# Advancing ultimate bond stress-slip model of UHPC structures through a novel hybrid machine learning approach

Ahad Amini Pishro, Shiquan Zhang, Qixiao Hu, Zhengrui Zhang, Mojdeh

Amini Pishro, Lili Zhang, Alain L'hostis, Farzad Hejazi, Yuetong Liu, Yuandi

Zhao

► To cite this version:

Ahad Amini Pishro, Shiquan Zhang, Qixiao Hu, Zhengrui Zhang, Mojdeh Amini Pishro, et al. Advancing ultimate bond stress-slip model of UHPC structures through a novel hybrid machine learning approach. Structures, 2024, 62, pp.106162. 10.1016/j.istruc.2024.106162. hal-05175148

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# Advancing ultimate bond stress-slip model of UHPC structures through a novel hybrid machine learning approach

Ahad Amini Pishro<sup>a</sup>, Shiquan Zhang<sup>b</sup>, Qixiao Hu<sup>b</sup>, Zhengrui Zhang<sup>a, *</sup>, Mojdeh Amini Pishro<sup>c</sup>, Lili Zhang<sup>a</sup>, Alain L'Hostis<sup>d</sup>, Farzad Hejazi<sup>e</sup>, Yuetong Liu<sup>b</sup>, Yuandi Zhao<sup>a</sup>

## ARTICLE INFO

Keywords:

Bond Stress

UHPC

PINN

GA

MLR

Machine Learning

## ABSTRACT

The significance of Ultimate Bond Stress-Slip (UBS-S) in reinforced Ultra-High Performance Concrete (UHPC) structures cannot be overstated, as it directly affects their load-carrying capacity, structural integrity, and long-term performance. A comprehensive analysis of the UHPC-Parallel Micro Element System (UHPC-PMES), including 144 specimens, evaluated the computational efficiency of the proposed UBS-S model. To this end, the most critical settings of steel bar diameter, concrete cover, bond length, and UHPC compressive strength ($d_b, c, l_b, f'_{UHPC}$) were directed to create this parametric research. Applying the hybrid approach of three different optimization techniques, namely Physics-Informed Neural Networks (PINN), Genetic Algorithms (GA), and Multiple Linear Regression (MLR), this study predicted the UBS-S at the interface of UHPC and steel bars. It formulated the hyper-parameters effect values $(\alpha, m, \beta, G)$. The presented research used these algorithms to solve an inverse problem in structural engineering. Comparing the results obtained from PINN, GA, and MLR demonstrated that machine learning techniques and the proposed PMES model could effectively and accurately investigate the ultimate bond stress-slip for reinforced UHPC structures.

## 1. Introduction

Ultra High-Performance Concrete (UHPC) has gained significant popularity over the past few decades due to its exceptional compressive strength. This has made it a preferred choice for many structural engineering projects. Various methods have been employed further to study the Ultimate Bond Strength (UBS)-Slip equations, including experimental studies, numerical analysis, and theoretical modeling using Multiple Linear Regression (MLR). Experimental studies have provided valuable insights into the behavior of UHPC under different loading conditions, while numerical analysis has allowed for the prediction of the material's response in complex structural systems. Theoretical models built through MLR have also helped predict the behavior of UHPC and its bond strength with reinforcing materials.

With the rising prominence of Ultra High-Performance Concrete (UHPC) owing to its remarkable compressive strength, research efforts have delved into comprehending the Ultimate Bond Strength (UBS)-Slip

equations through diverse methodologies, encompassing experimental studies, numerical analysis, and theoretical modeling using Multiple Linear Regression (MLR). The ultimate bond strength-slip model is a popular analytical tool used to predict the behavior of reinforced concrete structures. The model assumes that a stress-slip relationship characterizes the bond between the reinforcement and the surrounding concrete. This model delineates how applied stress influences the relative slip between the reinforcement and concrete. Key parameters include the ultimate bond strength, representing the maximum bond capacity before failure, and factors like concrete strength, reinforcement surface conditions, and development length that influence the bond behavior. The lateral displacement, deflections, and overall structural capacities of reinforced concrete (RC) members are intricately linked to the ultimate bond stress-slip behavior exhibited at the interface between the steel reinforcing bars and the surrounding concrete. Understanding the nuanced interplay between the applied stress on the reinforcement and the resulting slip is crucial for predicting the lateral behavior and

<footnote><sup>a</sup> School of Civil Engineering, Sichuan University of Science and Engineering, Zigong 643000, China</footnote>

<footnote><sup>b</sup> School of Mathematics, Sichuan University, Chengdu 610065, China</footnote>

<footnote><sup>c</sup> School of Architecture and Design, Southwest Jiaotong University, Chengdu 610031, China</footnote>

<footnote><sup>d</sup> École des Ponts, LVMT, Université Gustave Eiffel, F-77454 Marne-la-Vallée, France</footnote>

<footnote><sup>e</sup> School of Environment and Technology, University of the West of England, Bristol BS16 1QY, UK</footnote>

<footnote>* Corresponding author.
E-mail address: zhangzhengrui@suse.edu.cn (Z. Zhang).</footnote>
<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 1. Specimen characteristics and pullout test based on RILEM standards [1–5,21].</figcaption></figure>

overall deformations of RC elements. Engineers utilize this model to assess and optimize the bond performance in structural elements, which ensures the integrity and safety of reinforced concrete structures under various loading conditions [1–4].

In this research, a parametric study was conducted to investigate the effective parameters on the behavior of reinforced concrete structures using the Ultimate Bond Stress–Slip model. The study utilized machine learning algorithms to generate predictive models for the stress-slip relationship, considering the various parameters. The study analyzed a dataset of experimental and numerical data obtained from previous studies [1–5], allowing machine learning algorithms to identify patterns and relationships between the parameters and the behavior of the structures.

Simulating bond stress-slip in concrete structures is crucial for optimizing performance, informed design, and safety. The Parallel

Micro-Element System (PMES) is a numerical model used to simulate the UBS-S relationship in reinforced UHPC structures. This model utilizes a parallel approach where a group of micro-elements is arranged to represent the concrete material and a group of fibers is added to model the reinforcement. The slip at the interface between the concrete and the reinforcement is then calculated using the bond-slip model. The main advantage of the parallel micro-element system is its ability to provide accurate predictions of the bond behavior in concrete structures. This model can also simulate the effects of various parameters, such as material properties, geometric dimensions, and loading conditions, on the bond behavior. The results obtained from this model can be used to design and optimize reinforcement detailing in concrete structures. Several studies have investigated the effectiveness of the parallel micro-element system in predicting the bond stress-slip behavior of concrete structures [5–7].

The parallel micro-element system's precision in simulating bond stress-slip relationships aligns with the contemporary proliferation of machine learning (ML) across industries, including structural engineering. The application of ML in structural engineering, specifically the Ultimate Bond Stress–Slip (UBS-S) model, has shown promising results in predicting the behavior of reinforced concrete structures. As machine learning advances, it is expected to play an increasingly important role in structural engineering and other industries. Predicting bond stress in

<table><caption>**Table 1**
The UHPC mix designs (kg/m³) [1–5,23].</caption><thead><tr><th>Cement</th><th>Quartz Sand</th><th>Quartz Powder</th><th>Micro Silica</th><th>Nano Silica</th><th>Water</th><th>Super plasticizer</th></tr></thead><tbody><tr><td>665</td><td>1020</td><td>285</td><td>156.775</td><td>43.225</td><td>178</td><td>23</td></tr></tbody></table>
<table><caption>**Table 2**
Specimens data [1–5].</caption><thead><tr><th>Specimen</th><th>d<sub>b</sub> (mm)</th><th>C (mm)</th><th>L (mm)</th><th>f<sub>c</sub> (MPa)</th><th>f<sub>ct</sub> (MPa)</th><th>Bond Force (kg)</th><th>UBS (MPa)</th></tr></thead><tbody><tr><td>R12C4L1F155</td><td>12</td><td>48</td><td>12</td><td>155.1</td><td>6.85</td><td>3949.75</td><td>85.65</td></tr><tr><td>R12C4L2F155</td><td>12</td><td>48</td><td>24</td><td>155.1</td><td>6.85</td><td>7644.68</td><td>82.89</td></tr><tr><td>R12C1L4F155</td><td>12</td><td>12</td><td>48</td><td>155.1</td><td>6.85</td><td>6972.96</td><td>37.80</td></tr><tr><td>R14C4L1F155</td><td>14</td><td>56</td><td>14</td><td>155.1</td><td>6.85</td><td>4977.83</td><td>79.31</td></tr><tr><td>R14C4L2F155</td><td>14</td><td>56</td><td>28</td><td>155.1</td><td>6.85</td><td>9634.50</td><td>76.75</td></tr><tr><td>R14C1L4F155</td><td>14</td><td>14</td><td>56</td><td>155.1</td><td>6.85</td><td>8787.94</td><td>35.00</td></tr><tr><td>R16C4L1F155</td><td>16</td><td>64</td><td>16</td><td>155.1</td><td>6.85</td><td>6239.49</td><td>76.11</td></tr><tr><td>R16C4L2F155</td><td>16</td><td>64</td><td>32</td><td>155.1</td><td>6.85</td><td>12,059.51</td><td>73.55</td></tr><tr><td>R16C1L4F155</td><td>16</td><td>16</td><td>64</td><td>155.1</td><td>6.85</td><td>10,979.08</td><td>33.48</td></tr><tr><td>R18C4L1F155</td><td>18</td><td>72</td><td>18</td><td>155.1</td><td>6.85</td><td>6878.00</td><td>66.29</td></tr><tr><td>R18C4L2F155</td><td>18</td><td>72</td><td>36</td><td>155.1</td><td>6.85</td><td>13,272.02</td><td>63.96</td></tr><tr><td>R18C1L4F155</td><td>18</td><td>18</td><td>72</td><td>155.1</td><td>6.85</td><td>11,747.93</td><td>28.31</td></tr><tr><td>R20C4L1F155</td><td>20</td><td>80</td><td>20</td><td>155.1</td><td>6.85</td><td>7782.97</td><td>60.76</td></tr><tr><td>R20C4L2F155</td><td>20</td><td>80</td><td>40</td><td>155.1</td><td>6.85</td><td>15,074.39</td><td>58.84</td></tr><tr><td>R20C1L4F155</td><td>20</td><td>20</td><td>80</td><td>155.1</td><td>6.85</td><td>13,411.94</td><td>26.18</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Research structure.</figcaption></figure>

concrete structures is critical in designing and analyzing reinforced concrete members. In recent years, ANNs have shown promising results in predicting bond stress due to their ability to learn complex nonlinear relationships from data. Xiaoming You et al. [8] applied a hybrid ML to estimate the bond stress at the interface of UHPC and steel bars. The analysis of several models has shown that the proportion between the depth of the embedded material and the diameter of the reinforcing bar plays a crucial role in determining the bond strength of UHPC. Their finding was consistent with the results obtained through experiments. Zhijie Li et al. [9] utilized machine learning techniques to predict the bond strength between reinforcing bars and UHPC. The statistical significance of various factors derived from tree models consistently indicated that the ratio between the depth of embedded material and the diameter of reinforcing bars substantially affected the bond strength of UHPC. This finding aligned with the experimental observations. Abdulwarith Ibrahim Bibi Farouk et al. [10] applied a mixed machine learning method to forecast and measure the degree of uncertainty of the Ultimate Bond Strength (UBS) between UHPC and reinforcing steel bars. The uncertainty of the UBS was quantified through Monte Carlo simulations that considered geometrical and material configuration randomness. The simulation results showed that the yield strength of the steel rebar had the most sensitivity, while the embedment length had the

most negligible impact on the bond strength. Xianlin Wang et al. [11] predicted the bond strength of steel structures encased in concrete using machine learning methods. Quantitatively, it was confirmed that the relative concrete cover had the most significant effect on bond strength, while the relative bonded length had a relatively minor impact.

