Bond Strength Evaluation of FRP–Concrete Interfaces Affected by Hygrothermal and Salt Attack Using Improved Meta-Learning Neural Network

Abstract

Fiber-reinforced polymer (FRP) laminates are popular in the strengthening of concrete structures, but the durability of the strengthened structures is of great concern. Due to the susceptibility of the epoxy resin used for bonding and the deterioration of materials, the bond performance of the FRP–concrete interface could be degraded due to environmental exposure. This paper aimed to establish a data-driven method for bond strength prediction using existing test results. Therefore, a method composed of a Back Prorogation Net (BPNN) and Meta-learning Net was proposed, which can be used to solve the implicit regression problems in few-shot learning and can obtain the deteriorated bond strength and the impact weight of each parameter. First, the pretraining database Meta1, a database of material strength degradation, was established from the existing results and used in the meta-learning network. Then, the database Meta2 was built and used in the meta-learning network for model fine-tuning. Finally, combining all prior knowledge, not only the degradation of the FRP–concrete bond’s strength was predicted, but the respective weights of the environment parameters were also obtained. This method can accurately predict the degradation of the bond performance of FRP–concrete interfaces in complex environments, thus facilitating the further assessment of the remaining service life of FRP-reinforced structures.

1. Introduction

In recent years, fiber-reinforced polymer (FRP) has seen extensive use in the externally bonded (EB) strengthening of concrete structures, owing to its advantages of a high strength-to-weight ratio, durability, and compatibility with concrete buildings [1]. However, prolonged service can lead to degradation in the performance of concrete structures strengthened with EB-FRP, particularly under harsh environmental conditions such as tidal conditions, which act as hygrothermal exposure and a salt attack. The combined effects of these parameters contribute to the uncertain evolution of the bond performance at the interface of EB-FRP-strengthened concrete [2]. Predicting the degradation of bond properties in EB-FRP-strengthened concrete is a significant challenge in engineering practice.

Several researchers have investigated the interfacial bond performance of EB-FRP-strengthened concrete structures under hygrothermal conditions [3,4,5,6,7,8,9]. It has been observed that moisture adversely affects both the strength of the FRP itself and its interlaminar shear strength [3]. Furthermore, its bond performance is compromised due to the susceptibility of epoxy resins to the adverse effects of water and high temperatures [4,5]. In engineering applications, aside from hygrothermal exposure, salt erosion emerges as another significant parameter contributing to the degradation of the interfacial bond performance of FRP-reinforced concrete. Studies indicate that concrete structures are susceptible to corrosive degradation by sulphate [10]. Although sulphate corrosion significantly impacts the bond performance of the FRP–concrete interface, it has minimal effects on the mechanical properties of the FRP material and adhesive [11]. Salt spray cycling environments have been shown to cause damage to concrete–adhesive interfaces [12]. Based on this observation, a bond-slip model for a carbon fiber-reinforced polymer (CFRP)-reinforced concrete interface under the influence of sulphate cycling has been proposed [13]. With the application of machine learning algorithms, research on the FRP-to-concrete interface’s performance is no longer limited to experimentation. Employing data-driven methodologies to address these complex prediction challenges is a pragmatic approach [14,15]. Researchers have advocated for the utilization of artificial neural network (ANN) models to forecast the strength of interfacial bonds, yielding superior predictive outcomes [16,17,18]. Additionally, a study has proposed an equation for interfacial bond strength that utilizes an ANN model based on artificial bee colony optimization [19]. Building upon this foundation, other researchers have introduced an innovative and interpretable machine learning technique to predict the interfacial bond strength of FRP concrete by integrating machine learning models with traditional physical models [20]. However, the majority of research, whether using experimental methods or machine learning methods, predominantly concentrates on assessing the durability of FRP-to-concrete interfaces with respect to a singular environmental factor [21,22]. There is a scarcity of research considering the coupling effect of environmental factors. This is because when considering the coupling effects of multiple factors, experimental design often becomes more complex and the scale larger. Additionally, due to limited experimental data, traditional machine learning methods, which rely on extensive databases, cannot be effectively applied.

However, for complex studies involving numerous influencing variables, few-shot learning, transfer learning, and meta-learning methods offer viable approaches [23]. Some researchers have accomplished the prediction of square steel bar bond-slip curves by utilizing experimental data on the bond-slip of circular-steel-reinforced concrete as prior knowledge and a small amount of experimental data on the bond-slip of square-steel-reinforced concrete as posterior knowledge [24]. Hence, numerous experimental studies have explored the effect of single environmental factors, generating prior knowledge for developing prediction models of bond performances under the coupling effect of multiple environmental factors. It is notable that multiple-factor attacks may occur in coastal, high-temperature, tidal-action environments. Consequently, by combining a small amount of posterior knowledge, meta-learning can integrate diverse prior knowledge to formulate a degradation model influenced by multiple factors, addressing the limitations of traditional neural networks.

This study employed a limited amount of experimental data that considered the collective impact of hydrothermal and salt attacks to forecast deterioration trends in bond performance. It tackled the implicit regression challenges inherent in few-shot learning. Through the establishment of a database tailored to the model-agnostic meta-learning (MAML) algorithm, this research predicted the decline in CFRP-to-concrete interface bond strength and determined the significance of various influencing factors. Consequently, it elucidated the degradation patterns of interface bond performances, overcoming hurdles such as scant experimental data, complex feature dimensionality, and prediction difficulty, thereby showcasing its generalizability.

2. Summary of the Existing Test Results

Extensive experiments have been conducted [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47] to investigate the bond strength of CFRP-to-concrete interfaces under diverse environmental conditions, proposing assessment methods of bonding strength using the ultimate load at interface failure. In this section, based on the assessment criteria, the bond strength is regarded as the target parameter of the neural network. Two databases, Meta1 and Meta2, are established for prior knowledge and posterior knowledge of the neural network, respectively. The Meta1 database contains experimental data on bond strength degradation that consider only the influence of a single factor, while the Meta2 database contains experimental data on bond strength degradation that consider the simultaneous influence of hygrothermal and salt attacks.

