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Citation: Tuya, D.; Nagaya, Y.
Estimation of Continuous
Distribution of Iterated Fission
Probability Using an Artificial Neural
Network with Monte Carlo-Based
Training Data. J. Nucl. Eng. 2023,4,
691–710. https://doi.org/10.3390/
jne4040043
Academic Editor: Shichang Liu
Received: 27 July 2023
Revised: 17 October 2023
Accepted: 19 October 2023
Published: 6 November 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
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4.0/).
Article
Estimation of Continuous Distribution of Iterated Fission
Probability Using an Artificial Neural Network with Monte
Carlo-Based Training Data
Delgersaikhan Tuya * and Yasunobu Nagaya
Nuclear Science and Engineering Center, Japan Atomic Energy Agency, Tokai-mura, Naka-gun,
Ibaraki 319-1195, Japan; nagaya.yasunobu@jaea.go.jp
*Correspondence: tuya.delgersaikhan@jaea.go.jp
Abstract:
The Monte Carlo neutron transport method is used to accurately estimate various quantities,
such as k-eigenvalue and integral neutron flux. However, in the case of estimating a distribution
of a desired quantity, the Monte Carlo method does not typically provide continuous distribution.
Recently, the functional expansion tally (FET) and kernel density estimation (KDE) methods have
been developed to provide a continuous distribution of a Monte Carlo tally. In this paper, we propose
a method to estimate a continuous distribution of a quantity in all phase-space variables using a fully
connected feedforward artificial neural network (ANN) model with Monte Carlo-based training data.
As a proof of concept, a continuous distribution of iterated fission probability (IFP) was estimated
by ANN models in two distinct fissile systems. The ANN models were trained on the training data
created using the Monte Carlo IFP method. The estimated IFP distributions by the ANN models
were compared with the Monte Carlo-based data that include the training data. Additionally, the
IFP distributions by the ANN models were also compared with the adjoint angular neutron flux
distributions obtained with the deterministic neutron transport code PARTISN. The comparisons
showed varying degrees of agreement or discrepancy; however, it was observed that the ANN models
learned the general trend of the IFP distributions from the Monte Carlo-based training data.
Keywords:
iterated fission probability; continuous distribution; Monte Carlo neutron transport;
artificial neural network; machine learning
1. Introduction
The Monte Carlo neutron transport method offers an accurate estimation of various
quantities of interest such as k-eigenvalue and neutron flux or reaction rate integrated
over a phase-space region. In order to estimate a distribution of a desired quantity (e.g.,
neutron flux and reaction rate) over the phase-space domain of a problem, one mostly
needs to divide the phase-space domain into a number of bins (e.g., mesh) and tally the
quantity in each bin, since the Monte Carlo method does not typically provide a continuous
distribution. With this method, it becomes challenging to obtain the desired distribution
over a finer bin structure because the uncertainty of the tally in a small bin becomes large
due to a smaller number of events contributing to the bin. On the other hand, a few
methods to estimate a continuous distribution of a desired quantity over the phase-space
domain of a problem have been studied. In particular, kernel density estimation (KDE) and
functional expansion tally (FET) methods have been developed and applied to estimate the
continuous spatial distribution of neutron flux [
1
–
5
]. Both methods provide an estimation
of a continuous distribution and corresponding uncertainty of a tally during a Monte Carlo
calculation. In particular, the FET method has recently been used for estimating spatial
distributions of neutron flux and power density [6–8].
However, the FET and KDE methods have been applied to one- to three-dimensional
spatial distribution problems; application of the KDE and FET methods to problems of
J. Nucl. Eng. 2023,4, 691–710. https://doi.org/10.3390/jne4040043 https://www.mdpi.com/journal/jne
J. Nucl. Eng. 2023,4692
estimating a continuous distribution of a Monte Carlo tally in all space, energy, and
direction variables is scarce or has not yet appeared in the literature, to the best of the
authors’ knowledge. Consequently, the estimation of a continuous distribution of a variable
in all phase-space variables still remains a challenging problem. This motivated the authors
to explore an alternative method for estimating a continuous distribution of variables in
space, energy, and direction.
In this paper, we propose a fully connected feedforward artificial neural network
(ANN) model-based method for estimating a continuous distribution of a quantity in space,
energy, and direction from Monte Carlo-based training data. An ANN model learns, albeit
approximately, a continuous distribution from discrete data via supervised learning. As
a proof of concept, we consider the estimation of a continuous distribution of iterated
fission probability (IFP), which is a quantity proportional to adjoint angular neutron flux,
via an ANN model in a given fissile system from Monte Carlo-based training data. The
IFP was selected in this study because the algorithm to calculate IFP is more suitable for
creating data for training an ANN model in a given system; the algorithm provides an
estimation of IFP at a desired phase-space location. The proposed ANN model-based
method estimates a continuous distribution from data after a Monte Carlo calculation is
finished, in contrast to the KDE and FET methods, which can estimate the continuous
distribution during the Monte Carlo calculation. Furthermore, at this preliminary stage,
the proposed ANN model-based method does not provide an estimation of uncertainty on
an obtained continuous distribution.
The IFP denoted as
R(r,E,Ω)
at phase-space location
P≡(r,E,Ω)
represents an
asymptotic population due to a source neutron introduced at that location in a given
system. It was shown [
9
–
11
] that IFP is proportional to a fundamental mode of adjoint
angular neutron flux. Since IFP distribution is governed by an integral equation that cannot
be solved directly in general, it is estimated via a Monte Carlo neutron transport method.
The methods to estimate IFP in a forward Monte Carlo eigenvalue neutron transport
calculation have been developed [
9
,
10
], implemented in Monte Carlo neutron transport
codes, and used for various applications [
12
–
22
]. Although unknown, there must be a
function describing the
R(r,E,Ω)
distribution in a given system. Let us assume that the
unknown function is
ft
such that
R=ft(r,E,Ω)
in a given system, which is characterized
by its geometry, material composition, temperature, etc. The function
ft
maps from phase-
space location
P
to its corresponding IFP value
R
, that is,
ft:R7→R
. This unknown
function can be learned, at least approximately, by an ANN model, which is trained via
supervised learning on data created by the Monte Carlo method. Once an ANN model is
trained successfully on the data in a given system, it can then be used to estimate an IFP at
any phase-space location in the system, i.e., it can provide a continuous IFP distribution in
the system.
In the previous study [
23
], we showed the preliminary approximation of the IFP
distribution via an ANN model for the first time using the Godiva system. In that study,
the data were mostly produced according to the fission neutron distribution in the system.
In other words, the spatial position was sampled from fission sites in the system, energy
was sampled from the fission neutron spectrum, and direction was sampled from isotropic
distribution. With these sampling distributions, a majority of the data was around the peak
of the fission neutron spectrum, reducing the amount of data for other energy regions. It
was shown that although the ANN model learned the IFP distribution approximately, it
failed to learn the dependence of resonance-like peaking of the IFP distribution on the
energy variable. In contrast to the previous study, we consider applying ANN models for
estimating continuous IFP distributions in two distinct fissile systems in this study: the fast
spectrum Godiva core and the thermal spectrum simplified STACY core. Furthermore, in
the current study, the data are produced via arbitrary sampling distributions intended for
enhancing the training of ANN models.
The purpose of this paper is to show the proof of concept of the ANN model-based
method for estimating a continuous distribution of a quantity from Monte Carlo-based
J. Nucl. Eng. 2023,4693
training data. To this end, an estimation of a continuous distribution of IFP via ANN
models in two distinct fissile systems in terms of neutron spectrum is considered. The
paper is structured as follows. In Section 2, the Monte Carlo method for estimating an
IFP at a given phase-space location is described. The procedure to create discrete data
is given in Section 3for the Godiva and simplified STACY cores. Section 4presents the
considered ANN models, their training, and estimated IFP distributions by the ANN
models. The estimated distributions by the selected ANN models are compared to the
Monte Carlo-based data that include the training data in Section 4. Furthermore, the
comparisons between the estimated IFP distributions by the selected ANN models and
the adjoint angular neutron flux distributions by the deterministic neutron transport code
PARTISN are provided. The concluding remarks are given in Section 5.
2. Iterated Fission Probability Method
In this section, the estimation of an IFP at a given phase-space location in a given
system using the Monte Carlo neutron transport method, which is based on the IFP
calculation method by Kiedrowski et al. [
9
], is presented. The theoretical discussion of the
IFP method and the proportionality of IFP to adjoint angular neutron flux can be found in
the references [9–11].
In the IFP method by Kiedrowski et al., entire active cycles (generations) of a forward
Monte Carlo eigenvalue calculation are divided into non-overlapping uniform intervals
called blocks. Each block consists of an original generation, latent generation(s), and an
asymptotic generation. The original generation is located at the beginning of the block and
is followed by Llatent generation(s). The last generation in the block is the asymptotic
generation. In each block, the IFP for each progenitor neutron in the original generation
is estimated as a total fission neutron production by all of its progeny neutrons in the
asymptotic generation. A sufficient number of latent generations allows the progenies of
the progenitor neutron to converge to their asymptotic population.
