A Data-Driven Approach Based on Deep Neural Network Regression for Predicting the Compressive Strength of Steel Fiber Reinforced Concrete

Journal of Soft Computing in Civil Engineering 9-2 (2025) 1-31
How to cite this article: Hoang ND, Tran VD. A data-driven approach based on deep neural network regression for predicting the
compressive strength of steel fiber reinforced concrete. J Soft Comput Civ Eng 2024;9(2):1–31. https://doi.org/10.22115/scce.
2024.430215.1765
2588-2872/ © 2025 The Authors. Published by Pouyan Press.
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A Data-Driven Approach Based on Deep Neural Network
Regression for Predicting the Compressive Strength of Steel Fiber
Reinforced Concrete
Nhat-Duc Hoang 1,2,*
; Van-Duc Tran 2,3
1. Lecturer, Institute of Research and Development, Duy Tan University, Da Nang, 550000, Vietnam
2. Lecturer, Faculty of Civil Engineering, Duy Tan University, Da Nang, 550000, Vietnam
3. Lecturer, International School - Duy Tan University, Da Nang, 550000, Vietnam
* Corresponding author: hoangnhatduc@duytan.edu.vn
https://doi.org/10.22115/SCCE.2024.430215.1765
ARTICLE INFO ABSTRACT
Article history:
Received: 12 December 2023
Revised: 22 March 2024
Accepted: 15 April 2024
Estimating the compressive strength of steel fiber reinforced
concrete (SFRC) is a crucial task required in mix design. Thus,
a reliable method that can deliver accurate estimations of the
compressive strength of SFRC is a practical need. This study
puts forward a new deep neural network-based regression model
for solving the task at hand. The state-of-the-art Nesterov
accelerated adaptive moment estimation (Nadam) is used to
optimize the deep neural computing model that learns the
functional mapping between the compressive strength and
concrete’s constituents. A dataset, consisting of 303 samples
and 12 predictor variables, is used to train the deep learning
approach. Notably, the current work has carried out a
comparative study to identify the suitable regularization strategy
for establishing a robust SFRC strength estimation model.
Experimental results show that the L1 regularization helps
achieve the most desired performance, with a coefficient of
determination (R2) of roughly 0.96. Notably, an asymmetric
loss function is used along with Nadam to decrease the
percentage of overestimated cases from 50.83% to 27.08%. In
general, the proposed method can be a promising tool to support
construction engineers in SFRC mix design.
Keywords:
Steel fiber;
Concrete mix;
Compressive strength;
Deep neural networks;
Model regularization.

2 ND. Hoang; VD. Tran/ Journal of Soft Computing in Civil Engineering 9-2 (2025) 1-31
1. Introduction
Concrete, as a cementitious material, is commonly known for its brittle failure caused by low
tensile stress [1]. Hence, the utilization of steel fibers can be an effective approach to mitigate
this disadvantage of this construction material [2]. The inclusion of steel fibers in the concrete
mix aims to resist cracks and postpone their propagation [3]. It has been shown that the use of
discontinuous steel fibers significantly enhance the crack resistance of concrete by enlarging its
critical cracking strength [1]. Additionally, steel fibers with standard size and shape are able to
meliorate the concrete’s crack behavior and enhance its durability. Therefore, steel fiber added
concrete is increasingly employed in practical applications, such as large slabs of factories,
tunnel lining, rock slope stabilization, dams, shell domes, and many other structures [4].
Since compressive strength is an essential parameter in structural design, this parameter of steel
fiber-reinforced concrete (SFRC) has been thoroughly investigated by various works. The study
in [5] shows that the inclusion of steel fibers results in a 5% improvement in compressive
strength compared to plain concrete. It is observed that using steel fibers with a volume fraction
(Vf) of 1.5% yields roughly a 15% improvement in concrete strength; meanwhile, Vf > 1.5%
cannot help gain significant strength enhancement [6]. The combining effect of steel fibers and
silica fume are investigated in [7]. The authors show that the toughness of concrete depends on
the content of silica fume, Vf, and the fiber aspect ratio. The results in [8] pointed out that the
incorporation of steel fiber into concrete blended with silica fume improved the impact resistance
and ductility of the concrete mixes. In addition, water-cement ratio is an important factor when
designing SFRC for use with silica fume.
Sivakumar and Santhanam [9] studied the properties of high strength concrete with the use of
fiber content up to a Vf value of 0.5%. Via experiments, the maximum increase in compressive
strength of the concrete mixes was found to be about 15%. This result is consistent with that
reported in [6]. The study in [10] investigated the compressive strength of concrete mixes with
different Vf (i.e. 0, 0.5, 1.0, and 1.5%). Based on 60 data samples, an empirical model was
constructed to estimate fiber-reinforced concrete strength. The authors show good correlations
between the actual and estimated strength values. However, the use of this empirical model
requires experiments to identify the 28-day cube compressive strength of plain concrete. In
addition, there are regression coefficients that need to be determined. This fact, to some degree,
limits the generalization capability of the empirical equation. An experimental study on the
mechanical properties of concrete blended with fly ash and steel fibers was carried out in [11].
This study reported an increase in compressive strength of up to 10%.
Soulioti et al. [12] studied concrete mixes blended with steel fibers at Vf of 0.5%, 1.0%, and
1.5% and found that only the fractions of 0.5% and 1.5% helped increase the compressive
strength. The authors argued that the inclusion of steel fibres into the mixtures brings about
difficulty in consolidating the concrete mix and an increase in the amount of entrapped air. The
results reported in [13] indicate that the addition of 1% steel fiber considerably improves the
concrete’s strength. On the other hand, Abbass et al. [14] reported an increase in the 28-day
compressive strength of up to 25% when the fiber content ranges from 0.5% to 1.5%. The