The confirmed impact of factors like relative concrete cover, bond length, steel bar size, and concrete compressive strength highlights the crucial role of interfacial bond behavior between fiber-reinforced polymers (FRPs) and concrete. This relationship becomes pivotal in assessing the overall effectiveness of FRP strengthening in diverse applications, including flexural strengthening of reinforced concrete (RC) beams and columns, where FRPs are increasingly used for reinforcement or retrofitting. Cheng Yuan et al. [12] studied ML techniques to develop efficient regressors to determine the bond-slip model for the interface between FRP and concrete. To validate the accuracy of the model predictions, a sophisticated numerical simulation using LS-DYNA was employed to simulate the interfacial fracture process. The defined bond-slip model was then used as input to the contact relationship between FRP and concrete, and the resulting load-slip curves were compared with experimental data to verify the accuracy of the proposed method. Feng Zhang et al. [13] predicted the UBS at the interface between FRP and concrete. A novel and interpretable machine learning
<table><caption>**Table 3**
Micro Element Systems under monotonic loading [5].</caption><thead><tr><th>Abstract Depiction</th><th>Conceptual Representation</th></tr></thead><tbody><tr><td>UBS</td><td>LBS<sub>UHPC</sub> = 12.056 √(cf<sub>UHPC</sub>/d<sub>b</sub>) - 0.152 (b<sub>d</sub>/d<sub>b</sub>) √(f<sub>UHPC</sub>) (1)</td></tr><tr><td>H(x)</td><td>Switch = H(s - Δ) = { if s < Δ⇒switch open, 0 (2) if s ≥ Δ⇒switch close, 1 (3) τ<sub>el</sub> = G.s.s < Δ (3)</td></tr><tr><td>UHPC-SMES<br>Elastic Phase<br>[Before the spring's failure]</td><td>τ<sub>r</sub> = βτ<sub>ul</sub> = βGΔ.s ≥ Δ (4)</td></tr><tr><td>UHPC-SMES<br>Inelastic Phase<br>[After the spring's failure]</td><td>τ = H(Δ - s)Gs + H(s - Δ)βGΔ (5)</td></tr><tr><td>UHPC-SMES<br>Comprehensive Phase<br>Phases [Elastic<br>+ Inelastic]</td><td>τ<sub>k</sub> = H[Δ(x<sub>k</sub>) - s]Gs + H[s - Δ(x<sub>k</sub>)]β<sub>k</sub>GΔ (6)</td></tr><tr><td>UHPC-PMES<br>General Concept<br>k-th micro-element<br>(x<sub>k</sub>)</td><td>τ<sub>el</sub> : UBS in elastic phase; G : UBS-S stiffness; β : Friction coefficient; τ<sub>ul</sub> : Spring's failure stress.</td></tr></tbody></table>
<figure><img src="image_5.png" /><figcaption>Fig. 3. The UBS-S relationship provided by PINN, GA, and pullout tests [5].</figcaption></figure>

method for predicting the interfacial bond strength between FRP and concrete was developed in their study by combining machine learning models with traditional physical models. Ahad A. Pishro et al. [14,15] applied the ANN to predict and verify the structural capacity of different FRP-retrofitted RC beams, including the interfacial bonding between the FRP and concrete. Their results revealed the ANN application's accuracy in estimating RC members' structural performance.

The concept of Partial Differential Equations (PDE), involving derivatives of an unknown function concerning multiple variables, underpins the mathematical formulations and models, like the UBS-S model, used to describe and analyze the intricate bond behavior in UHPC structures. One application of PDEs in structural engineering is to model the bond stress between concrete and steel bars. By solving the PDE, we can predict the bond stress behavior and optimize the design of reinforced concrete structures. Recently, Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving PDEs in engineering applications. PINNs integrate neural networks with physics-based models to accurately predict the behavior of complex systems. PINNs can be trained on a dataset of experimental measurements to learn the underlying physics and predict the bond stress behavior under different loading conditions in the context of bond stress between concrete and steel bars. Therefore, combining PDEs and PINNs offers a promising approach to accurately modeling and predicting the bond stress behavior in reinforced concrete structures [16–19].

The application of PINNs has gained widespread acceptance and recognition in structural engineering, emerging as a well-established and effective computational tool for modeling and simulating complex physical behaviors, optimizing structural designs, and facilitating the analysis of diverse structural systems under varying loading conditions. However, researchers have also recognized the effectiveness of the Genetic Algorithm (GA) in engineering studies. Genetic Algorithms (GA) have been increasingly explored as a powerful tool for predicting bond

stress in concrete structures. GA is a search and optimization technique inspired by natural selection and genetics, which can efficiently explore the solution space and find the optimal solution. GA-based models can efficiently explore the solution space and optimize the input parameters of machine learning or regression models, resulting in improved prediction accuracy. These methods can potentially improve the design and assessment of concrete structures and enhance their safety and durability [16]. Xiaoyong Lv et al. [7] have utilized the GA to search for unknown parameters and develop a stochastic damage constitutive model for investigating the bond stress-slip relationship. They created a micro-element model consisting of spring, friction, and switch elements and introduced a parallel system of these micro-elements to model the Representative Interface Element (RIE). This approach reflects the randomness of the interface's micro-fracture and considers the spring element's fracture threshold as a random variable with a given Probability Density Function (PDF) and a given Cumulative Distribution Function (CDF). This approach aims to search for the unknown parameters of a given CDF, which is an inverse problem with available data. In a separate study [6], they utilized a two-parameter Weibull CDF and used algorithms for hyper-parameter search to estimate the unknown parameters.

Regarding the promising approach of ML in structural engineering, this research aims to apply the MLR method to create the regression equations for the UBS-S model. To this end, a comprehensive database from a series of experimental pullout tests based on ASTM C234–91a and RILEM standards [20,21] was verified by the FE software ABAQUS [22] to train our PINN and GA algorithms. Applying the PEMS model gave a clear presentation of the UBS-S mechanism at the UHPC-Steel bar interface. Finally, we investigated the hyper-parameters' effectiveness and proposed a series of MLR equations for the applied PINN and GA.

In this investigation, a parametric analysis was carried out to explore the influential factors affecting the performance of reinforced UHPC
<figure><img src="image_6.png" /><figcaption>Fig. 4. PINN and GA loss function curves [5].</figcaption></figure>

structures, employing the UBS-S model. Leveraging machine learning algorithms, predictive models for the stress-slip relationship were developed, considering a range of parameters. This approach provides valuable insights into the intricate dynamics of the bond stress-slip relationship and contributes to a more nuanced understanding of the behavior of reinforced UHPC structures, ultimately informing optimized design practices.

## 2. Micro element systems for UHPC structures

This study applied a dataset of 144 specimens tested experimentally and numerically following the ASTM and RILEM standards [1–5]. The specimens were named based on critical parameters such as steel bar diameter, concrete cover, bond length, and concrete compressive strength. The designations of the specimens correspond to specific random parameters, illustrated by codes like R20C2L3F155. In this code, R signifies the rebar diameter ($d_b = 20mm$), C indicates the concrete cover on the rebar ($c = 2 \times 20 = 40mm$), L represents the bond length ($l_b = 3 \times 20 = 60mm$), and F denotes the compressive strength of UHPC ($f'_c = 155MPa$). These parameters were thoughtfully selected to establish a comprehensive dataset for examining the bond strength between steel bars and UHPC. The specimens underwent progressive loading until failure, and their effectiveness was assessed through a pullout test. This approach thoroughly understood the bond strength between steel bars and UHPC under monotonic loading conditions. Fig. 1 illustrates the specimen structure and the pullout test according to the RILEM standards [1–5,21].

The UHPC was produced at the civil engineering laboratory of Sichuan University. After checking and testing different materials and designs for this study, we adopted the mix design proposed by Schneider Jianxin [23] for UHPC materials, as shown in Table 1. The polycarboxylate-based superplasticizer and Portland cement Type I with a strength class of 525 kg/cm² were used. The micro-silica and

nano-silica had average particle sizes of approximately 1 µm and 60 nm, respectively. The steel bars utilized in this investigation had a yield strength of 551 MPa. Achieving the desired compressive strength of UHPC involved thermal curing for 72 h at a temperature of 60 °C and 95% moisture [1–5].

Table 2 presents a general overview of the metrics and the UBS values corresponding to some random specimens.

This study conducted pullout tests to investigate the Ultimate Bond Strength (UBS). The data was analyzed using parametric Finite Element Method (FEM) and theoretical modeling, including Physics-Informed Neural Network (PINN) and Genetic Algorithm (GA). Multiple Linear Regression (MLR) was used to establish relationships between parameters. The study provided insights into the mechanical characteristics of the UHPC-Parallel Micro Element System (UHPC-PMES) and validated the results against experimental data [5]. Fig. 2 presents the research structure applied in this study.

The study introduces a Single Micro Element System (SMES) model to replicate the mechanical characteristics of the interface between steel bars and Ultra-High-Performance Concrete (UHPC). The SMES consists of a spring, friction, and switch element, representing interlocking, cracking, and friction between the materials. It enables analysis of the interface under monotonic loading and provides insights into its mechanical properties. The UHPC-SMES system transitions from elastic deformation to friction through a switch controlled by the Heaviside function algorithm. The random variable theory is employed to characterize failure events, ensuring controlled behavior. The UHPC-SMES helps understand the steel bar-UHPC interface mechanics and develop strategies to prevent debonding and cracking. The UHPC-Parallel Micro Element System (UHPC-PMES) extends this model, offering a detailed understanding of the Ultimate Bond Stress-Slip (UBS-S) relationship. Table 3 outlines the whole structure of micro-element systems for the UHPC structures subjected to monotonic loading [5].