2.1. Meta1 Database of Prior Knowledge

In this section, drawing on 22 references [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46], the Meta1 database is established, comprising 436 samples of CFRP-to-concrete interfacial ultimate loads under hygrothermal or salt attack after filtering out invalid experimental data and averaging across replicate groups, as shown in

Table 1. Data X in the Meta1 database.

Serial Number and Factors	Parameter Range	1	2	…	436
1—Chloride Concentration (%)	0–55.00	0	0	…	55.000
2—Sulfate Concentration (%)	0–88.00	0	0	…	7.68
3—Cycles (d)	0–1400.00	0	14.00	…	150.00
4—Soaking Time (h)	0–48.00	0	0	…	24.00
5—Drying Time (h)	0–16.00	0	0	…	0
6—Heating Time (h)	0–14.00	0	0	…	0
7—Temperature ($\mathrm{℃}$)	0–60.00	24.00	5.00	…	0
8—Humidity (%)	50.00–100.00	77.00	95.00	…	100.00
9—Soaking Temperature ($\mathrm{℃}$)	0–40.00	0	0	…	25.00
10—Drying Temperature ($\mathrm{℃}$)	0–25.00	0	0	…	0
11—Heating Temperature ($\mathrm{℃}$)	0–65.00	0	0	…	0
12—Paste Length (mm)	60.00–720.00	520.00	520.00	…	300.00
13—Paste Layers	1.00–2.00	1.00	1.00	…	1.00
14—CFRP Tensile Strength (MPa)	1560.00–4750.00	4750.00	4750.00	…	3355.00
15—CFRP Elastic Modulus (GPa)	104.00–306.700	230.00	230.00	…	230.10
16—CFRP Elongation (%)	1.1760–3.050	1.500	1.500	…	1.600
17—CFRP Surface Density ($\mathrm{g}/{\mathrm{m}}^2$)	216.00–3600.00	404.00	404.00	…	300.60
18—CFRP Dimensions (${\mathrm{m}\mathrm{m}}^3$)	360.00–20,000.00	5980.00	5980.00	…	7515.00
19—Concrete Strength (MPa)	21.00–71.60	35.40	35.40	…	63.60

This table is for illustration purposes only, with more data presented in Table S1.

and

Table 2. Data Y in the Meta1 database.

Factor	Parameter Range	1	2	…	436
Ultimate Load (KN)	3.9900–132.3833	132.3833	122.7373	…	23.1000

This table is for illustration purposes only, with more data presented in Table S1.

. The data serial numbers from 1 to 19 in Table 1 represent the input factors of the neural network, while the data in Table 2 serve as the output factors of the neural network. Moreover, to eliminate the influence of different experimental conditions on the parameter values, a normalization method is adopted to process the factor values and ultimate loads. This study employs feature scaling normalization. It involves utilizing the feature values of each parameter as standard values against which the different sample values of the parameter are measured. The normalized data are derived by dividing the sample values by their corresponding standard values.

The data in the Meta1 database are tailored to similar target tasks. Predominantly, these samples were acquired through double shear tests, single shear tests, and a number of four-point bending tests on CFRP-reinforced concrete specimens. Because various experimental studies present different parameters, necessitating the establishment of unified standards to streamline the marking process, the fundamental material properties of CFRP (serial numbers from 14 to 18) are ascertained via the pertinent literature [25,35] and technical specifications [48] (GB/T 21490-2008). The density of the CFRP laminate stands at $\rho =1.8\times {10}^6\mathrm{g}/{\mathrm{m}}^3$, while its thickness is set at $\delta =0.167\mathrm{m}\mathrm{m}$. Equations (1)–(3), respectively, compute the elongation, tensile strength, and surface density of the CFRP laminate, with Equation (4) determining the ultimate load at interface failure. The specifications of the CFRP laminate used in the test (serial numbers from 12 to 13) are obtained according to the technical specification [49]. The experimental conditions comprise two categories: environmental temperature and humidity treatments and wetting–drying cycles. Serial numbers 7 to 8 denote the temperature and humidity of the former. Serial numbers 4 to 6 and 9 to 11 indicate the time and temperature of the soaking, subsequent drying, and subsequent heating which form the description of the wetting–drying cycles. Serial number 3 signifies the recurring cycle times of the same environmental attack, while the salt-attacked environment (serial numbers from 1 to 2) is obtained from technical specifications [50].

$$\epsilon =\frac{\sigma }{E_f}$$

(1)

$$\sigma =E_f\times \epsilon$$

(2)

$$\xi =\rho \times \delta$$

(3)

$$P_u=\sigma \times S$$

(4)

where $\epsilon$ is the elongation of the CFRP laminate, $\sigma$ is the tensile strength of the CFRP laminate, $E_f$ is the elasticity modulus of the CFRP laminate, $\xi$ is the areal density of the CFRP laminate, $\rho$ is the volume density of the CFRP laminate, $\delta$ is the thickness of the CFRP laminate, $P_u$ is the ultimate load, and $S$ is the interfacial bond area.

Due to the lack of a unified and comprehensive definition of failure modes across various literature sources, this model does not incorporate failure modes as input factors. Moreover, researches have demonstrated the superior shear and tensile strength of epoxy resin adhesive layers compared to concrete, with organic adhesives effectively permeating CFRP fibers [32,35]. Consequently, the material properties of the epoxy resin adhesive are similarly omitted as factors.

The data within Meta1 are segmented into 50 prediction tasks. Within each prediction task, samples are further subdivided into training and test sets in a 4:1 ratio. To ensure an adequate distribution of samples across tasks, an assumption of equal sample distribution is made, mitigating the risk of inadequate task samples.

It is noteworthy that the Meta1 database integrates all data individually affected by hydrothermal and salt attacks. Therefore, for the former, the salt-related factors are set to 0, and for the latter, the hydrothermal-related data are set to 0. Consequently, for conventional neural networks, this dataset exhibits significant data gaps.