Kiedrowski’s method is generally designed for estimating adjoint-weighted quantities
such as kinetic parameters and sensitivity coefficients in steady-state fissile systems; hence,
it is implemented for forward Monte Carlo eigenvalue calculations. Consequently, phase-
space locations of source (progenitor) neutrons are sampled from fission sites: energy is
sampled from a fission neutron spectrum, the spatial position is set to the fission site, and
direction is sampled from isotropic distribution. On the contrary, in this study, we are
interested only in the estimation of IFP itself; a forward Monte Carlo eigenvalue problem
is not needed, and source neutrons can be sampled arbitrarily as in a fixed-source Monte
Carlo calculation.
Figure 1shows the general procedure, which is based on Kiedrowski’s method, for
estimating an IFP at a given phase-space location in this study with an example. A source
neutron (equivalent to a progenitor neutron in Kiedrowski’s method) is introduced at the
phase-space location
(r,E,Ω)
and is tracked in the source generation. Fission neutrons
(progeny neutrons in Kiedrowski’s method) caused by the source neutron are then tracked
over the next Llatent generations. Eventually, an IFP is estimated as a total fission neutron
production by all progeny neutrons in the asymptotic generation as below using the track-
length estimator.
R(r,E,Ω)=∑K
k=1∑Mk
m=1rk,m=∑K
k=1∑Mk
m=1νΣ fwk,mlk,m, (1)
where
k
denotes a progeny neutron index in the asymptotic generation,
m
denotes the
m-th collision of a progeny neutron,
K
is the total number of progeny neutrons in the
asymptotic generation,
Mk
is the total number of interactions (collision or surface crossing)
by a progeny neutron
k
,
νΣ f
is fission neutron production at an interaction (collision or
surface crossing) site, and
wk,m
and
lk,m
are the weight and track length of progeny neutron
kat its m-th interaction, respectively.
J. Nucl. Eng. 2023,4694
J. Nucl. Eng. 2023, 4, 4
or surface crossing) site, and 𝑤, and 𝑙, are the weight and track length of progeny
neutron 𝑘 at its 𝑚-th interaction, respectively.
Figure 1. Monte Carlo method for estimating IFP at phase-space location 𝒓,𝐸,𝜴.
The procedure described in Equation (1) and Figure 1 estimates an IFP for one ran-
dom sample at a given phase-space location. For the purpose of training ANN models
later in this paper, we calculate the mean IFP at a given phase-space location. A source
neutron at phase-space location 𝒓,𝐸,𝛀 is cloned 𝑁 times and each clone is given a
unique initial random number to estimate the mean IFP as
𝑅
𝒓,𝐸,𝛀=
∑𝑅𝒓,𝐸,𝛀|𝜉
, (2)
where 𝜉 is the initial random number for the 𝑖-th clone.
The Monte Carlo IFP method described in this section has been newly implemented
in continuous-energy Monte Carlo neutron transport solver Solomon [24]. In the rest of
this paper, the Monte Carlo IFP method described in this section and the Solomon solver
are used interchangeably.
3. Creation of Monte Carlo-Based Data
3.1. General Approach
In order to create data for a given system, a phase-space domain of the system needs
to be sampled according to some sampling distributions. In general, all variables (position,
energy, and direction) can be sampled from a uniform distribution in the domain. Typi-
cally, the geometrical domain of a fissile system is limited, i.e., on the order of 10
0
m, and
the angular domain is bounded in [−1.0, 1.0] for each component of the direction. On the
other hand, the energy domain spans over the range [0.0, 2 × 10
7
] eV. If energy is sampled
from a uniform distribution in the entire range, a majority of the data then will consist of
samples with high energy. Alternatively, if energy is sampled uniformly on a logarithmic
scale in the entire range, then a majority of the data consists of neutrons with energy less
than 1.0 eV. In order to include samples from various energy regions more equally, we
consider dividing the entire range into G intervals and sampling from a uniform distribu-
tion in each interval as illustrated in Figure 2. Furthermore, an arbitrary number of sam-
ples can be drawn in each interval. This division facilitates a more flexible sampling and
improves the overall sampling of various energy ranges; however, the boundaries of each
Figure 1. Monte Carlo method for estimating IFP at phase-space location (r,E,Ω).
The procedure described in Equation (1) and Figure 1estimates an IFP for one random
sample at a given phase-space location. For the purpose of training ANN models later in
this paper, we calculate the mean IFP at a given phase-space location. A source neutron at
phase-space location
(r,E,Ω)
is cloned
Nc
times and each clone is given a unique initial
random number to estimate the mean IFP as
R(r,E,Ω)=1
Nc∑Nc
i=1R(r,E,Ω|ξi), (2)
where ξiis the initial random number for the i-th clone.
The Monte Carlo IFP method described in this section has been newly implemented
in continuous-energy Monte Carlo neutron transport solver Solomon [
24
]. In the rest of
this paper, the Monte Carlo IFP method described in this section and the Solomon solver
are used interchangeably.
3. Creation of Monte Carlo-Based Data
3.1. General Approach
In order to create data for a given system, a phase-space domain of the system needs
to be sampled according to some sampling distributions. In general, all variables (position,
energy, and direction) can be sampled from a uniform distribution in the domain. Typically,
the geometrical domain of a fissile system is limited, i.e., on the order of 10
0
m, and the
angular domain is bounded in [
−
1.0, 1.0] for each component of the direction. On the other
hand, the energy domain spans over the range [0.0, 2
×
10
7
] eV. If energy is sampled from a
uniform distribution in the entire range, a majority of the data then will consist of samples
with high energy. Alternatively, if energy is sampled uniformly on a logarithmic scale in the
entire range, then a majority of the data consists of neutrons with energy less than 1.0 eV. In
order to include samples from various energy regions more equally, we consider dividing
the entire range into G intervals and sampling from a uniform distribution in each interval
as illustrated in Figure 2. Furthermore, an arbitrary number of samples can be drawn in
each interval. This division facilitates a more flexible sampling and improves the overall
sampling of various energy ranges; however, the boundaries of each energy interval need
to be decided. In the next two sections, we will introduce such energy structures for the
Godiva and simplified STACY cores. It should be noted that an energy structure is only
needed for creating the data containing discrete samples; an IFP distribution to be learned
J. Nucl. Eng. 2023,4695
by an ANN model from the data is continuous and generally independent of the energy
structure provided that the training is successful.
J. Nucl. Eng. 2023, 4, 5
energy interval need to be decided. In the next two sections, we will introduce such energy
structures for the Godiva and simplified STACY cores. It should be noted that an energy
structure is only needed for creating the data containing discrete samples; an IFP distri-
bution to be learned by an ANN model from the data is continuous and generally inde-
pendent of the energy structure provided that the training is successful.
Figure 2. Energy structure for energy sampling.
With the described sampling procedure, the spatial position, energy, and direction of
each source neutron are sampled randomly from uniform distributions in each energy inter-
val. An IFP corresponding to a given source neutron is then estimated via the Solomon solver
as discussed in Section 2. Data in a given system then can be created by storing the phase-
space location (input) of a source neutron and the estimated IFP (output) as illustrated in Fig-
ure 3. An ANN model can be trained to estimate a continuous distribution of IFP via super-
vised learning from such data containing discrete phase-space locations and IFP pairs.
It should be noted that the phase-space location sampling procedure described in this
section (and in Sections 3.2 and 3.3) is only necessary for the estimation of IFP distribution.
In the case of other quantities such as forward neutron flux and reaction rates, phase-space
locations would be sampled according to collision and surface crossing events during a
Monte Carlo calculation. Data then would be created by storing phase-space locations of
the events and corresponding estimations of a quantity.
Figure 3. Monte Carlo-based data creation procedure.
3.2. Data in Godiva Core
The Godiva [25] is a bare spherical core of highly enriched metallic uranium with
about 94.73 wt.%
235
U. Its radius is 8.741 cm and its physical density is 18.74 g/cm
3
. The
atomic number densities of
234
U,
235
U, and
238
U are 4.9184 × 10
−4
, 4.4994 × 10
−2
, and 2.4984 ×
10
−3
atoms × barn
−1
× cm
−1
, respectively.
The phase-space domain of the Godiva core is bounded in [−8.741, 8.741] cm for space,
[−1.0, 1.0] for angular component, and [0.0, 2.0 × 10
7
] eV for energy. As described in the previ-
ous section, we divide the energy range into a number of intervals to improve the sampling.
Although the energy structure of such division can be decided arbitrarily, we used the 73-
group energy structure from the SLAROM-UF code [26] in order to make a comparison of an
estimated IFP distribution by an ANN model and adjoint angular neutron flux distribution by
a deterministic code, in which the same energy group structure is used, simpler. The 73-group
energy structure is the smallest fine group structure provided by the SLAROM-UF code. The
Figure 2. Energy structure for energy sampling.
With the described sampling procedure, the spatial position, energy, and direction
of each source neutron are sampled randomly from uniform distributions in each energy
interval. An IFP corresponding to a given source neutron is then estimated via the Solomon
solver as discussed in Section 2. Data in a given system then can be created by storing
the phase-space location (input) of a source neutron and the estimated IFP (output) as
illustrated in Figure 3. An ANN model can be trained to estimate a continuous distribution
of IFP via supervised learning from such data containing discrete phase-space locations
and IFP pairs.
It should be noted that the phase-space location sampling procedure described in this
section (and in Sections 3.2 and 3.3) is only necessary for the estimation of IFP distribution.
