providing the best results.
TABLE III. INVERSE PROBLEM 1. TRAINING TESTS FOR MLP
ARCHITECTURES WITH
RELU AS ACTIVATION FUNCTION
/
TABLE IV. INVERSE PROBLEM 1. TRAINING TESTS OF MLP
ARCHITECTURES WITH DATA PREPROCESSING
Following the previous numerical tests, we have chosen
to work with an MLP model (two inputs,
four hidden layers with neurons each, and one output
neuron), the and the identity as activation functions in
the hidden and in the output layers, respectively. With
approximately
, the model reaches a mean squared
error
and coefficient of determination
. The application of the trained model to the
test data gave results with
and
(see Fig. 4).
Fig. 4. Inverse problem 1. Expected versus estimated . Train: circles.
Test: starts. Line fitted to test data results: dashed line
C. Inverse problem 2 – heterogeneous medium
In the inverse problem 2, we assume a heterogeneous
medium with piecewise constant absorption coefficients
()
The inverse problem consists of estimating
from detectors measurements of the scalar fluxes at
,
and at the times
and
. The
initial and boundary conditions, as well as the source, are the
same as for inverse problem 1.
The ANN inverse model has the detector measurements
,
as inputs and outputs the estimated absorption coefficients
and
. For its training, we have used the direct solver to
compute the training set
of
samples (patterns) with
. The test set
has been generated with
uniformly distributed random choices
.
For this inverse problem, we tested MLP architectures
4
( inputs,
neurons in each hidden layer
, and outputs) with Standard Scaler preprocessing the
input data. The training was stopped when the loss function
. Due to the stochasticity of the training method,
each test has been repeated three times. Table V presents the
results with the and the identity as activation functions
in the hidden and in the output layers, respectively. It is
tabulated the required average total number of epochs
and
computational time
. Like the inverse problem 1, the MLP
architecture 4 provided the best results, which
we now set to report the results to follow.
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LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024
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N. Roman, P. dos Santos and P. Konzen,
“ANN-MoC Method for Inverse Transient Transport Problems in One-Dimensional Geometry”,
Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.
TABLE V. INVERSE PROBLEM 2. TRAINING TESTS FOR MLP
ARCHITECTURES WITH DATA PREPROCESSING
With approximately
, the model reaches a mean
squared error
and coefficient of
determination
. The application of the
trained model to the test data gave results with
and
. Figs. 5 and 6 show the expected versus
estimated absorption coefficients for the training and test
samples. The fitted least square line is also shown for the test
data.
Fig. 5. Inverse problem 2. Expected versus estimated
. Train: circles.
Test: starts. Line fitted to test data results: dashed line
Fig. 6. Inverse problem 2. Expected versus estimated
. Train: circles.
Test: starts. Line fitted to test data results: dashed line
Sensitivity test. To validate the robustness and stability of
the proposed MLP model in this problem, a sensitivity test
was applied, which involves adding uniformly distributed
noise into the input data, more specifically in detectors
and
. Table VI shows the results of the mean squared error
and the mean absolute squared error () for different
levels of noise.
The results indicate that the MLP model is relatively
robust to low and moderate levels of noise in the input data.
The noise is propagated to the output by a factor of times.
The
is reached even with a noise level up to .
Figs. 7 and 8 show the expected versus estimated
and
of the test data set with noise levels of , and . In the
figures, the identify line is plotted as a dashed line as a guide.
We observe the absence of outliers, which also indicates a
good generalization of the ANN-MoC method.
TABLE VI. INVERSE PROBLEM 2. SENSITIVITY TESTS
Noise (%)
(%)
Fig. 7. Sensibility test for inverse problem 2. Comparison between expected
versus estimated
for different levels of input data noise
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Fig. 8. Sensibility test for inverse problem 2. Comparison between expected
κ
2
versus estimated
2
for different levels of input data noise
IV. CONCLUSIONS
In this paper, the ANN-MoC approach has been proposed
to solve the inverse transient transport problem of estimating
the absorption coefficient from scalar flux measurements at
the boundaries of the model domain. The central idea is to fit
an Artificial Neural Network (ANN) using samples generated
by direct solutions computed by a Method of Characteristics
(MoC) solver.