ND. Hoang; VD. Tran/ Journal of Soft Computing in Civil Engineering 9-2 (2025) 1-31 3
authors explained that the confining effect provided by the fibers to the concrete is the major
cause of the increase in the compressive strength. A study in [15] investigated the effects of steel
fiber content and coarse aggregate size on the properties of high-strength concrete. Experimental
results show that the influence of Vf on the mechanical properties of the concrete mixes is greater
than that of the aggregate size.
Although various studies have been dedicated to constructing data-driven models for estimating
the compressive strength of concrete mixes [16,17], the task of predicting this mechanical
property of SFRC is still challenging. It is because the number of influencing variables for
compressive strength in SFRC is higher than that in normal concrete. Besides the usual
constituents of a concrete mix (e.g., cement, water, aggregate, slag, silica fume, etc.), additional
parameters (e.g., Vf and geometric properties of the fiber) must be taken into account. Hence, it
is very difficult to estimate the compressive strength of SFRC with conventional regression
analysis models [18].
Accordingly, researchers and practitioners have increasingly relied on advanced machine
learning approaches to tackle the problem of interest. Altun et al. [19] compared the
performances of artificial neural network (ANN) and multiple linear regression (MLR) and
showed that the former significantly outperformed the latter. The study in [20] relied on ANN,
support vector machine (SVM), and M5 model tree to estimate the splitting tensile strength of
SFRC. Awolusi et al. [21] compared different ANN’s training algorithms to model properties of
SFRC; the authors found that back propagation algorithms outperformed the genetic algorithm-
based training approach. Kang et al. [18] resorted to boosting- and tree-based models to predict
the compressive and flexural strengths of SFRC; this study also revealed that the water-to-
cement ratio and silica fume content are the most crucial factors.
The results, reported in [22], investigated the capability of neural computing models and fuzzy
inference systems in predicting the compressive strength of lightweight concrete reinforced by
steel fibers. This work confirms that the results obtained from all three machine learning models
are acceptable in terms of predictive accuracy. The adaptive boosted SVM (AdaBoost-SVM) is
found to be the most capable method for predicting the 28-day compressive strength of SFRC
[23] with a coefficient of determination (R2
) of 0.96. SVM, random forest (RF), and decision tree
ensemble are used in [24]; the used machine learning models can help attain good prediction
performance with R2
of 0.92. Khan et al. [4] recently compared the performances of gradient
boosting-based ensemble learning and RF; the authors demonstrated the superiority of the RF
model, which helped achieve the most desired performance. Via analysis on variables’
importance, this study shows that content of cement imposes the highest positive influence on
the compressive strength value.
Recently, deep learning-based regression models, or deep neural networks (DNNs) used for
nonlinear estimation, have increasingly attracted the attention of researchers and practitioners in
the task of modeling the mechanical properties of concrete, including the compressive strength
[16]. DNNs are characterized by a hierarchical organization of hidden layers. Each hidden layer
acts as a feature engineering operator to extract informative representations of data [25]. Hence,
DNNs are generally superior to shallow neural networks in modeling complex functional

relationships between the compressive strength of concrete and its influencing variables [26].
Nevertheless, previous studies have rarely investigated the capability of DNNs to predict the
compressive strength of SFRC. Moreover, although DNNs have been employed for compressive
strength estimation of concrete mixes [16,26], few studies have fully employed and compared
the effectiveness of different regularization approaches for training the deep neural computing
models. Hence, the current work aims to fill this gap in the literature by proposing a novel deep
learning solution to cope with the problem at hand. The main contribution of the current work
can be summarized as follows:
(i) DNNs with stacked hidden layers and different activation functions are employed to model
the mapping relationship between the compressive strength of SFRC and its predictor
variables.
(ii) A comprehensive dataset, including 303 samples, has been collected from the literature to
train and verify the DNNs.
(iii) The Nesterov accelerated adaptive moment estimation (Nadam) [27] is used to optimize
the DNNs’ structure.
(iv) To deal with overfitting, various network regularization methods are used.
(v) Although various machine learning models have been proposed for estimating the
compressive strength of SFRC, the existing models only focus on minimizing the
prediction error in general. To the best of the writers’ knowledge, none of the previous
works have emphasized the importance of overestimation reduction in compressive
strength estimation. Therefore, this study aims to enhance the reliability of the prediction
model by restricting overestimated results. To achieve this goal, the current work employs
an asymmetric loss function to train the DNNs.
(vi) Based on extensive experiments with different configurations of DNNs, the most suitable
deep neural network (DNN) model can be identified. The performance of the proposed
DNN model is compared to that of the benchmark machine learning models, including
Levenberg–Marquardt Artificial Neural Network (LM-ANN), RF, and AdaBoost-SVM.
Subsequently, a MATLAB-based graphical user interface (GUI) for the deep learning
approach can be developed to facilitate its practical application.
2. Research method
2.1. Nadam-optimized deep neural network regression model
A deep neural network for regression (DNNR) model is characterized by a hierarchical
arrangement of hidden layers where the output of a preceding layer is the input of the succeeding
one [28]. DNNR models are capable of providing good fits to various datasets due to their ability
to learn and generalize functional mappings between the dependent variables and predictor
variables [16]. As pointed out in [29], the success of DNN can be explained by the observation
that this machine learning approach is able to separate the input domain (also called feature
space) into an exponentially larger number of linear regions than shallow machine learning
methods. As stated in [30], DNN is capable of carrying out effective feature extraction phase
with low computational cost. In addition, the increased depth of DNN can also be considered a
form of regularization due to the fact that an extracted feature in a later layer has a tendency to
comply with the type of structure specified by its preceding layer [31].

A DNNR model, potentially used for estimating the compressive strength of SFRC, can be
illustrated in Fig. 1. The model typically includes an input layer, a structure of stacked hidden
layers, and an output layer. The input layer receives input information regarding the concrete
constituents and curing age. Herein, the number of nodes in the first layer (denoted as D) is equal
to the number of concrete strength’s influencing factors. This input information is subsequently
processed by the neurons in the hidden layers. The number of hidden layers is denoted as M.
These hidden layers act as autonomous feature engineering operators in which increasingly
informative representations of the original data are constructed and used for predicting the
compressive strength of SFRC in the output layer. Since the target output is the estimated value
of the compressive strength, the output layer of the network contains one node.
...
x1
x2
xD
Σ
Σ
Σ
1
1
1
fA
fA
fA
...
...
Input layer
...
Σ
Σ
Σ
1
1
1
fA
fA
fA
...
Σ
1
...
Σ
Σ
Σ
1
1
1
fA
fA
fA
...
...
...
...
y
Estimated compressive
strength of SFRC
Hidden layers Output layer
Influencing
factors
fA denotes an activation function.
W1
... WM+1
Nadam Optimizer
Matrices of synaptic weights
Fig. 1. The structure of a DNNR model.
As can be seen in Fig. 1, the structure of the DNNR model used for predicting the compressive
strength of SFRC includes a stacking of successive hidden layers. The first hidden layer receives
and processes signals from the input layer. The subsequent hidden layers act as feature
engineering operators used for filtering and transforming their input data. Accordingly,
increasingly abstract and informative features, which are useful for strength prediction, can be
constructed. In addition, the organization of stacked hidden layers also plays a role as a network
regularization scheme. It is because the model’s parameters in a hidden layer must comply with
those in the adjacent layers. This structure of DNNR is potentially suitable for the task of
interest. The reason is that predicting the compressive strength of SFRC involves the
consideration of complex interplays among the mixture’s constituents. In addition, the functional
mapping between the compressive strength value and its influencing factors is often nonlinear
and sophisticated [32].
The learning process of a DNNR model aims to adapt its matrices of synaptic weights, which
connect successive layers in the network. For instance, the matrix W(1)
connects the input layer

and the 1st
hidden layer. The last matrix, denoted as W(M+1)
, stores the synaptic weights between
the last hidden layer and the output layer. Generally, a network that contains M hidden layers
requires the setting of M+1 matrices of synaptic weights. The capability of DNN to capture
nonlinear functional mappings depends on the use of nonlinear activation functions (AVFs) in
the neurons of the hidden layers. The commonly-employed AVFs for DNNR are the logistic
sigmoid (Sigmoid), hyperbolic tangent (Tanh), and rectified linear unit (ReLU). The equations of
those AVFs can be expressed as follows:
Sigmoid:
)
exp(
1
1
)
(
x
x
f


 and ))
(
1
(
)
(
)
(
' x
f
x
f
x
f 

 (1)
Tanh:
)
exp(
)
exp(
)
exp(
)
exp(
)
(
x
x
x
x
x
f




 and )
(
1
)
(
' 2
x
f
x
f 
 (2)
ReLU:








0
,
0
0
,
)
(
x
if
x
if
x
x
f and








0
,
0
0
,
1
)
(
'
x
if
x
if
x
f (3)
A DNNR model is trained in a supervised manner. Thus, the error (e) committed by a DNNR
model is required, and it can be expressed by:
e = t – y (4)
where t and y denote the observed and estimated values of the compressive strength of SFRC,
respectively.
Based on the error term, the commonly-used squared error loss (SEL) can be computed to
represent the goodness-of-fit of a DNNR model. The SEL is expressed by:
2
)
(
2
1 2
2 e
y
t
L 

 (5)
The derivative of L with respect to each synaptic weight (w), also called the gradient (g), can be
computed as follows:
w
L
g 

 / (6)
Based on such gradient, the back-propagation framework and the gradient-descent algorithm can
be used to adapt the synaptic weights of a DNNR model. Notably, the generalized delta rule can
be applied to compute the gradients of neurons in hidden layers [25]. Typically, the error term at
a neuron in a hidden layer is calculated as the sum of the errors of the neurons in the succeeding
layer weighted by their synaptic weights. In this study, the state-of-the-art Nadam optimizer [27],
a novel variant of the gradient-descent approach, is used to train the DNNR model used for
estimating the compressive strength of SFRC.
The Nadam optimizer incorporates the advantages of Nesterov momentum and the adaptive
moment estimation approach [33]. The Nesterov momentum helps increase the effectiveness of
the optimization process via the utilization of the gradient at the projected future position [34].
This optimizer computes the momentum term based on the Nesterov Momentum approach,
which has good capability in dealing with noisy gradients and to quickly converge to the
minimum of the loss function. Hence, the Nadam optimize is particularly helpful to deal with

regions of the loss function where its gradient with respect to w is flat [26]. Recent work has
demonstrated the superiority of Nadam in training deep neural network structures over other
methods [35].
Using this optimizer, the synaptic weight (w) at iteration uth
is adapted as follows:
1
1
1
1
1
1
)
1
(









 u
u
u
u
t
u b
v
g
m
w
w



 (7)
where  denotes the learning rate; 8
1 
 e
 is a small number to ensure numerical stability.
The values of mu+1, vu+1and bu+1can be expressed by:
u
u
u g
m
m )
1
( 1
1
1 
 


 (8)
2
2
2
1 )
1
( u
u
u g
v
v 
 



(9)
)
1
/(
1 1
1
1
2
1


 

 u
u
u
b 
 (10)
In addition, to cope with the problem of overfitting during the network’s training phase, this
study has incorporated network regularization approaches into the Nadam optimizer. The
employed network regularization approaches include L1, L2, and weight decay.
For the cases of L1 and L2 regularization, the loss function used during the model adaptation
process is revised as follows [36]:
L1-norm: 1
||
||
)
,
(
)
,
( w
y
t
L
y
t
LR 

 (11)
L2-norm:
2
2
||
||
)
,
(
)
,
( w
y
t
L
y
t
LR 

 (12)
where λ denotes the regularization coefficient.
Meanwhile, if the weight decay approach is used, the equation used for updating the synaptic
weight can be expressed by [37]:
)
)
1
(
( 1
1
1
1
1
1 t
u
u
u
u
t
u w
b
v
g
m
w
w 






 


 



 (13)
All of the L1, L2, and weight decay methods are effective schemes for restricting overfitting in
deep learning. These methods are used to construct capable prediction models with appropriate
levels of complexity in the model structure. The L1 and L2 regularization strategies include an
additional term (called a penalty term) to the loss function. Meanwhile, the weight decay method
directly modifies the equation used for updating the synaptic weights of DNNR. Notably, the L1
regularization includes the sum of the absolute values of the synaptic weights. Therefore, this
method allows the values of a certain number of weights to shrink towards zeros. This basically
helps construct sparse model structures. On the other hand, the penalty term used in L2
regularization is the sum of the squares of the synaptic weights. This regularization attempts to
minimize the model’s weights; however, it doesn’t have the tendency to shrink the weights
towards zeros. In addition, L1 regularization is more robust to outliers than L2 regularization. It is
because the former takes into account the absolute values of the synaptic weights, and the latter
considers the square of the synaptic weights. Therefore, in the presence of an outlier, the loss

function of L1 regularization increases linearly; meanwhile, that of L1 regularization enlarges
exponentially. Furthermore, the weight decay directly adds a term to reduce the magnitude of the
synaptic weight (as shown in Eq. (13)). This is a straight-forward method to shrink the neural
network’s weights and fend off overfitting.
2.2. Asymmetric loss function for training deep neural network regression
Although the aforementioned SEL helps reduce the training error of a DNNR model, it cannot
take into account the problem of overestimated values of compressive strength. In such
circumstances, the estimated value (y) surpasses the observed one (t). In the task of predicting
the compressive strength of concrete mixes, overestimation of the outcomes should be avoided to
enhance the reliability of the machine learning model. Therefore, this study proposes to
incorporate the asymmetric SEL, denoted as ASEL, into the Nadam-optimized model. The
ASEL’s equation can be expressed by [38]:










0
,
2
0
,
2
1
)
(
2
2
e
if
e
e
if
e
e
LA

(14)
where  is a parameter that determines the degree of asymmetry.
Using ASEL, it is only required to modify the partial derivative of L() with respect to w that
connects the last hidden layer to the output layer. The revised partial derivative of the loss
function is given by [38]:

































0
)
(
0
)
(
)
(
)
(
)
(
)
(
y
t
e
if
w
y
y
t
y
t
e
if
w
y
y
t
w
y
y
L
w
L
M
i
M
i
M
i
M
i
A

(15)
2.3. The benchmark machine learning models
2.3.1. The Levenberg–Marquardt artificial neural network
The Levenberg–Marquardt artificial neural network (LM-ANN) [39] combines the steepest
descent algorithm and the Gauss-Newton method to train the network used for nonlinear function
approximation. The applicability of ANN models in estimating the mechanical properties of
SFRC [40,41] and other crucial tasks in structural engineering [42–45] was reported in various
studies. The coupling effect of the steepest descent algorithm and the Gauss-Newton method
helps the LM-ANN approach to converge quickly to an acceptable solution. However, LM-based
optimization necessitates the computation and storage of Jacobian matrices, which are the
matrices of all first-order partial derivatives of the loss function (L) with respect to the synaptic
weights (w) [46]. The memory cost for Jacobian matrix storage is the main hindrance of the LM
method used for training deep learning models with a huge number of tuning parameters [47].