2.2. Meta2 Database of Posterior Knowledge

The Meta2 database is generated under the combined effect of hydrothermal and salt attacks to obtain the ultimate interfacial load for posterior data analysis [40,47]. Five groups of 15 CFRP-to-concrete interface double shear tests considered wetting–drying cycles and sixty groups of 180 CFRP-to-concrete interface double shear tests considered wetting-drying cycles, CFRP bond length, the water-to-cement ratio, fly ash admixture, and sulfate concentrations. Each group comprised duplicate samples.

Table 3. Data X in the Meta2 database.

Serial Number and Factors	Parameter Range	1	2	…	80
1—Chloride Concentration (%)	0–0	0	0	…	0
2—Sulfate Concentration (%)	0–10.00	10.00	10.00	…	5.00
3—Cycles (d)	0–150.00	30.00	30.00	…	90.00
4—Soaking Time (h)	6.00–12.00	12.00	12.00	…	6.00
5—Drying Time (h)	1.00–4.00	4.00	4.00	…	1.00
6—Heating Time (h)	5.00–8.00	8.00	8.00	…	5.00
7—Temperature (°C)	0–0	0	0	…	0
8—Humidity (%)	100.00–100.00	100.00	100.00	…	100.00
9—Soaking Temperature (°C)	25.00–30.00	25.00	25.00	…	30.00
10—Drying Temperature (°C)	25.00–25.00	25.00	25.00	…	25.00
11—Heating Temperature (°C)	40.00–40.00	40.00	40.00	…	40.00
12—Paste Length (mm)	60.00–180.00	60.00	80.00	…	100.00
13—Paste Layers	1.00–1.00	1.00	1.00	…	1.00
14—CFRP Tensile Strength (Mpa)	3106.0000–3659.0000	3659.00	3659.00	…	3106.00
15—CFRP Elastic Modulus (Gpa)	256.0000–275.6000	265.00	265.00	…	275.60
16—CFRP Elongation (%)	1.5860–1.7140	1.714	1.714	…	1.586
17—CFRP Surface Density (g/m2)	205.0000–300.6000	205.00	205.00	…	300.60
18—CFRP Dimensions (mm3)	115.2000–835.0000	115.20	153.60	…	835.00
19—Concrete Strength (Mpa)	18.3500–54.4000	34.16	34.16	…	31.60

This table is for illustration purposes only, with more data presented in Table S2.

illustrates 65 sets of test details labeled according to the format of the data in Meta1.

Table 4. Data Y in the Meta2 database.

Serial Number	Parameter Range	1	2	…	80
Ultimate Load (KN)	5.80–22.90	16.60	18.10	…	6.40

This table is for illustration purposes only, with more data presented in Table S2.

displays the interfacial ultimate loads for 65 sets of specimens. Consequently, we constructed the database Meta2 under the combined effects of hydrothermal and salt attacks.

The databases (Meta1 database and Meta2 database) essential for the MAML algorithm used in this study are nearly finalized. The forthcoming section will introduce the application of the MAML algorithm.

3. Application of Model-Agnostic Meta-Learning (MAML) Model

When employing neural networks for prediction, their accuracy heavily relies on the size of the training dataset. Insufficient training samples lead to a significant decrease in predictive accuracy. However, in the era of big data, similar prediction tasks can always be found to provide some prior knowledge to the network. In this study, for the task of predicting the degradation of the bonding strength between CFRP and concrete under the combined effect of hydrothermal and salt attacks, similar prediction tasks would involve predicting the degradation of the bonding strength between CFRP and concrete under the influence of either a hydrothermal or salt attack individually. These tasks share common inputs (such as the material properties of CFRP and concrete) and common outputs (bonding strength), despite differences in their parameters related to temperature, humidity, and salt. These shared characteristics provide a solid knowledge foundation for neural network learning. Therefore, when the sample size for a target prediction task is small, learning prior knowledge on large-scale datasets similar to the target task is recommended, and then the neural network can be transferred to the target task. In the field of deep learning, this problem is called few-shot learning, and the MAML algorithm has been proposed [51] to solve this problem.

3.1. MAML Model

Unlike conventional neural networks, which optimize learning outcomes based on individual samples, the MAML algorithm focuses on tasks as its training target, thereby creating the learning process. Instead of directly acquiring a mathematical model for prediction, it aims to learn “how to learn a mathematical model more efficiently and effectively.” MAML is primarily divided into two components: meta-learning and fine-tuning processes. Meta-learning involves designing model characteristics that can be shared across various tasks, facilitating these characteristics’ adjustments. Its objective is to train the model on a series of related tasks to gather useful general knowledge for tackling new tasks. The meta-learning training process entails iterative tasks, wherein the model utilizes its current parameters for each task and computes the loss specific to that task. Subsequently, the model parameters are updated by calculating a weighted average of the losses across these tasks. Fine-tuning takes place after meta-learning (as delineated in the fine-tune steps in Algorithm 1), during which the model’s parameters are fixed and further adjustments are made to adapt to specific new tasks. This fine-tuning stage resembles traditional machine learning training. Throughout fine-tuning, the model is trained using data from new tasks, adjusting parameters via gradient descent to minimize the loss function associated with the new tasks. Since the model’s initial parameters have been optimized during meta-learning, the fine-tuning training process typically requires fewer iterations and can swiftly adapt to new tasks. Consequently, even with limited existing data, MAML can still achieve high accuracy.

Algorithm 1. Algorithm of MAML.