In the case of other quantities such as forward neutron flux and reaction rates, phase-space
locations would be sampled according to collision and surface crossing events during a
Monte Carlo calculation. Data then would be created by storing phase-space locations of
the events and corresponding estimations of a quantity.
J. Nucl. Eng. 2023, 4, 5
energy interval need to be decided. In the next two sections, we will introduce such energy
structures for the Godiva and simplified STACY cores. It should be noted that an energy
structure is only needed for creating the data containing discrete samples; an IFP distri-
bution to be learned by an ANN model from the data is continuous and generally inde-
pendent of the energy structure provided that the training is successful.
Figure 2. Energy structure for energy sampling.
With the described sampling procedure, the spatial position, energy, and direction of
each source neutron are sampled randomly from uniform distributions in each energy inter-
val. An IFP corresponding to a given source neutron is then estimated via the Solomon solver
as discussed in Section 2. Data in a given system then can be created by storing the phase-
space location (input) of a source neutron and the estimated IFP (output) as illustrated in Fig-
ure 3. An ANN model can be trained to estimate a continuous distribution of IFP via super-
vised learning from such data containing discrete phase-space locations and IFP pairs.
It should be noted that the phase-space location sampling procedure described in this
section (and in Sections 3.2 and 3.3) is only necessary for the estimation of IFP distribution.
In the case of other quantities such as forward neutron flux and reaction rates, phase-space
locations would be sampled according to collision and surface crossing events during a
Monte Carlo calculation. Data then would be created by storing phase-space locations of
the events and corresponding estimations of a quantity.
Figure 3. Monte Carlo-based data creation procedure.
3.2. Data in Godiva Core
The Godiva [25] is a bare spherical core of highly enriched metallic uranium with
about 94.73 wt.%
235
U. Its radius is 8.741 cm and its physical density is 18.74 g/cm
3
. The
atomic number densities of
234
U,
235
U, and
238
U are 4.9184 × 10
−4
, 4.4994 × 10
−2
, and 2.4984 ×
10
−3
atoms × barn
−1
× cm
−1
, respectively.
The phase-space domain of the Godiva core is bounded in [−8.741, 8.741] cm for space,
[−1.0, 1.0] for angular component, and [0.0, 2.0 × 10
7
] eV for energy. As described in the previ-
ous section, we divide the energy range into a number of intervals to improve the sampling.
Although the energy structure of such division can be decided arbitrarily, we used the 73-
group energy structure from the SLAROM-UF code [26] in order to make a comparison of an
estimated IFP distribution by an ANN model and adjoint angular neutron flux distribution by
a deterministic code, in which the same energy group structure is used, simpler. The 73-group
energy structure is the smallest fine group structure provided by the SLAROM-UF code. The
Figure 3. Monte Carlo-based data creation procedure.
3.2. Data in Godiva Core
The Godiva [
25
] is a bare spherical core of highly enriched metallic uranium with
about 94.73 wt.%
235
U. Its radius is 8.741 cm and its physical density is 18.74 g/cm
3
. The
atomic number densities of
234
U,
235
U, and
238
U are 4.9184
×
10
−4
, 4.4994
×
10
−2
, and
2.4984 ×10−3atoms ×barn−1×cm−1, respectively.
The phase-space domain of the Godiva core is bounded in [
−
8.741, 8.741] cm for space,
[
−
1.0, 1.0] for angular component, and [0.0, 2.0
×
10
7
] eV for energy. As described in the
previous section, we divide the energy range into a number of intervals to improve the
sampling. Although the energy structure of such division can be decided arbitrarily, we
used the 73-group energy structure from the SLAROM-UF code [
26
] in order to make a
comparison of an estimated IFP distribution by an ANN model and adjoint angular neutron
flux distribution by a deterministic code, in which the same energy group structure is used,
simpler. The 73-group energy structure is the smallest fine group structure provided by
the SLAROM-UF code. The number of clones was set to 100. In each energy interval,
8×104samples
were created using the IFP method in the Solomon solver, giving a total of
5.84
×
10
6
samples in the data. The number of latent generations in the IFP method was set
to 10, a value recommended for most problems.
J. Nucl. Eng. 2023,4696
3.3. Data in Simplified STACY Core
The simplified STACY [
27
] is an infinite cylindrical core of enriched uranyl nitrate
solution surrounded by a light water reflector. The radius of the inner infinite cylinder
containing the uranyl nitrate solution is 22.0 cm and the outer radius is 23.0 cm, giving the
light water reflector a thickness of 1.0 cm. Table 1lists the atomic number densities of the
constituent nuclides.
Table 1. Atomic number densities of simplified STACY core.
Material Nuclide Atomic Density
(atoms ×barn−1×cm−1)
Fuel solution
235U 7.92122 ×10−5
238U 7.06258 ×10−4
1H 5.69525 ×10−2
14N 2.87772 ×10−3
16O 3.80270 ×10−2
Light water reflector * 1H 6.66566 ×10−2
16O 3.33283 ×10−2
* Free gas treatment is used for 1H in Solomon.
The data in the simplified STACY core are created similarly to those in the case of
the Godiva core. The phase-space domain of the simplified STACY core is bounded in
[−23.0, 23.0] cm
for X- and Y-axis, [
−∞
,
∞
] for Z-axis, [
−
1.0, 1.0] for angular component,
and [0.0, 2.0
×
10
7
] eV in energy. The range in the Z-axis is sampled in [
−
50.0, 50.0];
however, this does not affect the IFP distribution at all as it does not vary in the Z-axis.
The energy is sampled from the uniform distribution in each energy interval as described
in Section 3.1. In this study, we used the 107-group energy structure from the SRAC
code [
28
] in order to make a comparison with adjoint angular neutron flux distribution
by a deterministic code later, in which the same energy structure is used. The 107-group
structure is the only energy structure provided by the SRAC code. Furthermore, it should
be noted that the 107-group energy structure is bounded in [0.0, 1.0
×
10
7
] eV, meaning that
with the created data, the continuous distribution of IFP in this range is supposed to be
estimated by ANN models. The number of clones was set to 100. In total, the data contain
2.14
×
10
6
samples created via the IFP method in Solomon; in other words,
2×104samples
were drawn in each energy interval. The number of latent generations in the IFP method
was set to 10.
4. Estimation of Continuous IFP Distribution via ANN Model
4.1. Descriptions of ANN Models and Training
A fully connected feedforward artificial neural network [
29
] (ANN) with a single
real-valued output is considered in this work. Figure 4a shows such an ANN. It consists
of an input layer,
d−
1 layers called hidden layers, and an output layer. The number of
neurons in each hidden layer is a model hyperparameter that needs to be decided arbitrarily
or via hyperparameter search. The number of possibilities of making ANN models with a
different number of neurons in each hidden layer grows exponentially with the depth
d
. In
order to simplify the hyperparameter search, we consider ANNs in which the number of
neurons in each hidden layer is the same, as shown in Figure 4b, where
n
is the number of
neurons in each hidden layer and is referred to as the width of the network. Consequently,
only the optimal values of width and depth need to be searched for.
J. Nucl. Eng. 2023,4697
J. Nucl. Eng. 2023, 4, 7
Figure 4. Fully connected feedforward ANN.
In order to find the optimal depth and width of the network for each system, we
performed a random search [30]. Ten combinations of width and depth were sampled
randomly from the ranges (7, 42) for width and (5, 21) for depth; these ranges were partly
identified as suitable in the previous study [23]. The sampled combinations are given in
Table 2. Consequently, ten ANN models were created to be trained for each system. The
rectified linear unit (ReLU) function is used as an activation function for each hidden layer.
In addition to the model hyperparameters, there are algorithmic hyperparameters
related to the training of an ANN model such as a learning rate and batch size (batch size
is the number of training samples used for updating model parameters). In this work, we
set the learning rate to 0.001, which is common in practice. The batch size was set to about
1% of the total number of samples in the data.
Table 2. Combinations of width and depth resulted from random search.
ANN Model Identification Width, 𝒏 Depth, 𝒅
ANN-1 41 12
ANN-2 30 14
ANN-3 24 8
ANN-4 18 17
ANN-5 13 20
ANN-6 23 9
ANN-7 20 7
ANN-8 8 5
ANN-9 19 6
ANN-10 17 20
To train an ANN model, a measure of the performance of the ANN model training is
needed for optimization purposes. Such a measure is called a cost function. As the current
task of estimating an IFP distribution via an ANN model is a non-linear regression task, a
commonly used cost function named Mean Squared Error (MSE), defined in Equation (3),
is used in this work.
𝐽𝜽≡𝑀𝑆𝐸=
∑𝑅
−𝑅
=
∑𝑅
−
𝑓
𝒓𝒊,𝐸,𝜴𝒊|𝜽
, (3)
where 𝜽 is the parameters of the ANN model, 𝑅
is the IFP corresponding to 𝑖-th input
in the data (i.e., the IFP calculated by the Solomon solver via Equation (2)), 𝑅
is the IFP
estimation by the ANN model for the 𝑖-th input, and 𝑁 is the number of samples in the
Figure 4. Fully connected feedforward ANN.