2.3.2. Random forest
The random forest (RF) [48] attempts to construct an ensemble of decision trees (DT) for pattern
recognition tasks. If individual DTs are regression trees, the RF is applicable for nonlinear
function approximation. The main difference between a RF model and a DT is the number of
trees [4]. A RF model concurrently builds multiple regression trees. Each tree may feature a
different number of input attributes and training samples. This is the method that a RF uses to
generate diversity in individual DTs. When all DTs in the ensemble are constructed, a RF yields
the final predicted value of the compressive strength via a majority vote. RF has been shown to
be a robust method for modeling complex problems in civil engineering [49,50], especially for
estimating the strength performance of concrete mixes [4]. This machine learning method has
recently been demonstrated to outperform backpropagation neural network and support vector
regression in predicting the CS of fiber reinforced concrete [51].
2.3.3. Adaptive boosted support vector machines
Adaptive Boosting, or AdaBoost, [52], is a method of constructing an ensemble of machine
learning models in a sequential manner. This method can be employed in conjunction with
various machine learning algorithms. The final output of the ensemble is derived by computing a
weighted sum that is the final outcome of the boosted machine learning model. AdaBoost uses
weighting values to express the importance of data instances so that poor prediction outcomes
committed by previous learners can be effectively addressed by subsequent ones. Although weak
learners (e.g., regression trees) are originally applied in AdaBoost, recent works show that this
method can also combine strong based models (e.g., SVM) to form capable function estimators
[23]. Previous works have affirmed the superiority of AdaBoost over individual machine learning
models (e.g., ANN and SVM) in predicting the compressive strength of concrete [53,54].
3. The collected dataset
To train and verify the generalization capability of the DNNR approaches, a comprehensive
dataset, consisting of 303 outcomes of concrete strength testing, was collected from the
literature. This study aims to establish a comprehensive database of SFRC used for constructing
robust machine learning-based strength estimation models. Therefore, a meticulous literature
review was conducted, and data samples from reliable sources (e.g., academic journal articles)
were examined. Data instances from previous experimental works that were reported with clear
information of the mixes’ constituents, curing age, and specimen size were collected. In addition,
since this study focuses on the use of steel fibers, other forms of fiber-based reinforcements are
excluded from the data collection process. Based on the literature review, the source and basic
description of the data samples are summarized in Table 1. Herein, laboratory tests obtained from
19 previous works are used to construct the historical dataset.
The contents of cement, water, fine aggregate (FA), coarse aggregate (CA), silica fume (SF), fly
ash, slag, and superplasticizer are used as basic constituents of concrete mixes. In addition, the
volume fraction of fiber, length of the fiber, and diameter of the fiber are used to characterize the

influence of steel fiber on the compressive strength of SFRC. In addition, the curing age of
concrete is also used as an influencing factor.
In total, there are 12 input variables used for estimating the compressive strength of SFRC. The
statistical descriptions of those input variables, including the minimum value, average, standard
deviation, skewness, and maximum value, are presented in Table 2. Moreover, distributions of
the predictor variables and the dependent variable of compressive strength are illustrated by the
histogram in Fig. 2.
In addition, since the testing samples reported in previous works have different sizes, correlation
factors are required to standardize the compressive strength values [38]. Using these correlation
factors, the compressive strength values of cubic and cylindrical samples with different
dimensions are converted into those of 100 mm cubes.
Table 1
The sources of the data samples.
Data source
Number of
samples
Type of specimens Note Ref.
1 30 100x200 mm cylinders
Studying the effect of steel fibers with various
lengths and diameters on the mechanical
properties of concrete
[14]
2 18 100 mm cubes
Investigate the effect of steel fibers on the
mechanical properties of high-strength concrete
[13]
3 60 150 mm cubes
Concrete mixes blended with fly ash and
reinforced with steel fibers
[11]
4 2 150 mm cubes Concrete mixes containing steel fibers [55]
Studying the influence of different steel fiber
dosages on the performance of concrete mixes
[15]
6 20 150x300 mm cylinders High strength concrete containing steel fiber [7]
Steel fiber-reinforced concrete mixes containing
silica fume and slag
[56]
8 12 100x200 mm cylinders Concrete mixes containing steel fibers [5]
9 36 100 mm cubes
Investigate the influence of steel fibers on the
mechanical properties of concrete blended with
silica fume
[8]
Slag-blended concrete mixes containing steel
fibers
[57]
High-performance steel fiber reinforced
concrete
[58]
12 21 150x300 mm cylinders Concrete mixes containing steel fibers [59]
14 2 100 mm cubes
Properties of high strength concrete mixes
containing metallic fibers
[9]
15 5 150x300 mm cylinders High-strength concrete containing steel fiber [6]
Compressive strength of steel fiber reinforced
high strength concrete
[61]
Studying the effect of fiber content on the
mechanical properties of steel-fiber-reinforced
concrete
[62]

Fig. 2. Distributions of input variables and CS of concrete.

Table 2
Statistical description of the variables in the collected dataset.
Variables Unit Notation Min Average Standard deviation Skewness Max
Cement content kg/m3
X1 230 402.07 99.82 0.98 680
Water content kg/m3
X2 133 176.53 98.13 5.65 762
FA content kg/m3
X3 400 788.36 125.80 -0.48 1225
CA content kg/m3
X4 356 965.15 162.94 -1.23 1284
SF content kg/m3
X5 0 14.84 23.38 1.41 102
Fly ash content kg/m3
X6 0 11.88 32.41 2.65 120
Slag content kg/m3
X7 0 28.15 71.86 2.23 285
Superplasticizer content kg/m3
X8 0 3.42 2.20 0.01 9
Volume fraction of steel fiber % X9 0 0.84 0.74 1.37 5
Length of steel fiber mm X10 0 43.11 14.82 -0.47 60
Diameter of steel fiber mm X11 0 0.62 0.18 -0.86 1
Age Day X12 7 49.21 75.27 3.67 365
Compressive strength MPa Y 32.95 68.68 21.19 0.40 122.30
4. Results and discussion
4.1. Experiment setting
As stated in the previous section, the dataset, including 303 samples, is employed to train and
test the proposed deep learning approach. The influencing factors, including the mixture’s
constituents and the age concrete, are used to predict the compressive strength of SFRC. To
enhance the learning performance, the current dataset was normalized by the Z-score method.
The purpose of the normalization is to avoid the circumstance in which variables with large
magnitudes dominate those with small magnitudes. The Z-score method aims to standardize the
input features so that their distributions are characterized by a mean of 0 and a standard deviation
of 1. This approach can help reduce the influence of outliers in the dataset and facilitate the
adaptation of synaptic weights in DNNR during the training process. Based on the comparative
work of [63], the performance of the Z-score method can be better than that of the minimum–
maximum value based normalization approach. In addition, Fei et al. [64] show that the Z-score
standardization is highly helpful for improving the training performance of deep learning
models. Hence, this study has selected this data normalization method for pre-processing the
collected dataset. The Z-score method can be expressed by:
X
X
O
Z
X
X