Algorithm 1 presents the calculation of the UBS-S damage factor at the steel bar-UHPC interface:
**Algorithm 1.** : Damage of UBS-S.

* **Step 1:** Formulate the average ultimate bond stress of UHPC-PMES ($\tau_{ave_{PMES}}$) for ($k = 1, 2, 3, ..., K$):

$$ \tau_{ave_{PMES}} = \frac{1}{K} \sum_{k=1}^{K} H[\Delta(x_k) - s]Gs + \frac{1}{K} \sum_{k=1}^{K} H[s - \Delta(x_k)]\beta_k G\Delta(x_k) \quad (7) $$

* **Step 2:** Establish the infinity concept for Equation (7).

$$ \tau_{PMES} = \int_{0}^{1} H[\Delta(x) - s]Gs \, dx + \int_{0}^{1} H[s - \Delta(x)]\beta G \Delta \, dx \quad (8) $$

* **Step 3:** Consider the damage to the steel bar-UHPC interface when the spring element fails. Create an equation to calculate the damage variable ($d$) in the UHPC-PMES system. The area of damage is indicated by $A_d$, while the total area of the PMES system is represented by $A_{PMES}$.

$$ d = \frac{A_d}{A_{PMES}} = \frac{1}{K} \sum_{k=1}^{K} H[s - \Delta(x_k)] \quad (9) $$

* **Step 4:** Establish a method for approaching infinity in Equation (9).

$$ d(s) = \int_{0}^{1} H[s - \Delta(x)]dx \quad (10) $$

* **Step 5:** Derive Equation (11) as the equation for calculating damage in the UBS-S relationship where ($\tau$) is the ultimate bond stress and ($s$) denotes the slip.

$$ \tau = (1 - d)Gs + \beta G \int_{0}^{1} H[s - \Delta]\Delta \, dx \\ \left\{ \begin{array}{l} \text{if } d = 0 \Rightarrow \tau = Gs \Rightarrow \text{non-cracked state} \\ \text{if } d = 1 \Rightarrow \tau = \beta G \int_{0}^{1} \Delta(x) \, dx \Rightarrow \text{full-cracked state} \end{array} \right. \quad (11) $$
### 3. Mathematical simulation

Equations (11) establishes the correlation between the ultimate bond stress ($\tau$) and the slip ($s$), but it is challenging to calculate $\tau$ accurately because the damage variable ($d$) and the fracture threshold ($\Delta$) are not known and cannot be estimated from available data. Consequently, to account for this limitation, we rely on the average value of $\tau$ as a substitute for the actual relationship between $\tau$, $d$, and $\Delta$. In other words, since the parameters that affect the calculation of $\tau$ cannot be determined precisely, we resort to approximating the relationship using the mean value of $\tau$. The parameter $\tau$ can be defined as [5]:

$$ \tau = (1 - \beta)Gse^{-\frac{\omega_m}{a}} + \beta G \int_{0}^{s} e^{-\frac{\omega_m}{a}} d\Delta \quad (12) $$

$\beta$ represents the friction coefficient, and $G$ denotes the stiffness of the relationship between the ultimate bond stress and slip. Additionally, $a$ and $m$ are hyper-parameters that play a role in determining the values of the four unknown parameters ($a, m, \beta$, and $G$). To find the optimal values of these parameters, we employ hyper-parameter search algorithms, which allow us to explore and evaluate different combinations of values for $a, m, \beta$, and $G$. Using these techniques, we can identify the best set of

parameter values that result in the most accurate and reliable predictions of the UBS-S and the relationship between the ultimate bond stress ($\tau$) and slip ($s$).

A Physics-Informed Neural Network (PINN) is an artificial neural network trained to solve problems involving partial differential equations (PDEs). PINNs combine the strengths of neural networks with the physical principles that govern the behavior of structures. The neural network is trained to predict the solution to a PDE while satisfying the physical laws governing the system's behavior. In recent years, Physics-Informed Neural Networks (PINNs) have become a powerful tool for solving complex problems in various fields, including structural engineering. PINNs combine the power of neural networks with the physical principles that govern the behavior of structures. In structural engineering, PINNs can be used to solve problems involving PDEs that describe the behavior of structures. For example, the deformation of a structure under a given load can be modeled using a PDE. The solution to this PDE can be approximated using a PINN. The advantage of using a PINN is that it can learn the behavior of the structure directly from the data rather than relying on a predetermined model. This makes PINNs a powerful tool for solving complex problems that are difficult to model using traditional methods.