In order to find the optimal depth and width of the network for each system, we
performed a random search [
30
]. Ten combinations of width and depth were sampled
randomly from the ranges (7, 42) for width and (5, 21) for depth; these ranges were partly
identified as suitable in the previous study [
23
]. The sampled combinations are given in
Table 2. Consequently, ten ANN models were created to be trained for each system. The
rectified linear unit (ReLU) function is used as an activation function for each hidden layer.
Table 2. Combinations of width and depth resulted from random search.
ANN Model Identification Width, nDepth, d
ANN-1 41 12
ANN-2 30 14
ANN-3 24 8
ANN-4 18 17
ANN-5 13 20
ANN-6 23 9
ANN-7 20 7
ANN-8 8 5
ANN-9 19 6
ANN-10 17 20
In addition to the model hyperparameters, there are algorithmic hyperparameters
related to the training of an ANN model such as a learning rate and batch size (batch size
is the number of training samples used for updating model parameters). In this work, we
set the learning rate to 0.001, which is common in practice. The batch size was set to about
1% of the total number of samples in the data.
To train an ANN model, a measure of the performance of the ANN model training is
needed for optimization purposes. Such a measure is called a cost function. As the current
task of estimating an IFP distribution via an ANN model is a non-linear regression task, a
commonly used cost function named Mean Squared Error (MSE), defined in Equation (3),
is used in this work.
J(θ)≡MSE =1
N∑N
i=1Ri−ˆ
Ri2=1
N∑N
i=1Ri−f(ri,Ei,Ωi|θ)2, (3)
where
θ
is the parameters of the ANN model,
Ri
is the IFP corresponding to
i
-th input
in the data (i.e., the IFP calculated by the Solomon solver via Equation (2)),
ˆ
Ri
is the IFP
estimation by the ANN model for the
i
-th input, and
N
is the number of samples in the
data. The parameters
θ
, which consist of the weights and biases, were initialized randomly
in the range [−2.0, 2.0] in this study.
J. Nucl. Eng. 2023,4698
The constructed ANN models are then trained on the data using the gradient descent
algorithm named Adam optimizer [
31
], which is a well-known stochastic gradient descent
algorithm. The gradient descent algorithm minimizes the cost function so that an ANN
model learns the underlying IFP distribution, albeit approximately, from the data. To help
ANN models generalize well and learn the underlying IFP distribution without overfitting
the training data, the early stopping technique [32] is used.
Before training, the data are transformed using the power transformer [
33
], which
tries to make each input variable have a normal distribution centered around 0 with unit
variance, to improve the training in each system. The data in each system are then divided
into training, validation, and test data with ratios of 0.67, 0.08, and 0.25, respectively; these
ratios are common in machine learning applications.
4.2. Results and Discussion
4.2.1. Godiva Core
Ten ANN models constructed according to the widths and depths given in Table 2
have been trained on the training data. Table 3lists the training, validation, and test
MSEs by each ANN model. It can be observed that MSEs by the different ANN models
did not vary significantly, giving the maximum relative difference of about 22% between
ANN-3 and ANN-8. Based on the training result, ANN-3 is selected as it provides the
smallest test MSE among the ten ANN models. ANN-3 consists of 4585 parameters in total
and 4201 of them are trainable parameters. The training time for ANN-3 was 1 hour on a
typical desktop computer with Intel(R) Core(TM) i7 (1.4 GHz) CPU and 32GB RAM.
Table 3. Training results of ANN models on Godiva data.
ANN Model
MSE
Training Validation Test
ANN-1 2.6 ×10−12.7 ×10−12.7 ×10−1
ANN-2 2.6 ×10−12.6 ×10−12.6 ×10−1
ANN-3 2.5 ×10−12.5 ×10−12.5 ×10−1
ANN-4 2.8 ×10−12.9 ×10−12.9 ×10−1
ANN-5 2.7×10−12.6 ×10−12.8 ×10−1
ANN-6 2.7 ×10−12.7 ×10−12.7 ×10−1
ANN-7 2.7 ×10−12.7 ×10−12.7 ×10−1
ANN-8 3.2 ×10−13.2 ×10−13.2 ×10−1
ANN-9 2.7 ×10−12.8 ×10−12.8 ×10−1
ANN-10 2.7 ×10−12.7 ×10−12.7 ×10−1
In order to compare the estimated IFP distribution by the selected ANN model with
the Monte Carlo-based data by the Solomon solver (refer to Section 3.2), a group-wise IFP
spectrum (i.e., mean IFPs over the energy structure) is considered because it is a commonly
used distribution for comparison. The group-wise mean IFP is calculated simply as below.
Rg=1
Ng∑E∈gR(r,E,Ω), (4)
where
R
is the IFP by either the Solomon solver or the selected ANN model and
Ng
is the
number of samples that have energy belonging to group
g
. The IFPs by the selected ANN
model are estimated at the same phase-space locations as the Solomon solver, which means
the IFPs are estimated by the selected ANN model at the phase-space locations of all the
data. The standard deviation of the mean IFP by the Solomon solver is estimated as
σg=s1
NgNg−1∑E∈gR2(r,E,Ω)−R2
g. (5)
J. Nucl. Eng. 2023,4699
Since only the shape of an IFP distribution is important, the IFP spectra by the selected
ANN model and Solomon solver are further normalized to their respective integral values
(i.e.,
∑G
g=1Rg
) for a given energy structure with
G
intervals. It should be noted that an
energy structure used for making an IFP spectrum is arbitrary and not necessarily the same
as the energy structures used for creating the data described in Section 3.
Figure 5shows the group-wise IFP spectra by the Solomon solver and the selected
ANN model using the SLAROM-UF-73g energy structure. It can be seen from Figure 5that
there are peak structures in the range [10
0
, 10
4
] eV related to the resonance cross-sections
of
235
U and
238
U. The distribution by the ANN model showed only a part of this structure.
J. Nucl. Eng. 2023, 4, 9
𝜎=
∑𝑅
𝒓,𝐸,𝜴−𝑅
∈ . (5)
Since only the shape of an IFP distribution is important, the IFP spectra by the se-
lected ANN model and Solomon solver are further normalized to their respective integral
values (i.e., ∑𝑅
) for a given energy structure with 𝐺 intervals. It should be noted that
an energy structure used for making an IFP spectrum is arbitrary and not necessarily the
same as the energy structures used for creating the data described in Section 3.
Figure 5 shows the group-wise IFP spectra by the Solomon solver and the selected
ANN model using the SLAROM-UF-73g energy structure. It can be seen from Figure 5 that
there are peak structures in the range [10
0
, 10
4
] eV related to the resonance cross-sections of
235
U and
238
U. The distribution by the ANN model showed only a part of this structure.
Figure 5. Group-wise normalized IFP spectra by Solomon solver and ANN model in Godiva core
with SLAROM-UF-73g energy structure (3𝜎
is given for Solomon).
To further clarify the agreement or discrepancy between the IFP spectra by the Solo-
mon solver and the selected ANN model, we use different energy structures varying from
coarse to fine. In particular, we consider LANL-30g [34] and SCALE-238g [35] energy
structures. The LANL-30g energy structure is relatively coarse and is mainly used for fu-
sion applications; however, it has the same lethargy width in the resonance range. On the
other hand, the SCALE-238g energy structure is used in various reactor physics applica-
tions. Figures 6 and 7 show the comparison between the IFP spectra by the Solomon solver
and the selected ANN model for the different energy structures.
Figure 5.
Group-wise normalized IFP spectra by Solomon solver and ANN model in Godiva core
with SLAROM-UF-73g energy structure (3σgis given for Solomon).
To further clarify the agreement or discrepancy between the IFP spectra by the Solomon
solver and the selected ANN model, we use different energy structures varying from
coarse to fine. In particular, we consider LANL-30g [
34
] and SCALE-238g [
35
] energy
structures. The LANL-30g energy structure is relatively coarse and is mainly used for
fusion applications; however, it has the same lethargy width in the resonance range. On the
other hand, the SCALE-238g energy structure is used in various reactor physics applications.
Figures 6and 7show the comparison between the IFP spectra by the Solomon solver and
the selected ANN model for the different energy structures.
J. Nucl. Eng. 2023, 4, 10
Figure 6. Group-wise normalized IFP spectra by Solomon solver and selected ANN model in Godiva
core on LANL-30g energy structure (3𝜎
is given for Solomon).
Figure 7. Group-wise normalized IFP spectra by Solomon solver and selected ANN model in Godiva
core on SCALE-238g energy structure (3𝜎
is given for Solomon).
To compare the group-wise IFP spectra quantitatively, we consider the average rela-
tive difference (Equation (6)) and the intersection (Equation (7)) via the Ruzicka similarity
coefficient [36].
𝑅𝐷
=
∑
,
,
(6)
𝑆 =∑
,
∑
,
,
(7)
where 𝑅
and 𝑅
are the normalized mean IFPs in energy group 𝑔 by the Solomon
solver and the ANN model, respectively. Note that the relative difference in Equation (6)
is relative to the maximum of two distributions. Table 4 lists the average and maximum
relative differences and the Ruzicka similarity coefficients between the group-wise IFP
spectra by the Solomon solver and the selected ANN model for each energy structure.
From Figures 5–7 and Table 4, it can be observed that the agreement between the IFP
distributions by the selected ANN model and the Solomon solver became worse as the
energy structure became finer. The comparison indicates that the continuous IFP distribu-
tion by the selected ANN model is not the same as the true IFP distribution, but is a rough
Figure 6.