 (16)
where Z
X and O
X are the standardized and the original variables, respectively. X
 and X

denote the mean and standard deviation of the original variable, respectively. Additionally, to
assess the predictive capability of the proposed DNNR models, the root mean square error
(RMSE), mean absolute percentage error (MAPE), and coefficient of determination (R2
) are
computed based on the models’ outcomes in the training and testing phases. The equations used
for calculating RMSE, MAPE, and R2
are given by:





N
i
i
i t
y
N
RMSE
1
2
)
(
1
(17)





N
i i
i
i
y
t
y
N
MAPE
1
|
|
100
(18)







 N
i
i
N
i
i
i
t
t
y
t
R
1
2
1
2
2
)
(
)
(
1 (19)
Where ti and yi are the actual and predicted compressive strength values, respectively. N is the
number of data points in the dataset at hand.
The Nadam algorithm is used to fit the DNNR models with respect to the standardized dataset.
Herein, the Sigmoid, Tanh, and ReLU are used as activation functions in the neurons in the
hidden layer. As mentioned earlier, the L1, L2, and weight decay are employed during the training
phases to regularize the magnitude of the synaptic weights. By doing so, the problem of network
overfitting can be alleviated. In addition, this study also investigated the performance of
combined network regularization strategies, including L1 combined with L2 (L1L2-DNNR), WD
combined with L1 (L1WD-DNNR), and WD combined with L2 (L2WD-DNNR). In total, there are
six DNNR models that employ different network regularization strategies. The DNNR models
were developed in the MATLAB programming environment and executed with the Dell G15
5511 (Core i7-11800H and 16 GB RAM).
Since Nadam belongs to the group of stochastic gradient descent algorithms, this optimizer
requires the specification of the batch size and the number of training epochs. The former
represents the number of data instances employed in one forward and backward pass through the
DNNR model. These data instances are necessary for computing the error gradient that is
subsequently used for updating the model’s synaptic weights. In general, larger batch sizes bring
about fast training speed, but the trained model may suffer from poor accuracy. On the contrary,
small batch sizes often yield good prediction accuracy. However, the training process using an
excessively small batch size may experience slow convergence due to random fluctuations in the
training data. In addition, one epoch contains a complete loop through all batches. The larger the
number of epochs, the more thoroughly the DNNR model is trained. Nevertheless, redundant
training epochs might only cause increases in computational expense without leading to
significant improvement in prediction accuracy. It is because excessively trained models often
suffer from overfitting. Notably, the selection of these two input parameters of the training
process is generally data-dependent. Therefore, in this study, the number of training epochs and
batch size are determined via trial-and-error runs with the collected dataset. Accordingly, the
deep learning models are trained over 1000 epochs with a batch size of 8.
It is noted that the tuning parameters of the DNNR models, including the number of hidden
layers, the number of neurons, and the activation function, are specified via five-fold cross-
validation processes. It is because cross validation has been proven to be a robust method for
model selection in machine learning [65]. Using the five-fold cross-validation process, the
dataset is randomly divided into five mutually exclusive folds. In each run, the model is trained

using four data folds; the other fold serves as testing samples. The quality of the model’s hyper-
parameters is evaluated by averaging the estimation errors obtained from the five testing folds.
Notably, the number of hidden layers in a deep learning model ranges from 2 to 6; the number of
neurons in a hidden layer varies from 6 to 20. In addition, the average RMSE obtained from the
five-fold cross-validation process is used to quantify the estimation errors.
Table 3
Parameter setting of the DL models.
The DNNR models Number of hidden layers Number of neurons Activation function
L1-DNNR 5 8 Sigmoid
L2-DNNR 4 10 Tanh
WD-DNNR 3 10 Sigmoid
L1L2-DNNR 3 16 ReLU
L1WD-DNNR 3 8 Tanh
L2WD-DNNR 3 10 ReLU
In addition, based on several trial experiments with the collected dataset, the suitable learning
rate and the regularization parameter are found to be 0.01 and 0.0001, respectively. The
configurations of the DNNR models are summarized in Table 3. As can be observed from the
results, the appropriate number of hidden layers ranges from 3 to 5. The number of neurons in
each hidden layer is between 8 and 16. The results also indicate that the suitable activation
function is dependent of the network regularization approach.
In the cases of the benchmark methods, similar cross-validation processes are employed to fine-
tune their hyper-parameters. The number of based models and the depth of a tree in RF are
searched in the ranges of [1, 100] and [1, 10], respectively. The regularization, kernel function,
and epsilon-tube parameters of an individual SVR vary in the ranges of [0.1, 1000], [0.01, 100],
and [0.001, 0.1], respectively. Moreover, the Adaboost based on SVR requires the setting of the
number of boosting iterations and the learning rate; these two hyper-parameters are fine-tuned in
the ranges of [10, 100] and [1, 10], respectively. For the case of LM-ANN, the maximum and
minimum numbers of neurons in the hidden layer are 6 and 20, respectively. Furthermore, the
learning rate of this neural computing model is searched in the range of [0.001, 0.1].
Based on the hyper-parameter setting processes, the RF model comprises 50 individual trees with
a tree depth of 5. The individual SVM model used in the Adaboost-SVR has the regularization
parameter C of 100, the kernel function parameter of 0.1, and the epsilon-tube parameter of 0.01.
Additionally, the Adaboost-SVR was built with 50 boosting iterations and a learning rate
parameter of 0.1. Accordingly, the LM-ANN consists of 12 neurons in the hidden layer and is
trained with a learning rate of 0.01.
4.2. Performance comparison
Based on the model configurations specified in the previous section, a repeated sampling
process, consisting of 20 independent runs, is executed. In each independent run, 10% of the
whole dataset is randomly extracted to create a testing set; the rest of the data is used for model
construction. This repeated sampling process aims to reduce the potential bias caused by