<table><caption>**Table 4**
MLR results for the PINN model.</caption><thead><tr><th rowspan="2">Specimens</th><th colspan="10">R<sub>MLR</sub> = ω<sub>0</sub> + ω<sub>1</sub>a + ω<sub>2</sub>m + ω<sub>3</sub>β + ω<sub>4</sub>G</th><th colspan="2">Predicted</th></tr><tr><th>ω<sub>0</sub></th><th>ω<sub>1</sub></th><th>a</th><th>ω<sub>2</sub></th><th>m</th><th>ω<sub>3</sub></th><th>β</th><th>ω<sub>4</sub></th><th>G</th><th>R<sub>MLR</sub></th><th>R<sub>PINN</sub></th></tr></thead><tbody><tr><td>R16C1L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.3125</td><td>1.2071</td><td>0.291</td><td>2.5874</td><td>0.0003</td><td>0.0038</td><td>929.6904</td><td>8.2184</td><td>5.481</td></tr><tr><td>R16C1L2F155</td><td>-5.7458</td><td>32.2790</td><td>0.3125</td><td>1.2071</td><td>0.5034</td><td>2.5874</td><td>0.2549</td><td>0.0038</td><td>453.1250</td><td>7.3266</td><td>8.9082</td></tr><tr><td>R16C1L3F155</td><td>-5.7458</td><td>32.2790</td><td>0.1250</td><td>1.2071</td><td>0.6287</td><td>2.5874</td><td>0.9298</td><td>0.0038</td><td>562.3396</td><td>3.5859</td><td>7.6928</td></tr><tr><td>R16C2L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.4777</td><td>1.2071</td><td>0.6070</td><td>2.5874</td><td>0.1251</td><td>0.0038</td><td>375.0006</td><td>12.1521</td><td>10.2030</td></tr><tr><td>R16C2L2F155</td><td>-5.7458</td><td>32.2790</td><td>0.5626</td><td>1.2071</td><td>0.6251</td><td>2.5874</td><td>0.0313</td><td>0.0038</td><td>343.7500</td><td>14.5533</td><td>20.2528</td></tr><tr><td>R16C2L3F155</td><td>-5.7458</td><td>32.2790</td><td>0.4256</td><td>1.2071</td><td>0.5308</td><td>2.5874</td><td>0.1251</td><td>0.0038</td><td>437.5001</td><td>10.6154</td><td>10.1104</td></tr><tr><td>R16C3L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.4688</td><td>1.2071</td><td>0.6154</td><td>2.5874</td><td>0.2530</td><td>0.0038</td><td>375.0000</td><td>12.2059</td><td>11.3365</td></tr><tr><td>R16C3L2F155</td><td>-5.7458</td><td>32.2790</td><td>0.4610</td><td>1.2071</td><td>0.6250</td><td>2.5874</td><td>0.2501</td><td>0.0038</td><td>376.9916</td><td>11.9658</td><td>12.2774</td></tr><tr><td>R16C4L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.0589</td><td>1.2071</td><td>2.2521</td><td>2.5874</td><td>0.9043</td><td>0.0038</td><td>140.8707</td><td>1.7479</td><td>1.0739</td></tr><tr><td>R16C4L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.1774</td><td>1.2071</td><td>0.6274</td><td>2.5874</td><td>0.953</td><td>0.0038</td><td>877.2338</td><td>6.5298</td><td>8.6925</td></tr><tr><td>R16C4L2F155</td><td>-5.7458</td><td>32.2790</td><td>0.0278</td><td>1.2071</td><td>2.8053</td><td>2.5874</td><td>0.9644</td><td>0.0038</td><td>297.8451</td><td>2.1625</td><td>4.1323</td></tr><tr><td>R18C1L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.3775</td><td>1.2071</td><td>0.5235</td><td>2.5874</td><td>0.1651</td><td>0.0038</td><td>333.9790</td><td>8.7649</td><td>3.7921</td></tr><tr><td>R18C1L2F155</td><td>-5.7458</td><td>32.2790</td><td>0.1876</td><td>1.2071</td><td>0.4999</td><td>2.5874</td><td>0.7481</td><td>0.0038</td><td>490.2346</td><td>4.7076</td><td>7.1834</td></tr><tr><td>R18C1L3F155</td><td>-5.7458</td><td>32.2790</td><td>0.1251</td><td>1.2071</td><td>0.9802</td><td>2.5874</td><td>0.8165</td><td>0.0038</td><td>249.9757</td><td>2.5359</td><td>4.4542</td></tr><tr><td>R18C2L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.4341</td><td>1.2071</td><td>0.6874</td><td>2.5874</td><td>0.2501</td><td>0.0038</td><td>312.4538</td><td>10.9281</td><td>14.3424</td></tr><tr><td>R18C2L2F155</td><td>-5.7458</td><td>32.2790</td><td>0.4504</td><td>1.2071</td><td>0.8242</td><td>2.5874</td><td>0.2501</td><td>0.0038</td><td>250.0000</td><td>11.3826</td><td>17.3554</td></tr><tr><td>R18C2L3F155</td><td>-5.7458</td><td>32.2790</td><td>0.5712</td><td>1.2071</td><td>0.6251</td><td>2.5874</td><td>0.1250</td><td>0.0038</td><td>250.0000</td><td>14.7178</td><td>10.5123</td></tr><tr><td>R18C3L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.6485</td><td>1.2071</td><td>0.6571</td><td>2.5874</td><td>0.1250</td><td>0.0038</td><td>249.9996</td><td>17.2516</td><td>17.8792</td></tr><tr><td>R18C3L2F155</td><td>-5.7458</td><td>32.2790</td><td>0.6480</td><td>1.2071</td><td>0.6417</td><td>2.5874</td><td>0.0914</td><td>0.0038</td><td>250.0000</td><td>17.1300</td><td>15.1669</td></tr><tr><td>R18C4L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.0683</td><td>1.2071</td><td>2.2427</td><td>2.5874</td><td>0.8384</td><td>0.0038</td><td>119.042</td><td>1.7867</td><td>1.0863</td></tr><tr><td>R18C4L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.1797</td><td>1.2071</td><td>0.6069</td><td>2.5874</td><td>0.9507</td><td>0.0038</td><td>790.9605</td><td>6.2462</td><td>6.6881</td></tr><tr><td>R18C4L2F155</td><td>-5.7458</td><td>32.2790</td><td>0.0486</td><td>1.2071</td><td>2.2672</td><td>2.5874</td><td>0.9869</td><td>0.0038</td><td>269.0213</td><td>2.1332</td><td>3.5728</td></tr><tr><td>R12C1L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.1634</td><td>1.2071</td><td>0.5553</td><td>2.5874</td><td>0.9944</td><td>0.0038</td><td>585.115</td><td>4.9903</td><td>6.0768</td></tr><tr><td>R12C1L2F155</td><td>-5.7458</td><td>32.2790</td><td>0.1894</td><td>1.2071</td><td>0.4768</td><td>2.5874</td><td>0.9716</td><td>0.0038</td><td>585.7532</td><td>5.6783</td><td>3.3478</td></tr><tr><td>R12C1L3F155</td><td>-5.7458</td><td>32.2790</td><td>0.1831</td><td>1.2071</td><td>0.5773</td><td>2.5874</td><td>0.9242</td><td>0.0038</td><td>497.2593</td><td>5.1380</td><td>2.6628</td></tr><tr><td>R12C1L4F155</td><td>-5.7458</td><td>32.2790</td><td>0.0533</td><td>1.2071</td><td>2.4864</td><td>2.5874</td><td>0.8329</td><td>0.0038</td><td>156.7859</td><td>1.7255</td><td>1.7838</td></tr><tr><td>R12C2L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.0075</td><td>1.2071</td><td>4.1575</td><td>2.5874</td><td>0.8753</td><td>0.0038</td><td>232.8569</td><td>2.6625</td><td>3.1887</td></tr><tr><td>R12C2L2F155</td><td>-5.7458</td><td>32.2790</td><td>0.0091</td><td>1.2071</td><td>3.9433</td><td>2.5874</td><td>0.8949</td><td>0.0038</td><td>218.9987</td><td>2.4537</td><td>2.2834</td></tr><tr><td>R12C2L3F155</td><td>-5.7458</td><td>32.2790</td><td>0.0088</td><td>1.2071</td><td>3.98</td><td>2.5874</td><td>0.8782</td><td>0.0038</td><td>222.059</td><td>2.4567</td><td>2.3061</td></tr><tr><td>R14C1L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.1728</td><td>1.2071</td><td>0.6264</td><td>2.5874</td><td>0.9906</td><td>0.0038</td><td>403.9241</td><td>4.6828</td><td>5.2517</td></tr><tr><td>R14C1L2F155</td><td>-5.7458</td><td>32.2790</td><td>0.1971</td><td>1.2071</td><td>0.5105</td><td>2.5874</td><td>0.9566</td><td>0.0038</td><td>460.6475</td><td>5.4543</td><td>3.8536</td></tr><tr><td>R14C1L3F155</td><td>-5.7458</td><td>32.2790</td><td>0.1802</td><td>1.2071</td><td>0.6736</td><td>2.5874</td><td>0.946</td><td>0.0038</td><td>360.5598</td><td>4.6988</td><td>2.6956</td></tr><tr><td>R14C1L4F155</td><td>-5.7458</td><td>32.2790</td><td>0.0209</td><td>1.2071</td><td>3.9921</td><td>2.5874</td><td>0.7541</td><td>0.0038</td><td>126.0828</td><td>2.1769</td><td>2.1921</td></tr><tr><td>R14C2L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.0095</td><td>1.2071</td><td>3.9142</td><td>2.5874</td><td>0.8822</td><td>0.0038</td><td>218.7204</td><td>2.3976</td><td>2.4734</td></tr><tr><td>R14C2L2F155</td><td>-5.7458</td><td>32.2790</td><td>0.0274</td><td>1.2071</td><td>2.7033</td><td>2.5874</td><td>0.9802</td><td>0.0038</td><td>222.7776</td><td>1.7827</td><td>2.3864</td></tr><tr><td>R14C2L3F155</td><td>-5.7458</td><td>32.2790</td><td>0.012</td><td>1.2071</td><td>3.6429</td><td>2.5874</td><td>0.8712</td><td>0.0038</td><td>213.3552</td><td>2.1020</td><td>2.546</td></tr><tr><td>R20C1L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.1871</td><td>1.2071</td><td>0.4973</td><td>2.5874</td><td>0.9935</td><td>0.0038</td><td>506.4761</td><td>5.3848</td><td>5.2835</td></tr><tr><td>R20C1L2F155</td><td>-5.7458</td><td>32.2790</td><td>0.1919</td><td>1.2071</td><td>0.493</td><td>2.5874</td><td>0.9766</td><td>0.0038</td><td>460.061</td><td>5.3149</td><td>2.6896</td></tr><tr><td>R20C1L3F155</td><td>-5.7458</td><td>32.2790</td><td>0.1574</td><td>1.2071</td><td>0.6099</td><td>2.5874</td><td>0.9871</td><td>0.0038</td><td>432.1334</td><td>4.2636</td><td>1.9052</td></tr><tr><td>R20C1L4F155</td><td>-5.7458</td><td>32.2790</td><td>0.0028</td><td>1.2071</td><td>4.8902</td><td>2.5874</td><td>0.906</td><td>0.0038</td><td>98.8257</td><td>2.9665</td><td>0.5532</td></tr><tr><td>R20C2L1F155</td><td>-5.7458</td><td>32.2790</td><td>0.0115</td><td>1.2071</td><td>3.6736</td><td>2.5874</td><td>0.9112</td><td>0.0038</td><td>206.9542</td><td>2.2022</td><td>2.1408</td></tr><tr><td>R20C2L2F155</td><td>-5.7458</td><td>32.2790</td><td>0.0277</td><td>1.2071</td><td>2.867</td><td>2.5874</td><td>0.9744</td><td>0.0038</td><td>190.9523</td><td>1.8543</td><td>1.6597</td></tr><tr><td>R20C2L3F155</td><td>-5.7458</td><td>32.2790</td><td>0.0144</td><td>1.2071</td><td>3.5059</td><td>2.5874</td><td>0.9017</td><td>0.0038</td><td>185.5813</td><td>1.9877</td><td>1.5481</td></tr></tbody></table>
<table><caption>**Table 5**
The variables effect (MLR and PINN models).</caption><thead><tr><th>Parameter</th><th>R<sub>MLR</sub></th><th>R<sub>PINN</sub></th><th>Difference</th></tr></thead><tbody><tr><td>a</td><td>0.986426</td><td>0.881427</td><td>0.104999</td></tr><tr><td>m</td><td>-0.609544</td><td>-0.544661</td><td>-0.064883</td></tr><tr><td>β</td><td>-0.859378</td><td>-0.767902</td><td>-0.091476</td></tr><tr><td>G</td><td>0.190656</td><td>0.170361</td><td>0.020295</td></tr><tr><td>R<sub>MLR</sub></td><td>1.00</td><td>-</td><td>N/A</td></tr><tr><td>R<sub>PINN</sub></td><td>-</td><td>1.00</td><td>N/A</td></tr></tbody></table>

(+) If the independent value increases, the dependent value also increases.

(-) If the independent value increases, the dependent value tends to decrease.

On the other hand, the Genetic Algorithm (GA) is a search algorithm that mimics the process of natural selection and genetic recombination to explore and identify the best combination of parameter values that minimize a given objective function. By iteratively testing different combinations of a, m, β, and G, GA can converge toward the set of parameter values that result in the most accurate predictions of the relationship between the ultimate bond stress and slip.

Concerning the experimental and numerical database, the variables

<table><caption>**Table 7**
The variables effect (MLR and GA models).</caption><thead><tr><th>Parameter</th><th>R<sub>MLR</sub></th><th>R<sub>GA</sub></th><th>Difference</th></tr></thead><tbody><tr><td>a</td><td>0.550441</td><td>0.364818</td><td>0.185623</td></tr><tr><td>m</td><td>0.137034</td><td>0.090822</td><td>0.046212</td></tr><tr><td>β</td><td>-0.248570</td><td>-0.164746</td><td>-0.083824</td></tr><tr><td>G</td><td>0.053282</td><td>0.035314</td><td>0.017968</td></tr><tr><td>R<sub>MLR</sub></td><td>1.00</td><td>-</td><td>N/A</td></tr><tr><td>R<sub>GA</sub></td><td>-</td><td>1.00</td><td>N/A</td></tr></tbody></table>

(+): If the independent value increases, the dependent value also increases.

(-): If the independent value increases, the dependent value tends to decrease.

and m are constrained to values within the range (0,1]. The parameter β represents the friction coefficient of the friction element after the spring element failure, and as such, it also assumes values within the range of (0,1]. The symbol G signifies the stiffness of the relationship between τ and s when the spring element is active, and its range is undetermined. Additionally, the magnitude of G might be considerable, allowing for its