Group-wise normalized IFP spectra by Solomon solver and selected ANN model in Godiva
core on LANL-30g energy structure (3σgis given for Solomon).
J. Nucl. Eng. 2023,4700
J. Nucl. Eng. 2023, 4, 10
Figure 6. Group-wise normalized IFP spectra by Solomon solver and selected ANN model in Godiva
core on LANL-30g energy structure (3𝜎
is given for Solomon).
Figure 7. Group-wise normalized IFP spectra by Solomon solver and selected ANN model in Godiva
core on SCALE-238g energy structure (3𝜎
is given for Solomon).
To compare the group-wise IFP spectra quantitatively, we consider the average rela-
tive difference (Equation (6)) and the intersection (Equation (7)) via the Ruzicka similarity
coefficient [36].
𝑅𝐷
=
∑
,
,
(6)
𝑆 =∑
,
∑
,
,
(7)
where 𝑅
and 𝑅
are the normalized mean IFPs in energy group 𝑔 by the Solomon
solver and the ANN model, respectively. Note that the relative difference in Equation (6)
is relative to the maximum of two distributions. Table 4 lists the average and maximum
relative differences and the Ruzicka similarity coefficients between the group-wise IFP
spectra by the Solomon solver and the selected ANN model for each energy structure.
From Figures 5–7 and Table 4, it can be observed that the agreement between the IFP
distributions by the selected ANN model and the Solomon solver became worse as the
energy structure became finer. The comparison indicates that the continuous IFP distribu-
tion by the selected ANN model is not the same as the true IFP distribution, but is a rough
Figure 7.
Group-wise normalized IFP spectra by Solomon solver and selected ANN model in Godiva
core on SCALE-238g energy structure (3σgis given for Solomon).
To compare the group-wise IFP spectra quantitatively, we consider the average relative
difference (Equation (6)) and the intersection (Equation (7)) via the Ruzicka similarity
coefficient [36].
RD =1
G∑G
g=1Rsol
g−Rann
g
maxRsol
g,Rann
g, (6)
SRuz =
∑G
g=1minRsol
g,Rann
g
∑G
g=1maxRsol
g,Rann
g, (7)
where
Rsol
g
and
Rann
g
are the normalized mean IFPs in energy group
g
by the Solomon
solver and the ANN model, respectively. Note that the relative difference in Equation (6)
is relative to the maximum of two distributions. Table 4lists the average and maximum
relative differences and the Ruzicka similarity coefficients between the group-wise IFP
spectra by the Solomon solver and the selected ANN model for each energy structure.
From Figures 5–7and Table 4, it can be observed that the agreement between the
IFP distributions by the selected ANN model and the Solomon solver became worse as
the energy structure became finer. The comparison indicates that the continuous IFP
distribution by the selected ANN model is not the same as the true IFP distribution, but is
a rough approximation of it. The discrepancy is particularly around the resonance energy
range and a low energy range, as shown for the finer energy structure, which is closer to a
continuous distribution, in Figure 7.
Table 4.
Relative differences and Ruzicka similarity coefficient between normalized IFP spectra by
Solomon solver and selected ANN model for different energy structures in Godiva core.
Energy Structure RD Maximum RD SRuz
LANL-30g 1.8 ×10−27.0 ×10−20.98
SLAROM-UF-73g 2.7 ×10−21.81 ×10−10.97
SCALE-238g 7.1 ×10−26.46 ×10−10.93
4.2.2. Simplified STACY Core
Similarly to the case of the Godiva core, ten ANN models constructed according to the
widths and depths given in Table 2have been trained on the training data. The training,
validation, and test MSEs by each ANN model are given in Table 5. It can be observed
that the MSEs by the different ANN models did not show significant differences from each
J. Nucl. Eng. 2023,4701
other; the maximum difference between the test MSE is about 5.6% between ANN-5 and
ANN-9. According to the training result, ANN-9 is selected as it gives the marginally
smaller test MSE. ANN-9 consists of 2148 parameters in total and 1920 of them are trainable.
The training time for ANN-9 was about 12 min on a typical desktop computer with Intel(R)
Core(TM) i7 (1.4 GHz) CPU and 32GB RAM.
Table 5. Training results of ANN models on STACY data.
ANN Model
MSE
Training Validation Test
ANN-1 3.78 ×10−13.81 ×10−13.83 ×10−1
ANN-2 3.78 ×10−13.79 ×10−13.81 ×10−1
ANN-3 3.77 ×10−13.78 ×10−13.80 ×10−1
ANN-4 3.77×10−13.78 ×10−13.80 ×10−1
ANN-5 3.97 ×10−14.00 ×10−14.02 ×10−1
ANN-6 3.78 ×10−13.78 ×10−13.80 ×10−1
ANN-7 3.78 ×10−13.78 ×10−13.80 ×10−1
ANN-8 3.83 ×10−13.83×10−13.85 ×10−1
ANN-9 3.77 ×10−13.77 ×10−13.79 ×10−1
ANN-10 3.78 ×10−13.79 ×10−13.81 ×10−1
As done in the previous section, to compare the IFP distributions by the Solomon
solver and the selected ANN model, we consider group-wise IFP spectra (see Equation (4)).
The IFP spectra by the selected ANN model and Solomon solver are normalized to their
respective integral values as described in Section 4.2.1. Figure 8shows the comparison
between the normalized IFP spectra by the Solomon solver and the selected ANN model
using the SRAC-107g energy structure. It can be observed that the ANN model learned the
general trend of the IFP distribution except for some discrepancies around the resonance
energy range.
J. Nucl. Eng. 2023, 4, 12
Figure 8. Group-wise normalized IFP spectra by Solomon solver and selected ANN model in sim-
plified STACY core on SRAC-107g energy structure (3𝜎
is given for Solomon).
To further see the agreement or discrepancy between the IFP spectra by the Solomon
solver and the selected ANN model, we used different energy structures varying from coarse
to fine. The LANL-30g, WIMS-69g [37], and SCALE-238g energy structures are considered.
Figures 9–11 show the comparison between the normalized IFP spectra by the Solomon solver
and the selected ANN model for each energy structure. It should be noted that the energy
structures above 10 MeV in LANL-30g and SCALE-238g have been truncated because the IFP
distribution up to 10 MeV was estimated by the ANN models in the simplified STACY core.
Table 6 presents the quantitative comparison between the normalized IFP spectra by
the Solomon solver and the selected ANN model in terms of the average relative difference
and Ruzicka similarity coefficient (see Equations (6) and (7)).
Figure 9. Group-wise normalized IFP spectra by Solomon solver and selected ANN model in sim-
plified STACY core on LANL-30g energy structure (3𝜎
is given for Solomon).
Figure 8.
Group-wise normalized IFP spectra by Solomon solver and selected ANN model in
simplified STACY core on SRAC-107g energy structure (3σgis given for Solomon).
To further see the agreement or discrepancy between the IFP spectra by the Solomon
solver and the selected ANN model, we used different energy structures varying from
coarse to fine. The LANL-30g, WIMS-69g [
37
], and SCALE-238g energy structures are
considered. Figures 9–11 show the comparison between the normalized IFP spectra by
the Solomon solver and the selected ANN model for each energy structure. It should be
J. Nucl. Eng. 2023,4702
noted that the energy structures above 10 MeV in LANL-30g and SCALE-238g have been
truncated because the IFP distribution up to 10 MeV was estimated by the ANN models in
the simplified STACY core.
Table 6presents the quantitative comparison between the normalized IFP spectra by
the Solomon solver and the selected ANN model in terms of the average relative difference
and Ruzicka similarity coefficient (see Equations (6) and (7)).
Table 6.
Relative differences and Ruzicka similarity coefficient between normalized IFP spectra by
Solomon solver and selected ANN model for different energy structures in simplified STACY core.
Energy Structure RD Maximum RD SRuz
LANL-30g 8×10−31.7 ×10−20.99
WIMS-69g 9×10−32.3 ×10−20.99
SRAC-107g 9×10−39.5 ×10−20.99
SCALE-238g 1.5×10−24.70 ×10−10.98
J. Nucl. Eng. 2023, 4, 12
Figure 8. Group-wise normalized IFP spectra by Solomon solver and selected ANN model in sim-
plified STACY core on SRAC-107g energy structure (3𝜎
is given for Solomon).
To further see the agreement or discrepancy between the IFP spectra by the Solomon
solver and the selected ANN model, we used different energy structures varying from coarse
to fine. The LANL-30g, WIMS-69g [37], and SCALE-238g energy structures are considered.
Figures 9–11 show the comparison between the normalized IFP spectra by the Solomon solver
and the selected ANN model for each energy structure. It should be noted that the energy
structures above 10 MeV in LANL-30g and SCALE-238g have been truncated because the IFP
distribution up to 10 MeV was estimated by the ANN models in the simplified STACY core.
Table 6 presents the quantitative comparison between the normalized IFP spectra by
the Solomon solver and the selected ANN model in terms of the average relative difference
and Ruzicka similarity coefficient (see Equations (6) and (7)).