Input: Point set X with 19 characteristics and 436 samples

Label set Y with 1 characteristic and 436 samples

Training parameter: $\theta$-weighs and bias in the net

Hyper parameter: epoch1-The number of times each task was trained in the MAML

epoch2-The number of times each task was trained in the Fine-tune

α-parameters to control the learning rate of the net

β-parameters to control the learning rate of the task

Data segmentation: 1. Divide the samples into training set and test

set2. Divide the training set into M tasks, the test set into N tasks

Steps in the MAML:

1. $\theta$ initialization

2. for j = 1, …, epoch1

3.  for batches of tasks in M tasks

4.    ${\theta }_{temp}{\theta }_{temp}$ initialization and get $\widehat{y}(X,\theta )$ by forward propagation

5.    ${\theta }_{temp}$ Calculate the loss of each task $l_{m_t}=\sum _{i=1}^m{(y-\widehat{y}(X,\theta \left)\right)}^2/m$

6.    ${\theta }_{temp}$ Update ${\theta }_{temp}$ for i times ${\theta }_{m_t}^i={\theta }_{m_t}^{i-1}-\alpha {\nabla }_{\partial {\theta }_{m_t}^{i-1}}{l}_{m_t}\left({\theta }_{m_t}^{i-1}\right)$

7.  end

8.  Calculate the loss function of each batch $L\left(\theta \right)={\sum }_{m_t=1}^{m_B}{l}_{m_t}\left({\theta }_{m_t}^i\right)={\sum }_{m_t=1}^{m_B}{l}_{m_t}({\theta }_{m_t}^{i-1}-\alpha {\nabla }_{\partial {\theta }_{m_t}^{i-1}}{l}_{m_t}({\theta }_{m_t}^{i-1}\left)\right)$

9.  Update $w_j^{\prime }$ in the θ by $w_j^{\prime }=w_j-\beta \frac{\partial L\left(\theta \right)}{\partial w}=w_j-\beta {\sum }_{m_t=1}^{m_B}\frac{\partial l_{m_t}\left({\theta }_{m_t}^i\right)}{\partial w_{{lm}_t}^i}$

10. end

Output: ${\theta }^{\prime }\left(w_j^{\prime }\right)$

Steps in the Fine-tune of a specific task in N tasks:

1. Froze the $w_j^{\prime }$ in the shallow layer of ${\theta }^{\prime }$

2. for j = 1, …, epoch2

3.  for batches of samples in the training set

4.  Get $\widehat{y}(X,\theta )$ with Forward propagation.

5.  Calculate the loss function: $L=\sum _{i=1}^m{(y-\widehat{y}(X,\theta \left)\right)}^2/m$

6.  Update θ to ${\theta }^{\prime }=\theta -\alpha {\nabla }_{\theta }L(X,Y,\theta )$

7.  end

8. end

Output: ${\theta }^{\prime }$

The steps outlined in Algorithm 1 describe the MAML process comprehensively. The notation ‘j’ is the current training time, ‘m’ represents the sample number, $\widehat{y}(X,\theta )$ is the predicted value, ${\theta }^{\prime }$ represents the updated weights and biases at the current training time, ‘y’ is the labeled value, ${\theta }_{temp}$ is a temporary parameter used to search for gradient descent directions, $m_B$ is the number of tasks in a batch, $m_t$ is the $m_t$-th task in the train, $\partial$ is the notation of a partial differential, $w$ is the component of array $\theta$ which represents a single weight or bias value, ‘L’ is the loss function of the regression problem, and $\nabla$ is the gradient of the variable.

The MAML model has three assumptions. The first is the fast adaptation assumption. Under this assumption, the MAML model can achieve good performance on a new task with just a few gradient updates, meaning that the initial parameters obtained through meta-training are close to the optimal parameters in the task space. The second is the shared parameter assumption. Under this assumption, although tasks are different, they can share the same model’s initialization parameters, namely its meta-parameters. These meta-parameters are learned through an optimization process across multiple tasks. The third is the finite sample assumption. Under this assumption, for each task only a small number of training samples (usually referred to as K-shot learning) are needed to adapt the model. This assumption is aligned with the goal of MAML, which is to achieve rapid learning.

This research utilizes two databases, Meta1 and Meta2, to develop the MAML model. The Meta1 database considers the CFRP-to-concrete interfacial ultimate loads under different environments (hygrothermal or salt attack) separately. The coupled database possesses features that influence the interfacial ultimate load under both wet–dry attacks and salt attacks, enabling the MAML model to perform transfer learning between these two domains during training. This enhances the model’s generalization ability and performance on new tasks, making it applicable to data obtained from both wet–dry attack and salt attack environments. The Meta2 database is entirely unrelated to the Meta1 database; it considers the interfacial ultimate load under the joint action of sulfate ions and wet–dry cycle environments. However, after training with Meta1, the MAML model is already suitable for environments where wet–dry and salt ions act together. Therefore, under Meta2 processing, the MAML model will demonstrate even stronger applicability.

3.2. MAML Training and Test Results

To ascertain the accuracy of the MAML prediction methodology, database Meta2, which is not involved in the training process, is utilized to evaluate its network performance. This posterior database is partitioned into distinct training and testing sets, and during the fine-tune process these sets are locked. However, throughout every meta-learning epoch, the assignment of tasks to training and testing sets remained dynamic; upon the completion of an epoch, tasks were reallocated, ensuring that both the training and testing sets were consistently resampled at a 4:1 ratio. The hyperparameters $\alpha$ and $\beta$ of the model were set to 0.01. In order to mitigate the possibility of incidental outcomes, a total of six repeated predictions were conducted, and the predictive results are illustrated in

Table 5. Prediction results of MAML.