<table><caption>**Table 6**
MLR results for the GA model.</caption><thead><tr><th rowspan="2">Specimens</th><th colspan="9">R<sub>MLR</sub> = ω<sub>0</sub> + ω<sub>1</sub>a + ω<sub>2</sub>m + ω<sub>3</sub>β + ω<sub>4</sub>G</th><th colspan="2">Predicted</th></tr><tr><th>ω<sub>0</sub></th><th>ω<sub>1</sub></th><th>a</th><th>ω<sub>2</sub></th><th>m</th><th>ω<sub>3</sub></th><th>β</th><th>ω<sub>4</sub></th><th>G</th><th>R<sub>MLR</sub></th><th>R<sub>GA</sub></th></tr></thead><tbody><tr><td>R16C1L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.3135</td><td>9.1666</td><td>0.3166</td><td>8.5370</td><td>0.0752</td><td>0.0170</td><td>798.3075</td><td>5.8084</td><td>4.1301</td></tr><tr><td>R16C1L2F155</td><td>-24.5887</td><td>42.3811</td><td>0.3016</td><td>9.1666</td><td>0.3035</td><td>8.5370</td><td>0.0511</td><td>0.0170</td><td>877.6057</td><td>6.3258</td><td>3.8567</td></tr><tr><td>R16C1L3F155</td><td>-24.5887</td><td>42.3811</td><td>0.3044</td><td>9.1666</td><td>0.3241</td><td>8.5370</td><td>0.0586</td><td>0.0170</td><td>804.9847</td><td>5.4632</td><td>3.2033</td></tr><tr><td>R16C2L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.4716</td><td>9.1666</td><td>0.5523</td><td>8.5370</td><td>0.0961</td><td>0.0170</td><td>404.7559</td><td>8.1598</td><td>9.0954</td></tr><tr><td>R16C2L2F155</td><td>-24.5887</td><td>42.3811</td><td>0.4633</td><td>9.1666</td><td>0.4940</td><td>8.5370</td><td>0.0417</td><td>0.0170</td><td>453.3193</td><td>7.6345</td><td>10.5460</td></tr><tr><td>R16C2L3F155</td><td>-24.5887</td><td>42.3811</td><td>0.4716</td><td>9.1666</td><td>0.5524</td><td>8.5370</td><td>0.0961</td><td>0.0170</td><td>387.3377</td><td>7.8647</td><td>8.3300</td></tr><tr><td>R16C3L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.4800</td><td>9.1666</td><td>0.5894</td><td>8.5370</td><td>0.2297</td><td>0.0170</td><td>382.4432</td><td>9.6173</td><td>10.8076</td></tr><tr><td>R16C3L2F155</td><td>-24.5887</td><td>42.3811</td><td>0.4872</td><td>9.1666</td><td>0.5936</td><td>8.5370</td><td>0.2062</td><td>0.0170</td><td>379.9851</td><td>9.7185</td><td>10.4195</td></tr><tr><td>R16C4L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.5001</td><td>9.1666</td><td>0.7704</td><td>8.5370</td><td>0.2775</td><td>0.0170</td><td>171.875</td><td>8.9579</td><td>3.1655</td></tr><tr><td>R16C4L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.2501</td><td>9.1666</td><td>0.8721</td><td>8.5370</td><td>0.6719</td><td>0.0170</td><td>500</td><td>8.2381</td><td>18.0328</td></tr><tr><td>R16C4L2F155</td><td>-24.5887</td><td>42.3811</td><td>0.2639</td><td>9.1666</td><td>0.9995</td><td>8.5370</td><td>0.6574</td><td>0.0170</td><td>406.107</td><td>8.2714</td><td>6.9084</td></tr><tr><td>R18C1L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.3450</td><td>9.1666</td><td>0.4754</td><td>8.5370</td><td>0.1716</td><td>0.0170</td><td>403.5818</td><td>2.7141</td><td>3.4961</td></tr><tr><td>R18C1L2F155</td><td>-24.5887</td><td>42.3811</td><td>0.1042</td><td>9.1666</td><td>0.9394</td><td>8.5370</td><td>0.9988</td><td>0.0170</td><td>273.5785</td><td>1.6145</td><td>3.3106</td></tr><tr><td>R18C1L3F155</td><td>-24.5887</td><td>42.3811</td><td>0.3088</td><td>9.1666</td><td>0.3635</td><td>8.5370</td><td>0.1501</td><td>0.0170</td><td>577.2648</td><td>2.9221</td><td>2.5369</td></tr><tr><td>R18C2L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.4508</td><td>9.1666</td><td>0.5379</td><td>8.5370</td><td>0.1419</td><td>0.0170</td><td>388.2582</td><td>7.2569</td><td>8.3868</td></tr><tr><td>R18C2L2F155</td><td>-24.5887</td><td>42.3811</td><td>0.4296</td><td>9.1666</td><td>0.4760</td><td>8.5370</td><td>0.1090</td><td>0.0170</td><td>416.2781</td><td>5.9863</td><td>8.7984</td></tr><tr><td>R18C2L3F155</td><td>-24.5887</td><td>42.3811</td><td>0.4406</td><td>9.1666</td><td>0.4641</td><td>8.5370</td><td>0.1001</td><td>0.0170</td><td>396.6052</td><td>5.9331</td><td>7.2704</td></tr><tr><td>R18C3L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.4718</td><td>9.1666</td><td>0.4699</td><td>8.5370</td><td>0.0975</td><td>0.0170</td><td>425.8775</td><td>7.7839</td><td>10.0858</td></tr><tr><td>R18C3L2F155</td><td>-24.5887</td><td>42.3811</td><td>0.4869</td><td>9.1666</td><td>0.4712</td><td>8.5370</td><td>0.0520</td><td>0.0170</td><td>407.0233</td><td>7.7269</td><td>9.2029</td></tr><tr><td>R18C4L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.3751</td><td>9.1666</td><td>0.7894</td><td>8.5370</td><td>0.428</td><td>0.0170</td><td>170.3346</td><td>5.0931</td><td>1.1544</td></tr><tr><td>R18C4L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.2501</td><td>9.1666</td><td>0.5748</td><td>8.5370</td><td>0.5938</td><td>0.0170</td><td>749.985</td><td>9.0944</td><td>7.56</td></tr><tr><td>R18C4L2F155</td><td>-24.5887</td><td>42.3811</td><td>0.3124</td><td>9.1666</td><td>0.6685</td><td>8.5370</td><td>0.507</td><td>0.0170</td><td>500.0758</td><td>7.6056</td><td>8.1186</td></tr><tr><td>R12C1L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.2931</td><td>9.1666</td><td>0.5001</td><td>8.5370</td><td>0.375</td><td>0.0170</td><td>496.6909</td><td>4.0596</td><td>6.0141</td></tr><tr><td>R12C1L2F155</td><td>-24.5887</td><td>42.3811</td><td>0.2149</td><td>9.1666</td><td>0.5001</td><td>8.5370</td><td>0.8651</td><td>0.0170</td><td>500.0029</td><td>4.9857</td><td>6.5276</td></tr><tr><td>R12C1L3F155</td><td>-24.5887</td><td>42.3811</td><td>0.2501</td><td>9.1666</td><td>0.52</td><td>8.5370</td><td>0.5756</td><td>0.0170</td><td>499.6425</td><td>4.1823</td><td>2.9324</td></tr><tr><td>R12C1L4F155</td><td>-24.5887</td><td>42.3811</td><td>0.3686</td><td>9.1666</td><td>0.3897</td><td>8.5370</td><td>0.0177</td><td>0.0170</td><td>563.5048</td><td>4.3326</td><td>4.0296</td></tr><tr><td>R12C2L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.3282</td><td>9.1666</td><td>0.9758</td><td>8.5370</td><td>0.5</td><td>0.0170</td><td>328.2861</td><td>8.1130</td><td>8.7841</td></tr><tr><td>R12C2L2F155</td><td>-24.5887</td><td>42.3811</td><td>0.3069</td><td>9.1666</td><td>0.6564</td><td>8.5370</td><td>0.5221</td><td>0.0170</td><td>439.4815</td><td>6.3608</td><td>8.5778</td></tr><tr><td>R12C2L3F155</td><td>-24.5887</td><td>42.3811</td><td>0.3019</td><td>9.1666</td><td>1</td><td>8.5370</td><td>0.5626</td><td>0.0170</td><td>304.6774</td><td>7.3534</td><td>5.6562</td></tr><tr><td>R14C1L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.25</td><td>9.1666</td><td>0.5412</td><td>8.5370</td><td>0.5673</td><td>0.0170</td><td>437.4981</td><td>3.2455</td><td>5.1543</td></tr><tr><td>R14C1L2F155</td><td>-24.5887</td><td>42.3811</td><td>0.1935</td><td>9.1666</td><td>0.5023</td><td>8.5370</td><td>0.9988</td><td>0.0170</td><td>470.0822</td><td>4.7318</td><td>3.8426</td></tr><tr><td>R14C1L3F155</td><td>-24.5887</td><td>42.3811</td><td>0.2665</td><td>9.1666</td><td>0.4752</td><td>8.5370</td><td>0.4986</td><td>0.0170</td><td>497.0126</td><td>3.7647</td><td>2.9522</td></tr><tr><td>R14C1L4F155</td><td>-24.5887</td><td>42.3811</td><td>0.3358</td><td>9.1666</td><td>0.9687</td><td>8.5370</td><td>0.5061</td><td>0.0170</td><td>187.634</td><td>6.0318</td><td>3.2252</td></tr><tr><td>R14C2L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.2813</td><td>9.1666</td><td>0.9956</td><td>8.5370</td><td>0.5885</td><td>0.0170</td><td>312.501</td><td>6.7941</td><td>5.6846</td></tr><tr><td>R14C2L2F155</td><td>-24.5887</td><td>42.3811</td><td>0.2671</td><td>9.1666</td><td>0.9688</td><td>8.5370</td><td>0.625</td><td>0.0170</td><td>312.5001</td><td>6.2582</td><td>4.8358</td></tr><tr><td>R14C2L3F155</td><td>-24.5887</td><td>42.3811</td><td>0.3634</td><td>9.1666</td><td>0.6875</td><td>8.5370</td><td>0.3711</td><td>0.0170</td><td>381.1879</td><td>6.7607</td><td>6.2212</td></tr><tr><td>R20C1L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.1918</td><td>9.1666</td><td>0.3799</td><td>8.5370</td><td>0.9309</td><td>0.0170</td><td>780.2699</td><td>8.2295</td><td>5.5718</td></tr><tr><td>R20C1L2F155</td><td>-24.5887</td><td>42.3811</td><td>0.2006</td><td>9.1666</td><td>0.4993</td><td>8.5370</td><td>0.8816</td><td>0.0170</td><td>454.4885</td><td>3.7397</td><td>2.7276</td></tr><tr><td>R20C1L3F155</td><td>-24.5887</td><td>42.3811</td><td>0.3126</td><td>9.1666</td><td>0.5083</td><td>8.5370</td><td>0.3431</td><td>0.0170</td><td>374.6428</td><td>2.6148</td><td>1.5741</td></tr><tr><td>R20C1L4F155</td><td>-24.5887</td><td>42.3811</td><td>0.2262</td><td>9.1666</td><td>0.8438</td><td>8.5370</td><td>0.6956</td><td>0.0170</td><td>185.2497</td><td>1.8192</td><td>2.5085</td></tr><tr><td>R20C2L1F155</td><td>-24.5887</td><td>42.3811</td><td>0.1942</td><td>9.1666</td><td>0.9713</td><td>8.5370</td><td>0.8555</td><td>0.0170</td><td>317.991</td><td>5.2526</td><td>6.3589</td></tr><tr><td>R20C2L2F155</td><td>-24.5887</td><td>42.3811</td><td>0.4479</td><td>9.1666</td><td>0.6876</td><td>8.5370</td><td>0.25</td><td>0.0170</td><td>308.5895</td><td>8.0752</td><td>5.3758</td></tr><tr><td>R20C2L3F155</td><td>-24.5887</td><td>42.3811</td><td>0.3048</td><td>9.1666</td><td>0.6996</td><td>8.5370</td><td>0.4999</td><td>0.0170</td><td>362.8307</td><td>5.1757</td><td>4.7013</td></tr></tbody></table>
<table><caption>**Table 8**

Descriptive statistics of PINN features.</caption><thead><tr><th>Descriptive statistics</th><th>a_PINN</th><th>m_PINN</th><th>β_PINN</th><th>G_PINN</th><th>R_PINN</th></tr></thead><tbody><tr><th>Min</th><td>0.0028</td><td>0.291</td><td>0.0003</td><td>98.8257</td><td>0.5532</td></tr><tr><th>Max</th><td>0.6485</td><td>4.8902</td><td>0.9944</td><td>929.6904</td><td>20.2528</td></tr><tr><th>Mean</th><td>0.208746</td><td>1.614967</td><td>0.687011</td><td>356.2998</td><td>6.070286</td></tr><tr><th>Std</th><td>0.196471</td><td>1.427541</td><td>0.36034945</td><td>192.9526</td><td>5.084939</td></tr><tr><th>CV</th><td>0.941196</td><td>0.883944</td><td>0.52451725</td><td>0.541545</td><td>0.837677</td></tr></tbody></table>