Figure 9. Group-wise normalized IFP spectra by Solomon solver and selected ANN model in sim-
plified STACY core on LANL-30g energy structure (3𝜎
is given for Solomon).
Figure 9.
Group-wise normalized IFP spectra by Solomon solver and selected ANN model in
simplified STACY core on LANL-30g energy structure (3σgis given for Solomon).
J. Nucl. Eng. 2023, 4, 13
Figure 10. Group-wise normalized IFP spectra by Solomon solver and selected ANN model in sim-
plified STACY core on WIMS-69g energy structure (3𝜎
is given for Solomon).
Figure 11. Group-wise normalized IFP spectra by Solomon solver and selected ANN model in sim-
plified STACY core on SCALE-238g energy structure (3𝜎
is given for Solomon).
The results shown in Figures 8–11 and Table 6 indicate that the agreement between
the IFP spectra by the Solomon solver and the selected ANN model increased as the en-
ergy structure became coarser. In the case of the finer energy structure of SCALE-238g,
which is closer to continuous distribution compared to the other energy structures, the
IFP spectrum by the selected ANN model shows increased discrepancies around the res-
onance energy range, indicating that the ANN model did not learn the resonance-like
structure of the IFP distribution fully. Despite the discrepancies around the resonance en-
ergy region, the ANN model appeared to have learned the general trend of the IFP distri-
bution in the simplified STACY core.
Tab le 6. Relative differences and Ruzicka similarity coefficient between normalized IFP spectra by
Solomon solver and selected ANN model for different energy structures in simplified STACY core.
Energy Structure 𝑹𝑫
Maximum 𝑹𝑫 𝑺𝑹𝒖𝒛
LANL-30g 8 × 10
−3
1.7 × 10
−2
0.99
WIMS-69g 9 × 10
−3
2.3 × 10
−2
0.99
SRAC-107g 9 × 10
−3
9.5 × 10
−2
0.99
SCALE-238g 1.5× 10
−2
4.70 × 10
−1
0.98
Figure 10.
Group-wise normalized IFP spectra by Solomon solver and selected ANN model in
simplified STACY core on WIMS-69g energy structure (3σgis given for Solomon).
J. Nucl. Eng. 2023,4703
J. Nucl. Eng. 2023, 4, 13
Figure 10. Group-wise normalized IFP spectra by Solomon solver and selected ANN model in sim-
plified STACY core on WIMS-69g energy structure (3𝜎
is given for Solomon).
Figure 11. Group-wise normalized IFP spectra by Solomon solver and selected ANN model in sim-
plified STACY core on SCALE-238g energy structure (3𝜎
is given for Solomon).
The results shown in Figures 8–11 and Table 6 indicate that the agreement between
the IFP spectra by the Solomon solver and the selected ANN model increased as the en-
ergy structure became coarser. In the case of the finer energy structure of SCALE-238g,
which is closer to continuous distribution compared to the other energy structures, the
IFP spectrum by the selected ANN model shows increased discrepancies around the res-
onance energy range, indicating that the ANN model did not learn the resonance-like
structure of the IFP distribution fully. Despite the discrepancies around the resonance en-
ergy region, the ANN model appeared to have learned the general trend of the IFP distri-
bution in the simplified STACY core.
Tab le 6. Relative differences and Ruzicka similarity coefficient between normalized IFP spectra by
Solomon solver and selected ANN model for different energy structures in simplified STACY core.
Energy Structure 𝑹𝑫
Maximum 𝑹𝑫 𝑺𝑹𝒖𝒛
LANL-30g 8 × 10
−3
1.7 × 10
−2
0.99
WIMS-69g 9 × 10
−3
2.3 × 10
−2
0.99
SRAC-107g 9 × 10
−3
9.5 × 10
−2
0.99
SCALE-238g 1.5× 10
−2
4.70 × 10
−1
0.98
Figure 11.
Group-wise normalized IFP spectra by Solomon solver and selected ANN model in
simplified STACY core on SCALE-238g energy structure (3σgis given for Solomon).
The results shown in Figures 8–11 and Table 6indicate that the agreement between
the IFP spectra by the Solomon solver and the selected ANN model increased as the energy
structure became coarser. In the case of the finer energy structure of SCALE-238g, which is
closer to continuous distribution compared to the other energy structures, the IFP spectrum
by the selected ANN model shows increased discrepancies around the resonance energy
range, indicating that the ANN model did not learn the resonance-like structure of the IFP
distribution fully. Despite the discrepancies around the resonance energy region, the ANN
model appeared to have learned the general trend of the IFP distribution in the simplified
STACY core.
4.2.3. Comparison between ANN Model and Deterministic Neutron Transport
Code PARTISN
To further analyze the estimated IFP distributions by the selected ANN models,
we consider various two-dimensional comparisons between the IFP distributions by the
selected ANN models and adjoint angular neutron flux distributions by the deterministic
neutron transport code PARTISN [38].
To obtain adjoint angular neutron flux distributions in the fissile systems, adjoint
eigenvalue calculations have been performed using the PARTISN code. For the Godiva
core, the PARTISN calculation has been performed with 100 meshes (
∆
r = 0.087 cm) and Sn
order of 180. The 73-group constants (e.g., cross-section) were obtained using the SLAROM-
UF code [
26
] with a Legendre order of 5 using the JENDL-4.0 nuclear data library [
39
]. Due
to the symmetry around the center of the core, a spherical geometry was used to produce
adjoint angular neutron flux in the form
ψ∗(r,g,µ)
, where
r
is the radial variable,
g
is the
energy group, and
µ
is the angular cosine along the radial direction. In the case of the
simplified STACY core, the PARTISN calculation has been done with 54 meshes (44 in the
fuel region giving
∆
r = 0.5 cm and 10 in the reflector region giving
∆
r = 0.1 cm) and an
Sn order of 16, which is the maximum. The 107-group constants were calculated by the
MOSRA-SRAC code [
40
] via its PIJ routine using the JENDL-4.0 nuclear data library. Due to
the XY symmetry around the center of the core, a cylindrical geometry was used to calculate
the adjoint angular neutron flux in the form
ψ∗(r,g,µ,η)
, where
r
is the radial variable in
XY-plane,
g
is the energy group,
µ
is the angular cosine along the radial direction, and
η
is
the angular cosine direction in the XY-plane perpendicular to the radial direction.
Before comparing the selected ANN models with the PARTISN code, we compare
the group-wise IFP spectra by the Solomon solver and adjoint neutron flux spectra by
the PARTISN code to confirm that the Monte Carlo and deterministic results are similar.
Figures 12 and 13 show the comparisons for the Godiva and simplified STACY cores,
respectively. In both Godiva and simplified STACY cores, the IFP spectrum by the selected
J. Nucl. Eng. 2023,4704
ANN model and adjoint neutron flux spectrum by the PARTISN code were normalized
to their respective integral values. Despite the agreement on the general trend in the
spectra, it can be seen that the comparisons show a slight discrepancy. However, the
discrepancy possibly results from the uniform sampling distribution of energy in each
group in calculating the IFP functions by the Solomon solver. In the case of the Godiva
core, it was confirmed in the previous study [
23
] that the IFP spectrum and adjoint angular
neutron flux spectrum are in agreement. On the other hand, the adjoint angular neutron flux
spectra are calculated with group constants produced using an energy spectrum (typically a
fission spectrum of
235
U) as a weighting function in SLAROM-UF and MOSRA-SRAC codes.
J. Nucl. Eng. 2023, 4, 14
4.2.3. Comparison between ANN Model and Deterministic Neutron Transport
Code PARTISN
To further analyze the estimated IFP distributions by the selected ANN models, we
consider various two-dimensional comparisons between the IFP distributions by the se-
lected ANN models and adjoint angular neutron flux distributions by the deterministic
neutron transport code PARTISN [38].
To obtain adjoint angular neutron flux distributions in the fissile systems, adjoint ei-
genvalue calculations have been performed using the PARTISN code. For the Godiva core,
the PARTISN calculation has been performed with 100 meshes (Δr = 0.087 cm) and Sn order
of 180. The 73-group constants (e.g., cross-section) were obtained using the SLAROM-UF
code [26] with a Legendre order of 5 using the JENDL-4.0 nuclear data library [39]. Due to
the symmetry around the center of the core, a spherical geometry was used to produce ad-
joint angular neutron flux in the form 𝜓∗𝑟,𝑔,𝜇, where 𝑟 is the radial variable, 𝑔 is the
energy group, and 𝜇 is the angular cosine along the radial direction. In the case of the sim-
plified STACY core, the PARTISN calculation has been done with 54 meshes (44 in the fuel
region giving Δr = 0.5 cm and 10 in the reflector region giving Δr = 0.1 cm) and an Sn order
of 16, which is the maximum. The 107-group constants were calculated by the MOSRA-
SRAC code [40] via its PIJ routine using the JENDL-4.0 nuclear data library. Due to the XY
symmetry around the center of the core, a cylindrical geometry was used to calculate the
adjoint angular neutron flux in the form 𝜓∗𝑟,𝑔,𝜇,𝜂, where 𝑟 is the radial variable in XY-
plane, 𝑔 is the energy group, 𝜇 is the angular cosine along the radial direction, and 𝜂 is
the angular cosine direction in the XY-plane perpendicular to the radial direction.