randomness in data selection and reliably evaluate the predictive capability of the machine
learning approaches. As mentioned earlier, the DNNR models were trained for 1000 iterations.
The training processes of the deep learning models using six regularization methods are
demonstrated in Fig. 3. In this figure, to better visualize the convergence property of each
method, the natural logarithm (log) of the loss function (L) is computed and shown in the y-axis.
The prediction outcomes of the DNNR models using different regularization methods are
depicted by scatter plots in Fig. 4. Observably, the DNNR models bring good fits to the
compressive strength of SFRC. The coefficient of determination R2
of the three models ranges
between 0.927 and 0.958. This means that the best deep learning model is able to explain roughly
96% variation in the compressive strength values.
Fig. 3. Training progresses of the deep learning models.
Table 4
Summary of DL-based prediction results.
DL models Indices
Phases
Training Testing
RMSE MAPE (%) R2
RMSE MAPE (%) R2
L1-DNNR
Mean 1.897 2.352 0.992 3.981 4.899 0.958
Std. 0.116 0.149 0.001 0.799 0.988 0.021
L2-DNNR
Mean 1.962 2.414 0.991 4.772 5.253 0.941
Std. 0.124 0.165 0.001 1.317 1.282 0.035
WD-DNNR
Mean 1.597 1.909 0.994 4.544 5.371 0.944
Std. 0.140 0.181 0.001 1.484 1.781 0.042
L1L2-DNNR
Mean 1.956 2.382 0.991 4.469 5.265 0.950
Std. 0.242 0.288 0.002 1.076 1.272 0.024
L1WD-DNNR
Mean 1.963 2.437 0.991 4.931 5.565 0.927
Std. 0.096 0.118 0.001 1.534 1.272 0.055
L2WD-DNNR
Mean 2.129 2.618 0.990 5.181 6.158 0.931
Std. 0.236 0.279 0.002 1.571 1.695 0.040

(a) (b)
(c) (d)
(e) (f)
Fig. 4. Prediction performances of the DL models using different regularization approaches: (a) L1, (b) L2,
(c) WD, (d) L1L2, (e) L1-WD, and (f) L2-WD.

Table 4 provides a summary of the predictive performance of the deep learning approaches. The
mean and standard deviation (Std.) of the aforementioned metrics (i.e., RMSE, MAPE, and R2
)
are computed for each model. Observed from Table 4, the L1 regularization method helps attain
the most desired performance with RMSE = 3.981, MAPE = 4.899%, and R2
= 0.958. The model
that uses both the L1 and L2 regularizations obtains the second-best outcome with RMSE =
4.469, MAPE = 5.265%, and R2
= 0.950. The performance of other methods is slightly worse
than that of the L1-DNNR.
Based on the experimental results, the L1 regularization deems best suited for the dataset at hand.
This outcome may be explained by the fact that this regularization scheme has the tendency to
construct sparse deep learning model, in which a certain number of the synaptic weights shrink
to small numbers that are close to zero. Sparse network architecture is less susceptible to noisy
data instances than densely-connected one [66]. Hence, this regularization method helps reduce
the unnecessary complexity of the model. Additionally, sparse network architecture is suitable
for modeling a dataset in which a certain number of data instances has similar characteristics
[31].
In other words, the features of the input data in different samples do not change drastically. The
dataset at hand is comprised of groups of data instances that contain information of SFRC mixes’
constituents. In many instances, underlying variables that are subjects to change are the ones
related to the dosage and the characteristics of the steel fibers. These facts may give insights into
the advantages of the L1 regularization for the current dataset.
Additionally, it is evident that the training errors are always larger than the testing errors. The
reason is that the testing sets comprise novel instances that were not encountered by the model.
However, the ratios of training to testing RMSE (RR) can be computed to express the
generalization property of each machine learning model. In general, excessively small values of
RR indicate the issue of overfitting. Overfitting occurs when the model perfectly learns the
patterns in the training dataset but performs poorly on the testing samples. On the contrary, a
value of RR close to 1 is highly desirable since it indicates that the overfitting problem is
alleviated. Accordingly, the higher the RR is, the more effective the regularization method is. The
results of RR attained from the deep learning models using the different regularization
approaches are summarized in Fig. 5. Observably, the L1 regularization has achieved the best RR
of 0.48. The second best result is RR = 0.48, which is attained by the hybrid L1L2 regularization.
Since the L1 regularization has attained the best performance, the DNNR model using this
regularization approach is selected for estimating the compressive strength of the SFRC. The
regularized deep learning method is then denoted as R-DNNR.
Moreover, the performance of the R-DNNR is compared with that of the benchmark approaches,
including the RF, Adaboost integrated with SVR (denoted as Adaboost-SVR), and LM-ANN.
The descriptions of those benchmark methods were provided in the previous sections of the
paper. The RF and AdaboostSVR were built with the functions provided in the Scikit-learn
toolbox [67]. The LM-ANN model was constructed via MATLAB’s neural network toolbox [68].

Fig. 5. Ratios of training to testing RMSE (RR) of different deep learning models.
The prediction results of the proposed R-DNNR and the benchmark models (RF, Adaboost-SVR,
and LM-ANN) are reported in Table 5. Generally, all of the models are capable of providing
fairly good fits to the observed compressive strength of SFRC. The coefficients of determination
of the model all surpass 0.85. In addition, the MAPE values are all less than 10%. This indicates
the good predictive capabilities of the employed machine learning models. As observed from this
table, the performance of the newly developed model in the testing phase is better than that of the
RF (RMSE = 5.793, MAPE = 7.817%, and R2
= 0.922), Adaboost-SVR (RMSE = 5.778, MAPE
= 7.004%, and R2
= 0.917), and LM-ANN (RMSE = 7.366, MAPE = 8.672%, and R2
= 0.853).
Table 5
Result comparison.
Phases Indices
RF Adaboost-SVR LM-ANN R-DNNR
Mean Std. Mean Std. Mean Std. Mean Std.
Training
RMSE 4.515 0.148 2.986 0.308 5.283 1.960 1.897 0.116
MAPE (%) 6.100 0.188 3.335 0.226 5.980 2.439 2.352 0.149
R2
0.954 0.004 0.980 0.004 0.930 0.050 0.992 0.001
Testing
RMSE 5.793 0.908 5.778 1.215 7.366 2.780 3.981 0.799
MAPE (%) 7.817 1.444 7.004 1.485 8.672 2.801 4.899 0.988
R2
0.922 0.024 0.917 0.034 0.853 0.121 0.958 0.021
In terms of RMSE, the proposed R-DNNR achieves roughly 31.3%, 31.1%, and 46%
improvements compared to the RF, Adaboost-SVR, and LM-ANN, respectively. Considering the
index of MAPE, the result improvement of the deep learning model compared with the
benchmark approach ranges between 30% and 43.5%. The aforementioned comparison is further
depicted in Fig. 6. Furthermore, the variations in the prediction errors of the models are
demonstrated by the boxplots in Fig. 7. It is noted that the median, denoted by a red line in Fig.
7, of the proposed deep learning method is significantly below that of the benchmark models.
Moreover, the Wilcoxon signed-rank test [69] with a p-value of 0.05 is also employed to verify
the significance of the result improvements. This is a non-parametric test that is widely used for
comparing prediction performance in machine learning. The pairwise comparisons of R-DNNR