<table><caption>**Table 9**

Descriptive statistics of GA features.</caption><thead><tr><th>Descriptive statistics</th><th>a_GA</th><th>m_GA</th><th>β_GA</th><th>G_GA</th><th>R_GA</th></tr></thead><tbody><tr><th>Min</th><td>0.1042</td><td>0.3035</td><td>0.0177</td><td>170.3346</td><td>1.1544</td></tr><tr><th>Max</th><td>0.5001</td><td>1</td><td>0.9988</td><td>877.6057</td><td>18.0328</td></tr><tr><th>Mean</th><td>0.329814</td><td>0.631081</td><td>0.414916</td><td>436.9848</td><td>6.085390698</td></tr><tr><th>Std</th><td>0.101019</td><td>0.219672</td><td>0.293989</td><td>165.5192</td><td>3.245445177</td></tr><tr><th>CV</th><td>0.306293</td><td>0.348088</td><td>0.70855028</td><td>0.378776</td><td>0.533317471</td></tr></tbody></table>

<table><caption>**Table 10**

MSE results.</caption><thead><tr><th>Method</th><th>R<sub>PINN</sub></th><th>R<sub>GA</sub></th></tr></thead><tbody><tr><td>Normal MLR</td><td>5.0904</td><td>5.7688</td></tr><tr><td>Customized MLR</td><td>3.3952</td><td>3.3903</td></tr></tbody></table>

specification within the interval (0,1000]. To enhance precision, each value is assigned four decimal places, leading to modified ranges for a, m, β as [0.0001,1] and for G as [0.0001,1000]. Consequently, the bit lengths for a, m, and β are established at 14 bits, while G is set to 24 bits. In summary, the cumulative gene length is determined to be 66 bits.

Therefore, both PINN and GA are powerful techniques that can be applied to determine the optimal hyper-parameter values to improve the accuracy of predictions for the UBS-S relationship.

Using the Mean Squared Error (MSE), the PINN loss and the GA fitness functions (R) can be formulated as Eq. (13), and the goal is to minimize it by finding the optimal hyper-parameters [5].

$$R = fitness = \frac{1}{M} \sum_{i=1}^{M} [\tau(s_i) - \tau_i]^2 \qquad (13)$$

This research applied a fully connected neural network called Multilayer Perceptron (MLP). The MLP architecture comprises at least three layers: an input layer, a hidden layer, and an output layer. Each node within a layer is linked with specific weights to every node in the subsequent layer. The MLP leverages nonlinear activation functions in all neurons, facilitating the mapping of weighted inputs to each neuron's output.

The input layer consists of M neurons representing M data points. Notably, the number of neurons in the initial hidden layer surpasses that of the input layer. The output layer comprises 4 neurons, corresponding to the 4 unknown parameters in this study. The pyramid structure extending from the hidden layers to the output layer is crucial for extracting essential information from the data.

The results encompass bond stress (τ) and slip (s), yielding a set of M points from a single experiment. The M points extracted from the dataset

$S_i(1 \le i \le M)$ serve as the input values for the Physics-Informed Neural Network (PINN), while the unknown hyperparameters a, m, β, and G constitute the output. Once the hyperparameters of the inverse problem are determined, the model value $\tau(s_i)$ corresponding to each $s_i$ is computed using PINN. Subsequently, R is calculated as per Eq. (13). The objective is to minimize R by focusing on the hyperparameters obtained. Ultimately, through multiple epochs of neural network training, the optimal values for parameters a, m, β, and G can be identified.

The effectiveness of the fitness function (R) lies in its ability to accurately depict the disparity between each gene and the optimal solution gene for the given problem, influencing both the speed and smoothness of optimization. A judicious selection of the (R) is crucial as it can enhance the optimization process. It is imperative, in the context of the optimization problem, to choose a fitness function that ensures non-negativity by establishing a mapping from the (R) to the objective function. Utilizing a set of M data pairs $(s_i, \tau_i)$ from pullout test observations on the relationship between τ and s, and employing the genetic algorithm's hyperparameters $(a, m, \beta, G)$, the model value $\tau(s_i)$ for each $s_i$ can be computed.

### 3.1. Multiple Linear Regression (MLR)

In this research study, we employed Multiple Linear Regression (MLR) as a statistical technique to explore and understand the associations and interdependencies among various influential parameters.

Algorithm 2 presents the steps of creating an MLR model applied in this research.

**Algorithm 2.** MLR model.
* **Step 1:** Determine the values of $a_1$ and $a_2$ in the following linear equation:

$$u = a_1x_1 + a_2x_2 \quad (14)$$

* **Step 2:** Apply the machine learning techniques to conduct linear regression on multiple variables.

* **Step 3:** Apply an n-dimensional feature sample data using linear regression:

$$h_{\theta}(x_1, x_2, \dots, x_n) = \theta_0 + \theta_1 x_1 + \dots + \theta_n x_n \quad (15)$$

* **Step 4:** Assume $x_0 = 1$ to simplify the equation:

$$h_{\theta}(x_0, x_1, x_2, \dots, x_n) = \sum_{i=0}^{n} \theta_i x_i \quad (16)$$

* **Step 5:** Provide a matrix form for a more concise understanding:

$$h_{\theta}(X) = X\theta, \quad (17)$$

$$\theta = \begin{bmatrix} \theta_0 \\ \theta_1 \\ \vdots \\ \theta_n \end{bmatrix}, \quad x = \begin{bmatrix} x_0 \\ x_1 \\ \vdots \\ x_n \end{bmatrix}, \quad Y = \begin{bmatrix} y_0 \\ y_1 \\ \vdots \\ y_n \end{bmatrix}, \quad X = \begin{bmatrix} x_{01} & \cdots & x_{0n} \\ \vdots & \ddots & \vdots \\ x_{m1} & \cdots & x_{mn} \end{bmatrix} \quad (18)$$

$m$ represents the number of samples, while $n$ denotes the number of features in each sample.

* **Step 6:** Adopt the Mean Square Error (MSE) as the chosen loss function to provide a mathematical framework for quantifying the error between predicted and actual values in the linear regression model.

* **Step 7:** Create the algebraic and matrix forms of the loss function.

$$J(\theta_0, \theta_1, \theta_2, \dots, \theta_n) = \sum_{i=0}^{m} (h_{\theta}(x_0, x_1, x_2, \dots, x_n) - y_i)^2 \quad (19)$$

$$J(\theta) = (X\theta - Y)^T(X\theta - Y) \quad (20)$$

* **Step 8:** Applying Equations (21) and (22) to estimate the MLR model parameters minimizes the corresponding loss function.

$$\frac{\partial J(\theta)}{\partial \theta} = 2X^T X\theta - 2X^T Y \quad (21)$$

$$\theta = (X^T X)^{-1} X^T Y \quad (22)$$

* **Step 9:** Create the Multiple Linear Regression (MLR) model:

$$f(x) = x^T (X^T X)^{-1} X^T Y \quad (23)$$
<figure><img src="image_7.png" /><figcaption>Fig. 5. Histograms of PINN features.</figcaption></figure>

## 4. Results and discussion

### 4.1. Ultimate Bond Stress-Slip (UBS-S)

The Ultimate Bond Stress-Slip (UBS-S) graph represents the bond stress and slip relationship in reinforced concrete structures, specifically for Ultra-High Performance Concrete (UHPC) structures. It helps engineers understand the maximum bond stress and slip values and the post-peak behavior of the bond stress and slip relationship. UBS-S graphs are

essential in designing and evaluating reinforced concrete structures' performance, enhancing reinforcing bars' design, and ensuring structural integrity and safety.

As shown in Fig. 3, comparing UBS-S predicted by PINN, GA, and experimental results is vital for validating and evaluating the accuracy of the computational models. The comparison helps to identify any discrepancies or errors in the predictions and provides insights into the behavior of the UHPC structures. Additionally, the comparison allows for the optimization of the models and parameters to improve the
<figure><img src="image_8.png" /><figcaption>Fig. 6. Histograms of GA features.</figcaption></figure>

accuracy of the predictions.

The loss function in PINN trains the neural network by minimizing the error between the predicted and observed values while ensuring that the model is physically consistent. In PINN, the loss function typically includes a data-driven loss term and a physics-based loss term. The data-driven term measures the discrepancy between the predicted values from the neural network and the observed data, while the physics-based term ensures that the predicted values satisfy the underlying physical laws governing the system being modeled.

In GA, the loss function is typically based on a fitness function that measures the quality of each candidate solution in the population. The fitness function in GA is used to evaluate the fitness of each candidate solution, select the best solutions, and evolve the population over successive generations. The fitness function is problem-specific and can be designed to optimize different objectives, such as maximizing performance or reducing error.

It is important to note that the loss function in both PINN and GA is a critical component of the optimization process. Minimizing the loss
<figure><img src="image_9.png" /><figcaption>Fig. 7. Correlation coefficient.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8. The relationship between four hyperparameters *a*, *β*, *G*, *m*, and *R<sub>PINN</sub>*.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 9. The relationship between four hyperparameters a, b, G, m, and R<sub>GA</sub>.</figcaption></figure>

function improves the model or solution until a satisfactory solution is obtained. Fig. 4 presents the PINN and GA loss function results.

As presented in Fig. 4, the PINN loss function graphs show how the total loss (which includes both the data-driven and physics-based loss terms) changes throughout training. This loss function reflects how well the model can fit the observed data while satisfying the underlying physical constraints. By monitoring the shape of the loss function graph, we can assess the effectiveness of the model architecture and hyper-parameters and identify any issues hindering the training process, such as overfitting or underfitting. Similarly, in GA, the fitness function graphs illustrate how the fitness of the candidate solutions evolves over successive generations. This fitness reflects how well each candidate solution (corresponding to a particular set of model parameters) can predict steel bars' ultimate bond stress-slip behavior in UHPC structures. By monitoring the shape of the loss function graph, we can assess the effectiveness of the genetic operators and parameters used in the algorithm and identify any issues hindering the optimization process, such as premature convergence or lack of diversity in the population.

Analyzing the loss function graphs of PINN and GA can help better understand the behavior and performance of their models for predicting the ultimate bond stress-slip behavior of steel bars in UHPC structures and make informed decisions about how to improve them.