Before comparing the selected ANN models with the PARTISN code, we compare
the group-wise IFP spectra by the Solomon solver and adjoint neutron flux spectra by the
PARTISN code to confirm that the Monte Carlo and deterministic results are similar. Fig-
ures 12 and 13 show the comparisons for the Godiva and simplified STACY cores, respec-
tively. In both Godiva and simplified STACY cores, the IFP spectrum by the selected ANN
model and adjoint neutron flux spectrum by the PARTISN code were normalized to their
respective integral values. Despite the agreement on the general trend in the spectra, it
can be seen that the comparisons show a slight discrepancy. However, the discrepancy
possibly results from the uniform sampling distribution of energy in each group in calcu-
lating the IFP functions by the Solomon solver. In the case of the Godiva core, it was con-
firmed in the previous study [23] that the IFP spectrum and adjoint angular neutron flux
spectrum are in agreement. On the other hand, the adjoint angular neutron flux spectra
are calculated with group constants produced using an energy spectrum (typically a fis-
sion spectrum of
235
U) as a weighting function in SLAROM-UF and MOSRA-SRAC codes.
Figure 12. IFP spectrum by Solomon solver and adjoint neutron flux spectrum by PARTISN in Go-
diva core (normalized).
Figure 12.
IFP spectrum by Solomon solver and adjoint neutron flux spectrum by PARTISN in Godiva
core (normalized).
J. Nucl. Eng. 2023, 4, 15
Figure 13. IFP spectrum by Solomon solver and adjoint neutron flux spectrum by PARTISN in sim-
plified STACY core (normalized).
Since adjoint angular neutron fluxes by the PARTISN code were produced already in
a group-wise manner, the IFP distributions in the given energy group by the ANN model
were estimated by averaging IFPs at 20 equal intervals between the lower and upper en-
ergy boundary of the group to make the comparisons. Figures 14 and 15 show the contour
plots revealing the dependence of the IFP distributions and adjoint angular neutron flux
distributions on energy and radial position for a given direction along the positive r in the
figure. The average relative differences between the IFP distributions and adjoint angular
neutron flux distributions were calculated using an expression similar to Equation (6) and
are given in the captions of the figures.
Figure 14. Contour plot of normalized adjoint angular neutron flux by PARTISN and normalized
IFP by selected ANN model (for 𝜇1 i.e., the direction is along r) in Godiva core, 𝑅𝐷
= 7 × 10
−2
.
Figure 13.
IFP spectrum by Solomon solver and adjoint neutron flux spectrum by PARTISN in
simplified STACY core (normalized).
Since adjoint angular neutron fluxes by the PARTISN code were produced already in
a group-wise manner, the IFP distributions in the given energy group by the ANN model
were estimated by averaging IFPs at 20 equal intervals between the lower and upper energy
boundary of the group to make the comparisons. Figures 14 and 15 show the contour
plots revealing the dependence of the IFP distributions and adjoint angular neutron flux
distributions on energy and radial position for a given direction along the positive rin the
figure. The average relative differences between the IFP distributions and adjoint angular
J. Nucl. Eng. 2023,4705
neutron flux distributions were calculated using an expression similar to Equation (6) and
are given in the captions of the figures.
J. Nucl. Eng. 2023, 4, 15
Figure 13. IFP spectrum by Solomon solver and adjoint neutron ux spectrum by PARTISN in sim-
plied STACY core (normalized).
Since adjoint angular neutron uxes by the PARTISN code were produced already in
a group-wise manner, the IFP distributions in the given energy group by the ANN model
were estimated by averaging IFPs at 20 equal intervals between the lower and upper en-
ergy boundary of the group to make the comparisons. Figures 14 and 15 show the contour
plots revealing the dependence of the IFP distributions and adjoint angular neutron ux
distributions on energy and radial position for a given direction along the positive r in the
gure. The average relative dierences between the IFP distributions and adjoint angular
neutron ux distributions were calculated using an expression similar to Equation (6) and
are given in the captions of the gures.
Figure 14. Contour plot of normalized adjoint angular neutron ux by PARTISN and normalized
IFP by selected ANN model (for 𝜇 ≈ 1 i.e., the direction is along r) in Godiva core, 𝑅𝐷
= 7 × 10−2.
Figure 14.
Contour plot of normalized adjoint angular neutron flux by PARTISN and normalized IFP
by selected ANN model (for µ≈1 i.e., the direction is along r) in Godiva core, RD = 7 ×10−2.
J. Nucl. Eng. 2023, 4, 16
Figure 15. Contour plot of normalized adjoint angular neutron ux by PARTISN and normalized
IFP by selected ANN model (for 𝜇 = 0.982, 𝜂 = 0.133) in simplied STACY core, 𝑅𝐷
= 7.1 × 10−2.
Figures 16–19 present several representative two-dimensional plots showing the de-
pendence of the IFP distribution by the selected ANN model and the adjoint angular neu-
tron flux distribution by the PARTISN code on the radial and angular variables for given
energy groups in both Godiva and simplified STACY cores. In the case of the Godiva core,
the contour polar plots visualize the cross-sectional view of the adjoint angular neutron flux
and IFP distributions for the given energy group and direction. It can be seen that the angu-
lar dependence of the adjoint angular neutron flux and IFP distribution decreases as the
energy decreases. On the other hand, in the case of the simplified STACY core, the contour
plots visualize the dependence of the adjoint angular neutron flux and IFP distributions on
the radial and cosine angular variable 𝜇 for the given energy group and 𝜂 direction.
The comparisons shown in Figures 17 and 19 appear similar, giving the average rel-
ative dierences of 5.4 × 10−2 and 2.8 × 10−2, respectively. However, the IFP distribution by
the selected ANN model shown in Figure 18 appears very irregular compared to its coun-
terpart by the PARTISN code. In general, the comparisons show varying degrees of dis-
crepancies between the adjoint angular neutron ux distributions by PARTISN and the
IFP distributions by the selected ANN models.
Overall, the comparisons indicate that the estimated IFP distributions by the ANN
models are only approximate representations of the true IFP distributions in both systems,
suggesting that further improvements are necessary to increase the accuracy and applica-
bility of the ANN model-based method. Despite the discrepancies, the obtained IFP dis-
tributions by the ANN models in this study suggest that the underlying IFP distributions
in ssile systems may be estimated, albeit approximately, in a continuous manner via
ANN model using data created from a Monte Carlo neutron transport calculation.
Figure 15.
Contour plot of normalized adjoint angular neutron flux by PARTISN and normalized IFP
by selected ANN model (for µ= 0.982, η= 0.133) in simplified STACY core, RD = 7.1 ×10−2.
Figures 16–19 present several representative two-dimensional plots showing the de-
pendence of the IFP distribution by the selected ANN model and the adjoint angular
neutron flux distribution by the PARTISN code on the radial and angular variables for
given energy groups in both Godiva and simplified STACY cores. In the case of the Go-
diva core, the contour polar plots visualize the cross-sectional view of the adjoint angular
neutron flux and IFP distributions for the given energy group and direction. It can be
seen that the angular dependence of the adjoint angular neutron flux and IFP distribution
decreases as the energy decreases. On the other hand, in the case of the simplified STACY
core, the contour plots visualize the dependence of the adjoint angular neutron flux and
IFP distributions on the radial and cosine angular variable
µ
for the given energy group
and ηdirection.
The comparisons shown in Figures 17 and 19 appear similar, giving the average
relative differences of 5.4
×
10
−2
and 2.8
×
10
−2
, respectively. However, the IFP distribution
by the selected ANN model shown in Figure 18 appears very irregular compared to its
counterpart by the PARTISN code. In general, the comparisons show varying degrees of
discrepancies between the adjoint angular neutron flux distributions by PARTISN and the
IFP distributions by the selected ANN models.
J. Nucl. Eng. 2023,4706
J. Nucl. Eng. 2023, 4, 17
Figure 16. (𝑟,𝜑) polar contour plot of normalized adjoint angular neutron ux distribution by
PARTISN and normalized IFP distribution by ANN model in Godiva core (for the highest energy
group and direction in 0° in the gure), 𝑅𝐷
= 4.4 × 10−2.
Figure 17. (𝑟,𝜑) polar contour plot of normalized adjoint angular neutron ux distribution by
PARTISN and normalized IFP distribution by ANN model in Godiva core (for the lowest energy
group and direction in 180° in the gure), 𝑅𝐷
= 5.4 × 10−2.
Figure 18. (𝑟,𝜇) contour plot of normalized adjoint angular neutron ux distribution by PARTISN
and normalized IFP distribution by ANN model in simplied STACY core (for the highest energy
group), 𝑅𝐷
= 1.1 × 10−1.
Figure 16. (r,ϕ)
polar contour plot of normalized adjoint angular neutron flux distribution by
PARTISN and normalized IFP distribution by ANN model in Godiva core (for the highest energy
group and direction in 0◦in the figure), RD = 4.4 ×10−2.
J. Nucl. Eng. 2023, 4, 17
Figure 16. (𝑟,𝜑) polar contour plot of normalized adjoint angular neutron ux distribution by
PARTISN and normalized IFP distribution by ANN model in Godiva core (for the highest energy
group and direction in 0° in the gure), 𝑅𝐷
= 4.4 × 10−2.