True value	0.18075	0.17955	0.22624	0.12481	0.14662
	0.15113	0.15639	0.0797	0.10752	0.09624
	0.0985	0.10752	0.04812	0.06917	0.12105
	0.06842	0.07293	0.0609	0.05338	0.04812
Test number	Comparation between true value and predicted value
1	Prediction	0.19447	0.18574	0.21813	0.11399	0.12917
	Error (%)	7.59%	3.45%	3.58%	8.67%	11.90%
	Prediction	0.1315	0.13119	0.10402	0.11414	0.11631
	Error (%)	12.99%	16.11%	30.51%	6.16%	20.85%
	Prediction	0.10779	0.10747	0.0575	0.10267	0.13875
	Error (%)	9.43%	0.05%	19.49%	48.43%	14.62%
	Prediction	0.0962	0.0962	0.08221	0.07765	0.07765
	Error (%)	40.60%	31.91%	34.99%	45.47%	61.37%
	Average value of error					21.41%
2	Prediction	0.1886	0.16743	0.21815	0.12658	0.12613
	Error (%)	4.34%	6.75%	3.58%	1.42%	13.97%
	Prediction	0.13867	0.17593	0.10879	0.08957	0.09052
	Error (%)	8.24%	12.49%	36.50%	16.69%	5.94%
	Prediction	0.08618	0.13234	0.04563	0.10025	0.08172
	Error (%)	12.51%	23.08%	5.17%	44.93%	32.49%
	Prediction	0.07311	0.07534	0.07534	0.07979	0.07979
	Error (%)	6.85%	3.30%	23.71%	49.48%	65.81%
	Average value of error					18.86%
3	Prediction	0.18796	0.17248	0.21364	0.10894	0.18536
	Error (%)	3.99%	3.94%	5.57%	12.72%	26.42%
	Prediction	0.1793	0.14326	0.08592	0.10988	0.10966
	Error (%)	18.64%	8.40%	7.80%	2.19%	13.94%
	Prediction	0.10735	0.12102	0.03915	0.08676	0.11362
	Error (%)	8.98%	12.56%	18.64%	25.43%	6.14%
	Prediction	0.08534	0.07927	0.04856	0.05382	0.06791
	Error (%)	24.73%	8.69%	20.26%	0.82%	41.13%
	Average value of error					13.55%
4	Prediction	0.17362	0.17145	0.20447	0.14038	0.1624
	Error (%)	3.94%	4.51%	9.62%	12.47%	10.76%
	Prediction	0.14833	0.15921	0.06257	0.10351	0.10385
	Error (%)	1.85%	1.80%	21.49%	3.73%	7.91%
	Prediction	0.12456	0.12149	0.06154	0.08116	0.12419
	Error (%)	26.46%	12.99%	27.89%	17.33%	2.59%
	Prediction	0.08554	0.07251	0.06977	0.05816	0.05051
	Error (%)	25.02%	0.58%	14.56%	8.95%	4.97%
	Average value of error					10.97%
5	Prediction	0.1575	0.1777	0.187	0.13353	0.10226
	Error (%)	12.86%	1.03%	17.34%	6.99%	30.26%
	Prediction	0.12764	0.13711	0.10588	0.11791	0.09229
	Error (%)	15.54%	12.33%	32.85%	9.66%	4.10%
	Prediction	0.0937	0.11422	0.04492	0.07759	0.11979
	Error (%)	4.87%	6.23%	6.65%	12.17%	1.04%
	Prediction	0.07659	0.08549	0.07472	0.07472	0.06124
	Error (%)	11.94%	17.22%	22.69%	39.98%	27.27%
	Average value of error					14.65%
6	Prediction	0.20051	0.17686	0.22602	0.10158	0.1814
	Error (%)	10.93%	1.50%	0.10%	18.61%	23.72%
	Prediction	0.13381	0.1324	0.08296	0.12441	0.12497
	Error (%)	11.46%	15.34%	4.09%	15.71%	29.85%
	Prediction	0.1153	0.09357	0.08316	0.08698	0.09428
	Error (%)	17.06%	12.97%	72.82%	25.75%	22.11%
	Prediction	0.07636	0.06469	0.06469	0.05541	0.0566
	Error (%)	11.60%	11.30%	6.22%	3.80%	17.62%
	Average value of error					16.63%

.

In the partitioned test set, as depicted in Table 5, the predictive results of each repeated prediction are compared against the true value, and the errors between them are calculated. Here the error is equal to (predicted value–true value)/true value. At the end of each test, the average error between the predicted value and the true value is calculated using the absolute value of the 20 predictions. It is evident that although the overall average error is relatively small, ranging from 10.97~21.41%, there are occasional instances of significant errors in individual data points. These may stem from incomplete features in those specific data instances. In the Meta2 database, the chloride concentration feature values and temperature feature values for all sample data are 0. This is because we are only considering the coupling effect of sulfate ion corrosion with the wet–dry cycle environment. As a result, certain data points may lack crucial features, rendering the model unable to fully comprehend the data, thereby leading to significant prediction errors. This is an inevitable bias in deep learning algorithms. However, since these larger biases occur only sporadically, it can still be seen that the network possesses good learning and prediction capabilities. Moreover, there has been no indication of overfitting when the input factors reach 19, which may be attributed to factors such as regularization, cross-validation, and the model architecture within the MAML algorithm. Furthermore, the average errors from these six repeated tests are not consistent, which may be attributed to the variability in the initialization of model weights and biases before each training process.

Figure 1 illustrates the evolution of the proximity between predicted values and true values during the training and testing processes of the MAML algorithm. In order to assess the prediction accuracy more comprehensively, we decided to no longer rely solely on simple relative errors but introduce the coefficient of determination, R-squared, to better characterize this relationship. The equation of R-squared is shown in Equation (5):

$$R^2=\left\{\begin{array}{ll}\hspace{1em}\hspace{1em}0& R^2\le 1\\ 1-\frac{\sum {\left(y_i-{\widehat{y}}_i\right)}^2}{\sum {\left(y_i-\overline{y}\right)}^2}& {0\le R}^2\le 1\\ \hspace{1em}\hspace{1em}1& R^2\ge 1\end{array}\right.$$

(5)

where $y_i$ is the ith test value, $\widehat{y_i}$ is the ith predicting value, and $\overline{y}$ is the average value of the test values.