vs. RF, Adaboost-SVR, and LM-ANN are found to be less than 0.05. The found p-values help
reject the null hypothesis of equal means and confirm the superiority of the R-DNNR model.
Fig. 6. Improvement in prediction accuracy achieved by the newly developed DNNR.
The regression error characteristic curve (REC), proposed in [70], is also used for appraising the
prediction performance of the proposed R-DNNR and the benchmark methods. A plot of REC
typically places the absolute residual on its x axis. The y axis shows the percentage of samples
having absolute errors smaller than the corresponding value on the x axis [71]. Hence, this plot
can be used to graphically assess the cumulative distribution of a machine learning model’s
prediction error [72]. Fig. 8 demonstrates the RECs of the R-DNNR, RF, Adaboost-SVR, and
LM-ANN models. As can be seen from this figure, the REC of the R-DNNR features the
uppermost position. This fact indicates that a large proportion of the prediction error has a small
magnitude. To quantify the model performance based on the REC, area under the curve (AUC)
can be computed. In general, the AUC varies from 0 to 1. In general, the larger the AUC is, the
better the predictive performance is. Based on AUC calculations, the R-DNNR has attained the
best outcome of 0.94. The AUCs of the Adaboost-SVR, RF, and LM-ANN are 0.92, 0.91, and
0.89, respectively.
Fig. 7. Box plots of models’ performance in terms of RMSE.

Fig. 8. Regression error characteristic curves of the proposed deep learning and the benchmark methods.
(a) (b)
(c) (d)
Fig. 9. Distribution of residual range (r): (a) R-DNNR, (b) RF, (c) Adaboost-SVR, and (d) LM-ANN.

In addition, more details regarding the distribution of the absolute residuals of the prediction
models are provided in Fig. 9. It can be seen from this figure that 65.83% of the deviations of the
R-DNNR are below the threshold value of 5%. Using such a threshold, the corresponding
proportions of prediction deviations obtained from the RF, Adaboost-SVR, and LM-ANN are
47.74%, 55.32%, and 46.33%, respectively. Moreover, the proportions of samples having
deviations smaller than 10% are 86.48%, 76.93%, 79.84%, and 74.33% for the R-DNNR, RF,
Adaboost-SVR, and LM-ANN, respectively. Notably, only 1.83% of the prediction deviations of
the R-DNNR exceed 20%. Meanwhile, this proportion of the data for the benchmark models
ranges between 6.29% and 8.67%. These results point out that the proposed R-DNNR is highly
suited for the task of estimating the compressive strength of SFRC.
Fig. 10. FAST-based variable importance.
In addition, the Fourier Amplitude Sensitivity Test (FAST) [73] is employed in this study to
evaluate the influences of predictor variables on the strength performance of SFRC. FAST is a
global sensitivity analysis approach; this method can be used to obtain the variation in the
model’s output caused by the changes in its input variables. The result of this analysis can be
presented as sensitivity values that quantify the relative importance of the variables. In this study,
FAST is implemented with the help of the MATLAB toolbox provided in [74]. The sensitivity
analysis outcomes are summarized in Fig. 10. Observed from the figure, the variables X2 (water
content), X3 (fine aggregate content), X12 (curing age), and X8 (dosage of superplasticizer) are in
the group of the most influential variables. Among the factors related to the steel fiber
reinforcement, the variable X9 (the volume fraction of steel fiber) has the strongest influence on
the target output. The contribution of X11 (the diameter of the steel fiber) and X10 (the length of
steel fiber) is less than that of the volume fraction of steel fiber. In addition, the contents of slag
(X7) and coarse aggregate (X4) exhibit minor influence on the deep learning model’s predictions.
Nevertheless, since the sensitivity values of all the variables are not null, they should be
incorporated into the compressive strength prediction model.
4.3. Analysis of the residuals of the proposed deep learning approach
This section is dedicated to investigating the characteristics of the residual of the R-DNNR
model. The magnitude of the residuals, which are demonstrated as their absolute values, is
presented in Fig. 11. Observed from this figure, the average residual is 2.94 MPa. The maximum

(max) and minimum (min) values are 22.79 MPa and 0.01 MPa, respectively. Furthermore, Fig.
12 demonstrates the residuals when their sign is taken into account. It is noted that the current R-
DNNR utilizes the SEL, which is an asymmetric loss function, during its training process.
Hence, the proportions of positive and negative residuals are expected to be similar. By
inspecting the residual values, it is found that the percentages of positive and negative cases are
49.17% and 50.83%, respectively. The distribution of residuals is further demonstrated in Fig.
13. The mean, standard deviation, and skewness of this distribution are 0.044, 4.059, and 0.462,
respectively.
Fig. 11. Absolute residuals of the DL method (average = 2.94, max. value = 22.79, min. value = 0.01).
Fig. 12. Residuals of the DL approach.
Fig. 13. Residual distribution of the DL method.
However, when the problem of predicting the compressive strength of SFRC is considered, the
cases in which the actual strength values (t) are greater than the predicted strength values (y) are

highly desirable. It is because they represent safe predictions. Because the residual e is calculated
as t – y, positive residuals are more desirable than negative residuals. Due to this reason, there is
a strong motivation to revise the learning process of the R-DNNR so that the model commits
fewer negative residuals. This study proposes the utilization of the asymmetric SEL, denoted as
ASEL, in the training phase of the R-DNNR. Based on the asymmetric nature of this loss
function, the structure of the deep learning model tends to reduce the number of overestimated
compressive strengths. The R-DNNR using the ASEL is then denoted as R-DNNR-ASEL. To
ease the model comparison process, the deep learning model using the conventional SEL is
denoted as R-DNNR-SEL in this section.
As demonstrated in the previous section (refer to Eq. 14 and Eq. 15), the ASEL depends on the
parameter , which controls its degree of asymmetry. This parameter should be set empirically
based on the dataset at hand [38]. An appropriate value of  is selected so that the number of
prediction outcomes having negative residuals is as small as possible and the overall predictive
capability of the R-DNNR is still satisfactory, i.e., the coefficient of determination (R2
) > 0.95.
Various values of this tuning parameter, which varies from 2 to 30, were used to train the R-
DNNR-ASEL model. Based on the pre-specified threshold of R2
, the suitable value of  is found
to be 5. Accordingly, the percentage of positive residuals significantly increases from 49.17% to
72.92%. In addition, the R2
of the R-DNNR-ASEL model is 0.954, which is slightly smaller than
that of the R-DNNR-SEL.
Fig. 14. Residuals of the R-DNNR using ASEL (percentage of positive residuals = 72.92% and
percentage of negative residuals = 27.08%).
Fig. 15. Residual distribution of the R-DNNR using ASEL (mean = 2.164, std. = 4.009, skewness =
0.616).