Applications of two different inverse transport problems
were reported, one with homogenous medium and the other
two region medium with piecewise constant absorption
coefficient. After several numerical tests, we found that small
MLPs could provide good estimations. Better results were
reached by preprocessing the input data with the Standard
Scaler. A sensitivity test was also reported for the second
problem. The results highlight the potential of the proposed
method to be applied to a broader range of inverse transport
problems.
Further developments should aim to improve the direct
solver. Improvements in the solution accuracy and, primarily,
in computational performance are important to provide the
ANN model with a higher-quality dataset. Solutions to more
complex inverse transport problems could also benefit from
the proposed approach, but once again, it will require
additional improvements in the direct solver. Finally, the use
of the proposed methodology for realistic problems depends
on how good the direct transport model is for the intended
application.
A
CKNOWLEDGMENT
The authors thank the Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior – Brasil
(CAPES) for partially financing this research (Finance Code
001).
R
EFERENCES
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N. Roman, P. dos Santos and P. Konzen,
“ANN-MoC Method for Inverse Transient Transport Problems in One-Dimensional Geometry”,
Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.
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AUTHORS
Nelson García Román was born on September 20, 1992, in Pinar del
Río, Cuba. He graduated as a Mechanical Engineer from the University
of Pinar del Rio (2011-2016). During his studies, he was involved as
a student assistant in Calculus and received two Honorable Mention
awards for the presentation of two papers on Applied Mathematics
in Engineering at the Scientific Conferences. After graduation, he
became a professor at the same university from 2016 to 2018, and
subsequently as a mathematics professor at the José Antonio
Echeverría Technological University (CUJAE) until 2022. Currently,
he is pursuing a Master's degree in the Graduate Program in Applied
Mathematics (PPGMAp) at the Federal University of Rio Grande do
Sul (UFRGS), where he holds a scholarship from CAPES, focusing on
numerical methods, computational modeling, and deep learning for
the numerical solution of particle neutral transport inverse problems.
Pedro Costa dos Santos was born on September 23, 1998, in Rio de
Janeiro - RJ, Brazil. During his secondary education, he successively
received Honorable Mentions in the Brazilian Public School Mathematics
Olympiad (OBMEP), and in 2015, he was awarded the Silver Medal in
the competition. From 2016 to 2018, he attended an undergraduate
course in Industrial Chemistry at the Federal University of Rio Grande
do Sul(UFRGS). Since 2018, he has been a student in the undergraduate
course of Applied Mathematics at UFRGS. Since 2022, he has been
granted a scholarship for research initiation in Applied Mathematics,
with an aim on Deep Learning applications to the numerical solution
of Inverse Particle Neutral Transport problems. In 2023, he received
the Honorable Mention for his research developments presented in
the Scientific Initiation Week (SIC) at UFRGS.
Pedro Henrique de Almeida Konzen was born on June 12, 1981, in Santa
Cruz do Sul - RS, Brazil. Doctor in Applied Mathematics from the Federal
University of Rio Grande do Sul (UFRGS, 2010), having conducted
doctoral research at Ruprecht-Karls-Universität Heidelberg/Germany
(Uni-HD, 2008-2010). Currently, Adjunct Professor at the Department
of Pure and Applied Mathematics (DMPA), Institute of Mathematics
and Statistics (IME), Federal University of Rio Grande do Sul (UFRGS,
since 2014). Permanent member of the Graduate Program in Applied
Mathematics (PPGMAp-UFRGS, since 2022). Has experience in the
field of applied mathematics, with emphasis on numerical methods,
computational simulation, mathematical modeling and deep learning.
Nelson Garcia Roman
Pedro Costas dos Santos
Pedro Henrique de Almeida Konzen
N. Roman, P. dos Santos and P. Konzen,
“ANN-MoC Method for Inverse Transient Transport Problems in One-Dimensional Geometry”,
Latin-American Journal of Computing (LAJC), vol. 11, no. 2, 2024.