## 4.2. Multiple Linear Regression (MLR)

Regarding Eq. (15), the MLR equation for the mentioned parameters would be as follows:

$$R_{MLR} = \omega_0 + \omega_1 a + \omega_2 m + \omega_3 \beta + \omega_4 G \quad (24)$$

where $\omega_0$ denotes the intercept of the MLR equation and $\omega_1$ to $\omega_4$ represent the coefficients. Applying the LinearRegression() function in Python, we can extract the results presented in Table 4 for the proposed PINN model.

Once an MLR equation has been trained on a dataset of hyper-parameters and corresponding loss values, it can be used to predict the loss value for a new set of hyperparameters without running the model. This can be useful for quickly assessing the quality of a new set of hyperparameters before running an expensive computation.

An MLR equation can also be used to optimize the hyperparameters of a model by finding the set of hyperparameters that minimize the loss function. This is typically done by iteratively adjusting the hyper-parameters and evaluating their performance using the MLR equation until the optimal set is found.

The MLR equation can provide insights into the relationship between the hyperparameters and the loss function. By examining the
<table><caption>**Table 11**

Improved fitting results.</caption><thead><tr><th>Specimens</th><th>R<sub>PINN</sub></th><th>Normal MLR</th><th>Customized MLR</th><th>R<sub>GA</sub></th><th>Normal MLR</th><th>Customized MLR</th></tr></thead><tbody><tr><td>R16C1L1F155</td><td>5.481</td><td>8.2184</td><td>4.5231081</td><td>4.1301</td><td>5.8084</td><td>4.5477693</td></tr><tr><td>R16C1L2F155</td><td>8.9082</td><td>7.3266</td><td>8.1556008</td><td>3.8567</td><td>6.3258</td><td>2.7406914</td></tr><tr><td>R16C1L3F155</td><td>7.6928</td><td>3.5859</td><td>4.9568419</td><td>3.2033</td><td>5.4632</td><td>3.8608539</td></tr><tr><td>R16C2L1F155</td><td>10.203</td><td>12.1521</td><td>13.159828</td><td>9.0954</td><td>8.1598</td><td>8.6678428</td></tr><tr><td>R16C2L2F155</td><td>20.2528</td><td>14.5533</td><td>15.002222</td><td>10.546</td><td>7.6345</td><td>8.2762781</td></tr><tr><td>R16C2L3F155</td><td>10.1104</td><td>10.6154</td><td>11.440158</td><td>8.33</td><td>7.8647</td><td>8.1409761</td></tr><tr><td>R16C3L1F155</td><td>11.3365</td><td>12.2059</td><td>13.362635</td><td>10.8076</td><td>9.6173</td><td>9.9859355</td></tr><tr><td>R16C3L2F155</td><td>12.2774</td><td>11.9658</td><td>13.28555</td><td>10.4195</td><td>9.7185</td><td>10.112843</td></tr><tr><td>R16C4L1F155</td><td>1.0739</td><td>1.7479</td><td>1.4452836</td><td>3.1655</td><td>8.9579</td><td>5.7467349</td></tr><tr><td>R16C4L1F155</td><td>8.6925</td><td>6.5298</td><td>9.1965951</td><td>18.0328</td><td>8.2381</td><td>12.040864</td></tr><tr><td>R16C4L2F155</td><td>4.1323</td><td>2.1625</td><td>3.9222473</td><td>6.9084</td><td>8.2714</td><td>8.7133553</td></tr><tr><td>R18C1L1F155</td><td>3.7921</td><td>8.7649</td><td>8.2484962</td><td>3.4961</td><td>2.7141</td><td>3.0791173</td></tr><tr><td>R18C1L2F155</td><td>7.1834</td><td>4.7076</td><td>4.5703737</td><td>3.3106</td><td>1.6145</td><td>1.2867251</td></tr><tr><td>R18C1L3F155</td><td>4.4542</td><td>2.5359</td><td>3.253747</td><td>2.5369</td><td>2.9221</td><td>4.6733813</td></tr><tr><td>R18C2L1F155</td><td>14.3424</td><td>10.9281</td><td>12.063332</td><td>8.3867</td><td>7.2569</td><td>7.4122252</td></tr><tr><td>R18C2L2F155</td><td>17.3554</td><td>11.3826</td><td>12.557002</td><td>8.7984</td><td>5.9863</td><td>6.2475534</td></tr><tr><td>R18C2L3F155</td><td>10.5123</td><td>14.7178</td><td>13.841553</td><td>7.2704</td><td>5.9331</td><td>5.8966842</td></tr><tr><td>R18C3L1F155</td><td>17.8792</td><td>17.2516</td><td>16.53642</td><td>10.0858</td><td>7.7839</td><td>7.9700155</td></tr><tr><td>R18C3L2F155</td><td>15.1669</td><td>17.13</td><td>16.213106</td><td>9.2029</td><td>7.7269</td><td>7.7196555</td></tr><tr><td>R18C4L1F155</td><td>1.0863</td><td>1.7867</td><td>1.5297083</td><td>1.1544</td><td>5.0931</td><td>2.2165417</td></tr><tr><td>R18C4L1F155</td><td>6.6881</td><td>6.2462</td><td>8.4662922</td><td>7.56</td><td>9.0944</td><td>7.6673803</td></tr><tr><td>R18C4L2F155</td><td>3.5728</td><td>2.1332</td><td>3.4167909</td><td>8.1186</td><td>7.6056</td><td>10.258569</td></tr><tr><td>R12C1L1F155</td><td>6.0768</td><td>4.9903</td><td>4.9388496</td><td>6.0141</td><td>4.0596</td><td>5.0950654</td></tr><tr><td>R12C1L2F155</td><td>3.3478</td><td>5.6783</td><td>4.3539728</td><td>6.5276</td><td>4.9857</td><td>5.5209184</td></tr><tr><td>R12C1L3F155</td><td>2.6628</td><td>5.138</td><td>5.116555</td><td>2.9324</td><td>4.1823</td><td>5.1999863</td></tr><tr><td>R12C1L4F155</td><td>1.7839</td><td>1.7255</td><td>2.194437</td><td>4.0296</td><td>4.3326</td><td>5.7726199</td></tr><tr><td>R12C2L1F155</td><td>3.1887</td><td>2.6625</td><td>3.0948368</td><td>8.7841</td><td>8.113</td><td>8.0530422</td></tr><tr><td>R12C2L2F155</td><td>2.2834</td><td>2.4537</td><td>2.6900846</td><td>8.5778</td><td>6.3608</td><td>8.531377</td></tr><tr><td>R12C2L3F155</td><td>2.3061</td><td>2.4567</td><td>2.8406338</td><td>5.6562</td><td>7.3534</td><td>5.9666789</td></tr><tr><td>R14C1L1F155</td><td>5.2517</td><td>4.6828</td><td>3.6749289</td><td>5.1543</td><td>3.2455</td><td>4.1301917</td></tr><tr><td>R14C1L2F155</td><td>3.8536</td><td>5.4543</td><td>3.6613285</td><td>3.8426</td><td>4.7318</td><td>5.0354868</td></tr><tr><td>R14C1L3F155</td><td>2.6956</td><td>4.6988</td><td>3.9866052</td><td>2.9522</td><td>3.7647</td><td>4.5942899</td></tr><tr><td>R14C1L4F155</td><td>2.1921</td><td>2.1769</td><td>1.6121476</td><td>3.2252</td><td>6.0318</td><td>2.1164241</td></tr><tr><td>R14C2L1F155</td><td>2.4734</td><td>2.3976</td><td>2.7503952</td><td>5.6846</td><td>6.7941</td><td>5.7903946</td></tr><tr><td>R14C2L2F155</td><td>2.3864</td><td>1.7827</td><td>2.2480478</td><td>4.8358</td><td>6.2582</td><td>6.108234</td></tr><tr><td>R14C2L3F155</td><td>2.546</td><td>2.102</td><td>2.6691408</td><td>6.2212</td><td>6.7607</td><td>8.6712374</td></tr><tr><td>R20C1L1F155</td><td>5.2835</td><td>5.3848</td><td>3.549771</td><td>5.5718</td><td>8.2295</td><td>5.2860694</td></tr><tr><td>R20C1L2F155</td><td>2.6896</td><td>5.3149</td><td>3.0269726</td><td>2.7276</td><td>3.7397</td><td>4.1022239</td></tr><tr><td>R20C1L3F155</td><td>1.9052</td><td>4.2636</td><td>3.4762569</td><td>1.5741</td><td>2.6148</td><td>2.5739636</td></tr><tr><td>R20C1L4F155</td><td>0.5532</td><td>2.9665</td><td>0.00173</td><td>2.5085</td><td>1.8192</td><td>-0.2400568</td></tr><tr><td>R20C2L1F155</td><td>2.1408</td><td>2.2022</td><td>2.3281487</td><td>6.3589</td><td>5.2526</td><td>5.06987</td></tr><tr><td>R20C2L2F155</td><td>1.6597</td><td>1.8543</td><td>1.7224182</td><td>5.3758</td><td>8.0752</td><td>8.4526127</td></tr><tr><td>R20C2L3F155</td><td>1.5481</td><td>1.9877</td><td>1.9381472</td><td>4.7013</td><td>5.1757</td><td>8.5791768</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>**Fig. 10.** The improved results by customized MLR.</figcaption></figure>
coefficients of the MLR equation, we can identify which hyper-parameters are most important for minimizing the loss function and how they interact with each other.

Table 5 provides the effects of hyper-parameters a, m, β, and G on the R predicted by the MLR Eq. (24) and the PINN. According to the differences between the effect values, it can be proved that the PINN and MLR could reach a logical precision on the proposed models. Table 6 also presents the MLR equation results for the GA model proposed in this study. Using MLR equations to model the relationship between hyper-parameters and loss functions leads to more efficient and effective optimization of the models and a deeper understanding of the factors that influence their performance.

In the ultimate bond stress-slip for reinforced UHPC structures, understanding the coordination between the hyperparameters of a PINN and an MLR model can be particularly useful. PINN can help capture the physics-based behavior of the UHPC structures, while MLR can help incorporate the data-driven behavior of the structures. By adjusting the hyperparameters of both models in a coordinated way, we can enhance the performance of the combined model and accurately predict the ultimate bond stress-slip behavior of reinforced UHPC structures. This can be useful for designing and evaluating the performance of UHPC structures, which have high strength and durability and are increasingly used in infrastructure and construction projects.

Table 7 shows the accuracy of our proposed MLR and GA models. Understanding the coordination between the hyperparameters of a genetic algorithm (GA) and multiple linear regression (MLR) models can help optimize the performance of a machine learning model. The coordination between the hyperparameters of GA and MLR can help maximize the performance of the machine learning models in structural engineering by adjusting the hyperparameters in a coordinated way to balance exploration and exploitation in the solution space and to avoid overfitting or underfitting the training data.