Figure 17. (𝑟,𝜑) polar contour plot of normalized adjoint angular neutron ux distribution by
PARTISN and normalized IFP distribution by ANN model in Godiva core (for the lowest energy
group and direction in 180° in the gure), 𝑅𝐷
= 5.4 × 10−2.
Figure 18. (𝑟,𝜇) contour plot of normalized adjoint angular neutron ux distribution by PARTISN
and normalized IFP distribution by ANN model in simplied STACY core (for the highest energy
group), 𝑅𝐷
= 1.1 × 10−1.
Figure 17. (r,ϕ)
polar contour plot of normalized adjoint angular neutron flux distribution by
PARTISN and normalized IFP distribution by ANN model in Godiva core (for the lowest energy
group and direction in 180◦in the figure), RD = 5.4 ×10−2.
J. Nucl. Eng. 2023, 4, 17
Figure 16. (𝑟,𝜑) polar contour plot of normalized adjoint angular neutron ux distribution by
PARTISN and normalized IFP distribution by ANN model in Godiva core (for the highest energy
group and direction in 0° in the gure), 𝑅𝐷
= 4.4 × 10−2.
Figure 17. (𝑟,𝜑) polar contour plot of normalized adjoint angular neutron ux distribution by
PARTISN and normalized IFP distribution by ANN model in Godiva core (for the lowest energy
group and direction in 180° in the gure), 𝑅𝐷
= 5.4 × 10−2.
Figure 18. (𝑟,𝜇) contour plot of normalized adjoint angular neutron ux distribution by PARTISN
and normalized IFP distribution by ANN model in simplied STACY core (for the highest energy
group), 𝑅𝐷
= 1.1 × 10−1.
Figure 18. (r,µ)
contour plot of normalized adjoint angular neutron flux distribution by PARTISN
and normalized IFP distribution by ANN model in simplified STACY core (for the highest energy
group), RD = 1.1 ×10−1.
J. Nucl. Eng. 2023,4707
J. Nucl. Eng. 2023, 4, 18
Figure 19. (𝑟,𝜇) contour plot of normalized adjoint angular neutron ux distribution by PARTISN
and normalized IFP distribution by ANN model in simplied STACY core (for the lowest energy
group), 𝑅𝐷
= 2.8 × 10−2.
5. Conclusions
The method to estimate a continuous distribution of a quantity (e.g., neutron ux and
reaction rate) in all phase-space variables using a fully connected feedforward articial
neural network (ANN) model with Monte Carlo-based training data has been proposed
in this study. As a proof of concept of this method, the estimation of a continuous distri-
bution of iterated ssion probability (IFP), which is the quantity proportional to adjoint
angular neutron ux, in all phase-space variables in a given ssile system has been per-
formed in this work. To this end, two distinct ssile systems were considered: the fast
spectrum Godiva core and the thermal spectrum simplied STACY core.
The ANN model was trained to learn the continuous distribution of IFP in each sys-
tem from the Monte Carlo-based training data containing a discrete list of phase-space
locations and corresponding IFP value pairs. The data were created by the IFP method
implemented in the continuous-energy Monte Carlo neutron transport solver Solomon.
The estimated continuous IFP distributions by the ANN models were compared
against the Monte Carlo-based data, which include the training data, by the Solomon solver
in the form of the IFP spectrum. The comparison has been performed using three or four
energy structures ranging from coarse to fine. The comparisons showed that across the en-
ergy structures, the average relative differences between the IFP spectra by the ANN models
and Solomon solver were about 1.8–7.1% in the case of Godiva core and 0.8–1.5% in the case
of the simplified STACY core. On the other hand, the maximum relative differences, which
occurred around the resonance energy regions, were about 7.0–64.6% in the Godiva core
and 1.7–47.0% in the simplified STACY core across the considered energy structures. The
comparisons also showed that the discrepancy between the ANN models and Solomon
solver increased as the energy structure became finer. Furthermore, the discrepancy is larger
around the resonance energy regions, where the true IFP distributions showed peak-like
structures while the estimated IFP distributions by the ANN models did not exhibit such
structures fully. The comparisons indicated that the estimated IFP distributions by the ANN
models were only approximate representations of the true IFP distributions in both systems.
The estimated continuous IFP distributions by the ANN models were further com-
pared to the adjoint angular neutron ux distributions obtained with the deterministic
neutron transport code PARTISN. To compare the ANN models and PARTISN, various
two-dimensional distributions have been considered. The comparisons showed that the
average relative dierences between the estimated IFP distributions by the ANN models
and the adjoint angular neutron ux distribution by the PARTISN code were about 2.8–
11%. The comparisons revealed again that the estimated IFP distributions by the ANN
models did not show resonance-energy-related peak structures fully, while the PARTISN
results showed such structures.
Figure 19. (r,µ)
contour plot of normalized adjoint angular neutron flux distribution by PARTISN
and normalized IFP distribution by ANN model in simplified STACY core (for the lowest energy
group), RD = 2.8 ×10−2.
Overall, the comparisons indicate that the estimated IFP distributions by the ANN
models are only approximate representations of the true IFP distributions in both systems,
suggesting that further improvements are necessary to increase the accuracy and appli-
cability of the ANN model-based method. Despite the discrepancies, the obtained IFP
distributions by the ANN models in this study suggest that the underlying IFP distributions
in fissile systems may be estimated, albeit approximately, in a continuous manner via ANN
model using data created from a Monte Carlo neutron transport calculation.
5. Conclusions
The method to estimate a continuous distribution of a quantity (e.g., neutron flux and
reaction rate) in all phase-space variables using a fully connected feedforward artificial
neural network (ANN) model with Monte Carlo-based training data has been proposed in
this study. As a proof of concept of this method, the estimation of a continuous distribution
of iterated fission probability (IFP), which is the quantity proportional to adjoint angular
neutron flux, in all phase-space variables in a given fissile system has been performed in
this work. To this end, two distinct fissile systems were considered: the fast spectrum
Godiva core and the thermal spectrum simplified STACY core.
The ANN model was trained to learn the continuous distribution of IFP in each system
from the Monte Carlo-based training data containing a discrete list of phase-space locations
and corresponding IFP value pairs. The data were created by the IFP method implemented
in the continuous-energy Monte Carlo neutron transport solver Solomon.
The estimated continuous IFP distributions by the ANN models were compared
against the Monte Carlo-based data, which include the training data, by the Solomon solver
in the form of the IFP spectrum. The comparison has been performed using three or four
energy structures ranging from coarse to fine. The comparisons showed that across the
energy structures, the average relative differences between the IFP spectra by the ANN
models and Solomon solver were about 1.8–7.1% in the case of Godiva core and 0.8–1.5% in
the case of the simplified STACY core. On the other hand, the maximum relative differences,
which occurred around the resonance energy regions, were about 7.0–64.6% in the Godiva
core and 1.7–47.0% in the simplified STACY core across the considered energy structures.
The comparisons also showed that the discrepancy between the ANN models and Solomon
solver increased as the energy structure became finer. Furthermore, the discrepancy is
larger around the resonance energy regions, where the true IFP distributions showed peak-
like structures while the estimated IFP distributions by the ANN models did not exhibit
such structures fully. The comparisons indicated that the estimated IFP distributions by
the ANN models were only approximate representations of the true IFP distributions in
both systems.
J. Nucl. Eng. 2023,4708
The estimated continuous IFP distributions by the ANN models were further com-
pared to the adjoint angular neutron flux distributions obtained with the deterministic
neutron transport code PARTISN. To compare the ANN models and PARTISN, various
two-dimensional distributions have been considered. The comparisons showed that the
average relative differences between the estimated IFP distributions by the ANN models
and the adjoint angular neutron flux distribution by the PARTISN code were about 2.8–11%.
The comparisons revealed again that the estimated IFP distributions by the ANN models
did not show resonance-energy-related peak structures fully, while the PARTISN results
showed such structures.
Despite the varying discrepancies in the estimated IFP distributions by the ANN mod-
els when compared to the Monte Carlo neutron transport Solomon solver and deterministic
neutron transport code PARTISN, this work showed that the underlying IFP distributions
may be estimated, at least approximately, in a continuous manner via an ANN model from
Monte Carlo-based training data. In the future, to make the ANN model-based method
more applicable and to improve its accuracy, several important works need to be performed.
First, an estimation of a continuous distribution of a forward angular or scalar neutron flux
using the ANN model-based method from Monte Carlo-based data must be investigated.
Second, alternative artificial neural network designs, cost functions, and different model
and algorithmic hyperparameters should be considered to more accurately capture the
finer details of IFP distributions such as resonance-energy related structures. Third, if the
prior works prove to be good, then the ANN model-based method should be applied to
more challenging problems in terms of geometry, material composition, and neutron flux
or reaction rate distributions. Lastly, methods to quantify the uncertainty of the ANN
model-based method need to be explored.
Author Contributions:
Conceptualization, D.T.; methodology, D.T. and Y.N.; software, D.T. and
Y.N.; validation, D.T. and Y.N.; formal analysis, D.T.; investigation, D.T.; resources, D.T. and Y.N.;
data curation, D.T.; writing—original draft preparation, D.T.; writing—review and editing, Y.N.;
visualization, D.T.; supervision, Y.N.; project administration, Y.N.; funding acquisition, Y.N. All
authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
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