The number of iterations, serving as a special hyperparameter, is also tested, as shown in Figure 1. In MAML, an epoch refers to the number of iterations as well as the operation number of the entire training and testing dataset. To achieve the best results, it is essential to properly set the maximum number of epochs. In this study, epochs in the range of 0 to 5000 are selected. As the R-squared value approaches 1.0, indicating better prediction results, it is evident that, with increasing training epochs, both during the training and testing processes, the predicted results converge to higher values, around 0.8. As depicted in Figure 1a, the convergence speed during the training process is notably rapid. In the initial iterations, the predicted values quickly approach the true value, reaching an R-squared value of approximately 0.8 at around 300 iterations. Subsequently, the convergence rate slows down, with a slight increase in the R-squared value, nearing stability around 1000 iterations. The subsequent trend suggests that the value will eventually approach 1.0; however, this would require significantly increased costs. In Figure 1b, the evolution curve of R-squared values during the testing process and with increasing iterations is illustrated. It is noticeable that there is a notable distinction between the testing and training processes. While some individual tests still exhibit a rapid convergence speed consistent with the training process and their R-squared values are initially high, the convergence speed of tests 1, 2, and 3 significantly decreases. Although the results of these tests converge by the 1000th iteration, their R-squared values slightly decrease, stabilizing between 0.75 and 0.82. This occurrence is likely due to the disparity in the amount of data used between the training and testing processes. Therefore, the final convergence results may be biased, but the primary objective of achieving rapid convergence while maintaining prediction accuracy on a small-sample database has been accomplished.

Hence, the MAML prediction approach proves suitable for forecasting the interfacial bond performance between CFRP and concrete amidst varied environmental influences. Moreover, given the limited sample size of Meta2 tasks and the favorable prediction outcomes achieved by MAML, this method is also applicable to tasks with scant samples.

3.3. Comparison between MAML and Previous Analytical Models

In recent years, with the increasing application of deep learning in various engineering fields, there has been a growing number of cases where scholars in the field of civil engineering tend to use traditional ANN models, namely, backward propagation algorithms, for implicit regression problems. As shown in Figure 2, traditional neural networks are mainly divided into two groups: forward propagation and backward propagation. During forward propagation, with m training samples available, where X represents input variables and y represents output results, these m samples can be represented as $\left\{\right(x^{\left(1\right)},y^{\left(1\right)}),(x^{\left(2\right)},y^{\left(2\right)}),...,(x^{\left(m\right)},y^{\left(m\right)}\left)\right\}$. Input variable X is represented as an m × n_x matrix, where m denotes the number of training samples and n_x represents the dimensionality of the input feature vectors. Similarly, output variable y is represented as an m × n_y matrix, with m denoting the number of training samples and n_y representing the dimensionality of the output feature vectors. The above is the preparation of the dataset before training.

It is clearly indicated in Figure 3 that the input matrix X undergoes computations through the hidden layers of the neural network, where each layer calculates its respective formulaic results and applies an activation function (typically ReLU) to generate the layer’s output results. Eventually, these computations culminate in the final output $\widehat{y}$; this is the process of forward propagation. Back propagation differs; forward propagation calculated the predicted value $\widehat{y}$ and then used the loss function to calculate the difference (L) between the predicted value $\widehat{y}$ and the actual value y. In the process of backpropagation, the gradients of each layer’s parameters ${\nabla }_{\theta }$, calculated based on the loss function, are computed in reverse order, starting from the last layer. Subsequently, the weights of each layer are adjusted layer by layer ($\theta =\theta -\alpha {\nabla }_{\theta }L$), ultimately updating the parameters. This iterative procedure leads to a gradual decrease in the loss function L and thereby constitutes the backpropagation process.

In this section, a comparison is made between the BPNN, a widely used algorithm in civil engineering for implicit regression problems, and the MAML algorithm in their predictions of the degradation level of CFRP-to-concrete interface bond strength under the coupling effect of hydrothermal and salt attacks. This aims to demonstrate that the MAML algorithm, suitable for few-shot learning, is more appropriate for addressing complex implicit regression problems. To ensure a more reliable comparison of the predictive results between BPNN and MAML, the same database is adopted. The following content compares the accuracy of the predictive results, the fitting capability of the network models, and the convergence speed and precision of the two algorithms.

Figure 3 illustrates the comparison between the prediction results of both the MAML and BPNN using the database Meta2. A total of two repeated tests are conducted to avoid randomness. The x-axis represents the sample number of the test set, while the y-axis denotes the normalized bond strength. The black line represents the true values of 20 test samples, the red line represents the predicted values of the MAML algorithm, and the blue line represents the predicted values of the BPNN algorithm.

As shown in Figure 3, it can be observed that the predicted results of the BPNN in both tests are significantly higher than the true values. Moreover, after regularization, most of the predicted values of the BPNN are distributed between 0.15 and 0.20. For samples with lower strength, the BPNN does not exhibit satisfactory learning results, indicating that the network’s learning ability is limited and the results are underfitting. The main reason for the underfitting is the low sample size, which is also one of the important limitations of neural network algorithms in practical engineering applications. For the MAML algorithm, which incorporates prior knowledge, as shown by the red line in Figure 3, the predictive results are significantly superior to those of the BPNN algorithm. Moreover, it achieves better fitting results for both larger and smaller values. Figure 4 provides a more intuitive comparison of the fitting capabilities of the two algorithms.

Figure 4 depicts the fitting of the predictions of two algorithms. The black line in the middle represents perfect fitting, where the predicted values match the actual values. The red portion represents prediction data from the MAML algorithm, while the blue portion represents prediction data from the BPNN algorithm. The R² value indicates the slope of the scatter fit to the line, with values closer to 1.0 indicating a closer approximation to perfection. It can be observed that the computational results of the MAML algorithm are primarily distributed near the curve of perfect fitting, with the fitting results from two repeated predictions being almost identical, with R-squared values of 0.9137 and 0.9472, respectively. On the other hand, the computational results of the BPNN mostly cluster in the top-left of the graph, indicating that the neural network struggles to predict results effectively as the actual values decrease. This discrepancy highlights the superior predictive performance of MAML, compared to BPNN, of this complex task.

The loss function reflects the difference between the predicted values and the actual values. The closer the loss function is to zero, the closer the predicted values are to the actual values. This function is a crucial variable in the learning process of neural networks, and its evolution pattern represents the learning path or process of the neural network. Figure 5 illustrates the comparison of loss functions during the training and testing processes of two neural network algorithms.