### 4.3. Descriptive Statistics

Descriptive statistics were performed on each characteristic of the data, including the calculation of the minimum (Min), maximum (Max), mean, standard deviation (Std), and coefficient of variation (CV) for each feature. The minimum and maximum values describe the range, the mean reflects the central tendency, the standard deviation represents the dispersion, and the coefficient of variation provides a dimensionless measure of dispersion.

As presented in Tables 8 and 9, analysis of the data reveals that the variation range is relatively small for variables a, m, and β, slightly more significant for R, and largest for G. Comparing the discrete degree of each feature, a, m, and R shows higher discreteness for PINN, while β and G exhibit lower discreteness. Conversely, for GA, the features show a lower degree of dispersion, with β being the most discrete and a, m, and G being less discrete.

A histogram is a graphical representation of the distribution of numerical data. It consists of a series of contiguous bars, each representing a range or interval of values, and the height of each bar corresponds to the frequency or count of data points falling within that range. Histograms are commonly used to visualize a dataset's shape, center, and spread. They provide insights into the distribution of the data, including its central tendency, variability, and any patterns or outliers present. As shown in Figs. 5 and 6, histograms were created for each feature to visualize the data distribution. In the case of PINN, most features are predominantly concentrated in one or two bins, with fewer data points in other bins. This indicates that while most data is clustered, some points vary within a specific range. Conversely, each feature exhibits a more uniform distribution for GA compared to PINN, although there are still two or four intervals with higher data counts. This observation highlights the correlation between data distribution and coefficient of variation (CV).

Next, we apply the simple correlation coefficient method to conduct

a multiple linear analysis of independent variables. This method enables us to assess the strength and direction of the linear relationship between independent variables by calculating the Pearson correlation coefficient (r). The Pearson correlation coefficient quantitatively measures the degree to which two variables vary linearly. Its value ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and 1 indicates a perfect positive linear relationship. A higher absolute value of r signifies a stronger linear relationship between variables, where x̄ and ȳ represent the mean of the data:

$$r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i - \bar{x})^2 \sum_{i=1}^{n} (y_i - \bar{y})^2}} \quad (25)$$

As mentioned before, the correlation coefficient is a statistical measure that describes the strength and direction of a relationship between two variables. It ranges from -1 to 1. The correlation coefficient quantifies the degree to which changes in one variable are associated with changes in another. It's important to note that correlation does not imply causation; it only measures the strength and direction of association between variables. Upon observing Fig. 7, it is evident that the correlation coefficient between most independent variables is very low. Only a few independent variables exhibit a correlation coefficient of about 0.8. Therefore, we conclude that the independent variables have no significant multiple linear relationship. We intend to use multiple linear regression (MLR) for fitting based on the descriptive statistics and multiple linear data analysis. This approach will enhance our ability to predict R more effectively.

### 4.4. Customized MLR

In the final section, we utilized Multiple Linear Regression (MLR) to model the four hyperparameters: a, β, G, m, and R, using the training data to obtain coefficients for the expression. This approach serves a dual purpose – firstly, the MLR equation enables the prediction of R for a new set of hyperparameters without executing the model. Secondly, we can optimize the model's hyperparameters to identify a set that minimizes R. As a result, we examine the interplay among the four hyperparameters, aiming for more accurate fitting results through MLR.

Upon observation, it becomes evident that the relationships among various hyperparameters, R<sub>PINN</sub>, and R<sub>GA</sub> exhibit diversity, ranging from linearity to inverse proportionality. As the connections among hyperparameters a, β, G, m, and R<sub>PINN</sub>, and R<sub>GA</sub> exhibit nonlinear relationships, relying solely on MLR for fitting is insufficient. Therefore, our approach involves deducing equations that capture the intricate relationships among each mentioned parameter, aiming to achieve enhanced accuracy in the modeling process.

Subsequently, we can identify the foundational functions governing the relationships among the four hyperparameters a, β, G, m, and R<sub>PINN</sub>:

$$\{a, \frac{1}{m}, \beta, \beta^2, G, G^2, 1\} \quad (26)$$

The correlation between R<sub>PINN</sub> and individual hyperparameters is characterized by its proximity to linearity with a, inverse proportionality with m, and quadratic tendencies with both β and G.

Similarly, for four hyperparameters a, β, G, m, and R<sub>GA</sub>:

$$\{a, m, m^2, m^3, \beta, G, G^2, 1\} \quad (27)$$

The association between R<sub>GA</sub> and each hyperparameter manifests as near linearity with a and β, cubic tendencies with m, and quadratic tendencies with G.

Using the data, we apply the least square method to ascertain the coefficients for the basis functions. The comprehensive algorithm is outlined below:

**Algorithm 3.** Improved method.
* Step 1: Determine the basis function B.

* Step 2: R = BW, where W is the coefficient.

* Step 3: Minimize $J(W) = ||R_{real} - BW||^2$.

* Step 4: Differentiate $J(W)$ and make it equal to 0. In other words, put the data into the equation $R = BW$ and solve the linear equations $B^T BW = B^T R$.

* Step 5: Get W and R.

Within this context, since R = BW forms an overdetermined system of equations, during Step 4, both sides of the equation set undergo multiplication by $B^T$ to attain a unique solution for the system.

Consequently, employing the least square method allows us to derive the following equations:

$$R_{PINN} = 29.747865a - \frac{5.074928}{m} + 4.868439\beta - 5.567125\beta^2 + 0.026213G - 1.197055e^{-5}G^2 - 1.358508 \quad (28)$$

$$R_{GA} = 38.083637a - 138.151074m + 250.945617m^2 - 130.449000m^3 + 6.816511\beta + 0.074001G - 5.509500e^{-5}G^2 - 9.143459 \quad (29)$$

By utilizing Eqs. (28) and (29), we can directly predict the respective $R_{PINN}$ and $R_{GA}$ based on a set of hyperparameters, including $a, \beta, G$, and $m$. Hence, by fine-tuning the four hyperparameters, we can reduce the values of the loss functions $R_{PINN}$ and $R_{GA}$, leading to a more accurate representation of the connection between bond stress and slip.

We can assess the fitting effectiveness of both MLR and the enhanced method. We employ Mean Squared Error (MSE) to compare the fitting outcomes of the two methods, as it effectively captures the overall disparity between the predicted and actual values. A comparison of the MSE between the two methods reveals a significant reduction in MSE for the enhanced method, leading to a substantial improvement in fitting effectiveness. In contrast to MLR, the enhanced method exhibits a notable improvement, with a reduction in MSE by 33.30% and 41.23%, respectively. This signifies significant progress. Consequently, the equation derived from the enhanced method better captures the relationship among the four hyperparameters $a, \beta, G, m$, and $R_{PINN}$ and $R_{GA}$.

Specifically, we can analyze the $R_{PINN}$ and $R_{GA}$ for each set of specimens under different methods. Upon examination, it becomes evident that the enhanced method enhances the fitting effect compared to MLR. Thus, it can be concluded that the customized MLR method is effective.

Through a comparison of MSE, it is evident that the overall fitting effectiveness of hyperparameters $a, \beta, G, m$, and $R_{PINN}$ and $R_{GA}$ has been enhanced. Analyzing Table 11 and Fig. 10 reveals that, in comparison to MLR, the improved method provides more accurate predictions for $R_{PINN}$ and $R_{GA}$ across most specimens based on hyperparameters $a, \beta, G, m$. For instance, in the case of specimen R16C1L1F155, the discrepancy between the true $R_{PINN}$ and the MLR-predicted value is 2.7374, whereas the improved method yields a difference of only 0.9578919. Similarly, the actual $R_{GA}$ varies by 1.6783 from the MLR-predicted value but only by 0.4176693 from the prediction made by the improved method. The enhanced method demonstrates substantial progress in fitting accuracy.

## 5. Conclusion

Integrating PINN, GA, and MLR can be particularly beneficial for studying the hyper-parameters and loss functions in predicting the ultimate bond stress-slip relationship for reinforced UHPC structures.

Combining the physics-based behavior captured by PINN and the data-driven behavior incorporated by MLR, this approach could optimize the model's performance and accurately predict the ultimate bond stress-slip relationship. The coordinated adjustment of the hyperparameters of the different models could balance the exploration and exploitation of the solution space, avoid overfitting or underfitting of the training data, and capture complex nonlinearities in the data. Such a predictive model can help design and evaluate the performance of UHPC structures in infrastructure and construction projects, which require high strength and durability.

* The study highlighted that PINN required less prior knowledge, such as setting the range of values for hyperparameters in advance, in contrast to GA. Despite the fluctuation in the early stage, PINN was found to converge to more accurate local minima with a reduced data error. The accuracy of the parameters identified by PINN surpassed those identified by GA.

* To enhance MLR by identifying relationships between each hyper-parameter and loss function, this research established equations between hyperparameters and loss functions using a data-driven least square method. Upon comparison, it became evident that the improved method achieved more accurate loss predictions through hyperparameters.

* The comparison of MSE indicated a significant overall improvement in the fitting effectiveness of hyperparameters $a, \beta, G, m$, $R_{PINN}$ and $R_{GA}$ through the customized MLR method, as demonstrated by more accurate predictions across various specimens compared to the standard MLR approach.

* Upon comparing the outcomes of MLR with the enhanced method, we observed a reduction in MSE by 33.30% and 41.23%, respectively. This indicated an overall improvement in the fitting effectiveness of both hyperparameters and loss functions through the enhanced method. Specifically, the customized MLR achieved predictions closer to the actual values for approximately 60% of the specimens compared to normal MLR.

## 6. Possible directions for future studies

Using machine learning techniques to predict the ultimate bond stress-slip relationship for reinforced UHPC structures presents several opportunities for future research. Some potential directions for future studies in this field include expanding the scope of the models to incorporate more complex and multi-dimensional factors that affect the behavior of UHPC structures, such as temperature, moisture, and corrosion. Also, considering seismic and dynamic loadings would prove advantageous for future studies. Additionally, incorporating more comprehensive datasets and other types of machine learning models, such as deep learning, could enhance the accuracy and predictive power of the models. Moreover, validating the models through extensive experimental and field testing and applying them to real-world
engineering projects can further demonstrate their effectiveness and applicability. Finally, exploring the potential for integrating machine learning techniques with other design and optimization methods, such as finite element analysis and topology optimization, could yield more efficient and practical approaches to UHPC structure design and evaluation.

## Fundings

This research was supported by: Bridge Non-destructive Testing (NDT) and Engineering Computation Sichuan Provincial University Key Laboratory (2022QYY01).

Science and Technology Department of Sichuan Province (2022YFWZ0010).

Science and Technology Department of Sichuan Province (2022NSFSC1992).

The Key Laboratory of Colleges and Universities in Sichuan Province (2020 QYJ01).

## Declaration of Competing Interest

The authors declare no conflict of interest.

## Data availability

The datasets used or analyzed during the current study are available from the corresponding author upon reasonable request.

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