Table 6
Comparison between the R-DNNR-SEL and R-DNNR-ASEL models.
Phase Indices
R-DNNR
Using SEL Using ASEL
Mean Std. Mean Std.
Training
RMSE 1.897 0.116 4.154 0.095
MAPE (%) 2.352 0.149 5.053 0.127
R2
0.992 0.001 0.961 0.002
Testing
RMSE 3.981 0.799 4.519 0.575
MAPE (%) 4.899 0.988 5.481 0.575
R2
0.958 0.021 0.954 0.011
The plot of residuals and their distribution yielded by the R-DNNR-ASEL are shown in Fig. 14
and Fig. 15, respectively. Observably, the mean of the new residual distribution changes from
0.044 to 2.164. The residual distribution of the R-DNNR-ASEL is shifted to the right so that
there are more cases of underestimations. Notably, the standard deviation of the new distribution
(4.009) is very close to that of the previous distribution (4.059). This fact implies that using the
ASEL does not substantially alter the amount of dispersion of the residual distribution.
A comparison of the two deep learning models in terms of prediction accuracy is summarized in
Table 6. In general, the prediction accuracy of the model slightly deteriorates when the ASEL is
applied. This result may be explained by the fact that the DNNR model using SEL only focuses
on minimizing the discrepancy between the actual and estimated compressive strength values.
However, the training phase of the DNNR model using ASEL has an additional goal, which is
restricting the occurrence of overestimations. Therefore, the use of ASEL in training phase
requires a certain degree of compromise in the overall prediction accuracy.
Fig. 16. The prediction results of the deep learning model trained with ASEL.
However, the prediction performance of the R-DNNR-ASEL is still acceptable, with RMSE =
4.519, MAPE = 5.481%, and R2
= 0.954. Notably, this prediction performance is still better than
that of the benchmark approaches. In addition, the proportion of overestimated cases was
reduced from 50.83% to 27.08%. This fact shows the effectiveness of the ASEL when it is used
in the training phase of the deep learning model. In addition, the prediction outcome of the R-
DNNR-ASEL is illustrated via the scatter plot in Fig. 16. Observably, the majority of the data

samples are under the line of best fit. This means that the model tends to underestimate the
compressive strength values instead of overestimating them. Moreover, to facilitate the
application of the proposed deep learning model, a graphical user interface (GUI) (refer to Fig.
17) was built and stored at the GitHub repository. As shown in the GUI, the user can provide
input information regarding the content of a concrete mix and concrete age. The program is able
to provide estimations of the compressive strength value corresponding to the selected loss
function (either SEL or ASEL).
Fig. 17. Graphical user interface of the DNNR program used for estimating the compressive strength of
SFRC.
5. Conclusions
This study proposes and verifies a novel deep learning method for predicting the compressive
strength of SFRC. Deep learning models, characterized by the stacking of multiple hidden layers,
have been found to be appropriate for the task at hand. The newly developed R-DNNR model is
capable of generalizing the complex functional mapping between compressive strength and its
influencing factors. A dataset, consisting of 12 input variables and 303 samples, was used to train
and test the deep learning model. The input variables include the basic constituents of a concrete
mix and information regarding the steel fiber content (i.e., the volume fraction, length, and
diameter of the fiber).
The state-of-the-art Nadam algorithm is employed to adapt the parameters in the DNNR.
Moreover, six regularization schemes are employed in the training phases of the deep learning
model. Experimental results, which are supported by statistical tests, show that the R-DNNR

achieves the most desired performance with a RMSE of 3.98, a MAPE of 4.9%, and a R2
of 0.96.
These outcomes are significantly better than those of the benchmark methods by a large margin.
The proposed R-DNNR achieves roughly 31% and 30% improvements in terms of the RMSE
and MAPE, respectively. Moreover, the implementation of the ASEL in the R-DNNR further
helps reduce the proportion of overestimated cases from 50.83% to 27.08%. Accordingly, the R-
DNNR can deliver accurate and reliable estimations of the compressive strength of SFRC
mixtures.
Nevertheless, the current work also has a number of limitations. The first drawback is that the
newly developed method can only deliver point estimations of the compressive strength values
of SFRC. It is noted that there are various sources of uncertainty associated with the prediction
outcomes, e.g., the variations in physical properties of the mix’s constituents and curing
conditions. In addition, considering the complexity of the problem at hand, the size of the current
dataset is still limited. Hence, the implementation of the proposed R-DNNR should be restricted
to the existing ranges of the variables in the collected dataset. The prediction results for input
variables beyond these ranges may not be reliable and should be taken with a grain of salt.
Since deep learning generally requires large datasets to sufficiently ensure robust model
developments, the generalization of R-DNNR can be considerably enhanced when more training
samples are available. It is because additional data instances associated with actual experimental
results can help the model to explore regions in the learning space that are currently sparse. It is
noted the current model’s operation, trained and tested with 303 samples, can be accomplish in
roughly 11 sec. With high computing power and the increasing development of graphics
processing unit (GPU), the current deep learning framework can maintain its performance as the
size of the dataset enlarges. The scalability of R-DNNR for real-world applications can be
affirmed.
Accordingly, the current study can be extended in a number of directions. First, the capabilities
of other advanced optimizers in training deep neural network models can be investigated. By
doing so, algorithms that are suitable for the task of estimating the compressive strength of SFRC
can be identified. Second, the number of samples in the current dataset is still limited; the dataset
should be enlarged to enhance the generalization and applicability of the machine learning
model. Third, the current model can only deliver point estimations of the compressive strength; it
is highly desirable for practitioners if the deep learning model can yield range estimations of the
compressive strength.
Thus, incorporating advanced methods to compute the prediction intervals of the concrete
strength can be another potential research direction for the current work. Fourth, the current deep
learning approach can be applied to predict toughness, which is a crucial parameter of SFRC. In
addition, experimental results regarding the use of other cementitious replacement materials
(e.g., ground granulated blast furnace slag or rice husk ash) in SFRC can be incorporated into the
current dataset. It is because predictions of the mechanical properties of SFRC using these
environmentally friendly materials can help promote the sustainability of the concrete production
industry.

Supplementary material
The dataset and the program used to support the findings of this study have been deposited in the
GitHub repository at https://github.com/NHDDTUEDU/Nadam_DNNR_CS_SFRC.
Funding
This research received no external funding.
Conflicts of interest
The authors declare no conflict of interest.
Authors’ contribution statement
NHD, VDT: Conceptualization; NHD: Data Collection; NHD: Methodology; NHD, VDT:
Software; NHD: Writing – original draft; NHD, VDT: Writing – review & editing.
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