Figure 5 provides a comparative analysis of the training and testing losses of MAML and BPNN across different numbers of repeated tests. Barring the initial phase, MAML consistently exhibits lower training losses than BPNN. The training loss of MAML swiftly approaches zero and steadily converges after a relatively short number of epochs, whereas BPNN’s training loss fluctuates with increasing epochs, displaying a tendency to remain largely unconverged and frequently showcasing outliers. It is not surprising that the BPNN algorithm fails to converge in its prediction on this database. After all, with only around 80 samples, and not all of them being high-quality, the dataset falls far short of the big data requirements of neural network algorithms. On the contrary, both in training and testing, the MAML algorithm converges rapidly on this small sample database.

In summary of this section, it is evident that MAML demonstrates superior applicability and accuracy in predicting the CFRP-to-concrete interfacial bond strength affected by the coupling effect of hydrothermal and salt ion attacks. However, this does not imply that the MAML algorithm universally outperforms the BPNN algorithm. When the database is sufficiently abundant and of high quality, the superiority of the BPNN algorithm has been demonstrated. Therefore, the significant role of the MAML algorithm lies in its ability to utilize low-quality databases generated by combining two or more unrelated factors to accomplish prediction tasks under multiple influences. This is something that the BPNN cannot achieve, as it is constrained by its requirements for a certain size and quality of database.

4. Correlation Analysis of Parameters Affecting Bond Strength

Through MAML, the association of uncorrelated factors affecting the same problem has been achieved. In this section, the built-in weight matrices of MAML are utilized to provide a ranking of the importance of the input factors on the outcome. Furthermore, after distinguishing between primary and secondary factors, the necessity of secondary factors and their impact on the prediction results are calculated, aiming to optimize and simplify the input structure of the model.

Table 6. Weight coefficients of input factors.

Test 1
number	1	2	3	4	5	6	7	8
weight	0.0441	0.1241	0.0180	0.0335	0.0503	0.2889	0.2048	0.0090
number	9	10	11	12	13	14	15	16
weight	0.0131	0.0794	0.0182	0.0275	0.0130	0.0022	0.0149	0.0055
number	17	18	19
weight	0.0227	0.0244	0.0068
Test 2
number	1	2	3	4	5	6	7	8
weight	0.0458	0.0916	0.0170	0.0554	0.0319	0.1682	0.0662	0.0155
number	9	10	11	12	13	14	15	16
weight	0.0337	0.0297	0.1270	0.0704	0.0258	0.0148	0.0094	0.0179
number	17	18	19
weight	0.0562	0.0979	0.0256
Test 3
number	1	2	3	4	5	6	7	8
weight	0.0861	0.0272	0.0172	0.0334	0.0532	0.2714	0.1075	0.0109
number	9	10	11	12	13	14	15	16
weight	0.0265	0.0850	0.0570	0.0324	0.0418	0.0082	0.0265	0.0147
number	17	18	19
weight	0.0078	0.0844	0.0088
Test 4
number	1	2	3	4	5	6	7	8
weight	0.0492	0.0899	0.0067	0.0124	0.1067	0.1697	0.0458	0.0094
number	9	10	11	12	13	14	15	16
weight	0.0119	0.1248	0.0802	0.0208	0.0525	0.0108	0.0323	0.0124
number	17	18	19
weight	0.0248	0.0765	0.0634
Average results
number	1	2	3	4	5	6	7	8
weight	0.0563	0.0832	0.0147	0.0336	0.0605	0.2245	0.1061	0.0112
number	9	10	11	12	13	14	15	16
weight	0.0213	0.0797	0.0706	0.0377	0.0333	0.0090	0.0208	0.0126
number	17	18	19
weight	0.0279	0.0708	0.0262

‘number’ means the serial number of each factor, as illustrated in Table 1. The results are computed based on the combination of the databases Meta1 and Meta2.

displays the weight coefficients of the 19 input factors in the MAML model, with a total of four repeated calculations. The final importance coefficient for each factor is obtained by averaging the results of the three calculations.

As shown in Table 6, the final ranking of the importance weights of the influencing factors is as follows: Heating Time > Temperature > Sulfate Concentration > Drying Temperature > CFRP Dimensions > Heating Temperature > Drying Time > Chloride Concentration > Paste Length > Soaking Time > Paste Layers > CFRP Surface Density > Concrete Strength > Soaking Temperature > CFRP Elastic Modulus > Cycles > CFRP Elongation > Humidity > CFRP Tensile Strength. It is worth noting that the lower significance of ‘Chloride Concentration’ compared to ‘Sulfate Concentration’ is attributed to the absence of the consideration of chloride ions in the validation dataset. The same situation also applies to missing factors in the test set. Due to these characteristics being missed or unchanged in the test set, a similar importance analysis should correspond to specific problems rather than being universally applicable.

For the problem of degradation in the bond performance of CFRP-to-concrete interfaces under the influence of hydrothermal and salt attacks, it is considered that factors related to hydrothermal and salt are direct factors, while the material properties of the CFRP and concrete are indirect factors. According to traditional neural network prediction algorithms, direct factors are often indispensable, while indirect factors can be pruned or ignored. However, based on the computed weight coefficients in Table 6, many direct factors do not have high weight coefficients, whereas indirect factors like ‘CFRP dimensions’ rank fifth in terms of their importance coefficients. Therefore, if there is a need to optimize the number of input parameters, it is not as straightforward as ignoring indirect factors in the MAML algorithm; rather, a certain pruning optimization must be carried out based on the ranking of the factors’ importance.

Due to the missingness of the database Meta2, the first factors to be eliminated are two factors not included in the test set, which are ‘Chloride Concentration’ and ‘Temperature’.

Table 7. Results of predictive accuracy.

Factors	R-Square of Training	R-Square of Testing	RMSE of Training	RMSE of Testing	Convergence Steps
F	0.914	0.878	0.096	0.098	489
F1-1-7	0.912	0.878	0.096	0.098	502
F2-14	0.909	0.870	0.097	0.099	500
F3-8	0.910	0.871	0.097	0.099	488
F4-16	0.857	0.832	0.106	0.113	553
F5-3	0.855	0.833	0.106	0.112	570
F6-15	0.848	0.833	0.108	0.112	704
F7-9	0.825	0.791	0.116	0.120	1080
F8-19	0.626	0.554	0.171	0.177	2223
F9-17	0.582	0.522	0.173	0.181	2200
F10-13	0.575	0.501	0.173	0.182	2466
F11-4	0.393	0.336	0.212	0.235	3110

‘F’ indicates the inclusion of all 19 factors, and ‘F1′ represents the parameters included in the first row of the table, and so forth. ‘-’ signifies the excluded factors. ‘R-square’ is represented by Equation (5), and ‘RMSE’ is represented by Equation (6). The convergence steps represent the number of iterations required during the testing process for the loss function to become less than 0.002.

displays the comparative results of the algorithm’s predictive accuracy after sequentially eliminating non-significant influential factors sorted by importance.

$$RMSE=\sqrt{\frac{1}{n}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}$$

(6)

where $y_i$ is the ith test value and ${\widehat{y}}_i$ is the ith predicting value.

A higher R-squared value, approaching 1, indicates a closer fit between the predicted and actual values, implying greater accuracy in predictions. In Table 7, it is evident that, as the number of input factors decreases, there is a consistent decline in the R-squared values, from the highest at 0.914 in the training set to 0.878 in the test set. Furthermore, the convergence difficulty increases in the test set, as reflected by the rise in training iteration steps and, consequently, increased computational expenses.

Based on the importance ranking of the input factors mentioned above, an optimization of the number of input factors was conducted. The performance of the MAML algorithm on the database Meta2 exhibited a trend of stepwise descent, where the exclusion of the first four factors (‘1—Chloride Concentration’, ‘7—Temperature’, ’14—CFRP Tensile Strength’, ‘8—Humidity’) do not yield significant impacts. Both the computational steps required for convergence and the accuracy of the predictions remained at satisfactory levels. Specifically, the convergence steps remained around 500, with an R-squared value above 0.91 during the training process, while the R-squared value for test results stayed above 0.87. The first occurrence of stepwise descent in the performance of the MAML algorithm appeared after the fifth computation result. During the training process, the R-squared value decreased from 0.91 to around 0.85, while, in the testing process, it decreased from 0.87 to approximately 0.83–0.80. The factors excluded during this stage included ‘16—CFRP Elongation’, ‘3—Cycles’, ’15—CFRP Elastic Modulus’, and ‘9—Soaking Temperature’. It is noteworthy that these factors encompassed two indirect and two direct factors, yet the computed results remained acceptable. The predictive accuracy did not decrease significantly, and the convergence speed remained relatively fast. This suggests that the optimization of direct factors is not necessarily ineffective, while the optimization of indirect factors may sometimes lead to a certain degree of decline in the predicted results. The subsequent reduction in input factors directly resulted in model underfitting, indicating that the excluded input parameters are somewhat necessary. Therefore, the approximate optimal number of parameters for optimization is around eight.

In summary, the importance analysis based on the databases Meta1 and Meta2 demonstrates a certain rationality, revealing the prioritization of all factors relevant to the prediction task. The higher the importance, the greater the factor’s impact on the prediction results. When analyzing the influence of each parameter, a certain degree of linear decline in computational results is observed when removing input factors according to their importance ranking. Up to a maximum reduction of eight input factors, good predictive accuracy can be maintained. However, exceeding eight factors leads to the underfitting of the network, resulting in a significant decrease in prediction accuracy.

5. Conclusions

In this study, a meta-learning neural network technique for a coupling factors-affected problem is proposed and based on the MAML algorithm. This neural network possesses the capability of coupling multiple non-associated factors that influence the same problem, enabling efficient regression predictions, with fewer samples, under the influence of compound factors. This capability is particularly applicable to the complexities faced in the field of civil engineering today, where problems are intricate, influencing factors are numerous and relatively independent, and the scarcity of samples often renders traditional neural networks inadequate for prediction tasks. The following conclusions are drawn:

(1)

The MAML algorithm has successfully achieved the prediction of the degradation of the performance of the interfacial bond between CFRP and concrete under the influence of multiple factors, based on prior knowledge of single-factor influences, using 65 samples. Its R-squared value in the test set reached 0.88, demonstrating its high accuracy and rapid convergence speed.

(2)

In contrast to the conventional neural network method BPNN used for regression problems in situations where multiple unrelated factors coexist, leading to dataset degradation, the MAML algorithm remains effective in its learning and prediction. In such scenarios, the BPNN algorithm only yields underfitting results due to its low network learning capability.

(3)

In the degradation of the interfacial bond strength between CFRP and concrete under hydrothermal and salt environments, a total of 19 influencing factors were identified. Among these, 11 are direct environmental factors, while 8 are indirect factors related to the materials or specimens themselves. Utilizing the weight matrix embedded within the MAML algorithm, the importance of each factor in predicting the outcome was ranked. The rank top five factors are as follows: Heating Time > Temperature > Sulfate Concentration > Drying Temperature > CFRP Dimensions, accounting for a combined total of 56.43% of the overall importance.

(4)

When pruning the input factors based on their importance ranking, it was observed that the predictive accuracy of the algorithm exhibited a stepwise decline. Initially, removing four factors resulted in minimal change in predictive accuracy. However, when the number of removed factors increased to 5–8, there was a noticeable decrease in computational accuracy, although it remained within an acceptable range. Beyond this threshold, as the number of removed factors continued to increase, the learning capability of the neural network model significantly declined, leading to underfitting results of predictive accuracy.

(5)

In addition to the factors with high importance rankings, special attention should be given to the factors that lead to a sudden drop in predictive accuracy when removed. These include “CFRP Elongation” during the first decrease, “Concrete Strength” during the second decrease, and “Soaking Time” during the final decrease. While these factors may not have large importance coefficients, they might be essential components linked to certain factors. Hence, their removal could cause significant fluctuations in predictive outcomes.

In this study, the application of the MAML algorithm represents its initial exploration. However, the uniqueness of this attempt requires further validation using other databases. Moreover, the MAML algorithm itself is subject to certain assumptions and limitations that warrant further optimization.