Data-Driven Prediction of Electrical Resistivity of Graphene Oxide/Cement Composites Considering the Effects of Specimen Size and Measurement Method

Abstract

The prediction of electrical resistivity of graphene oxide (GO) reinforced cement composites (GORCCs) is essential to promote the application of the composites in civil engineering. Traditional experiments find it challenging to capture the effect of various features on the electrical resistivity of the GORCCs. In this work, machine learning (ML) techniques are employed to explore the complex nonlinear relationships between different influencing factors and the electrical resistivity of the GORCCs. A total of 171 datasets are utilized for training and testing the ML models. It is demonstrated that the applied ML models are effective and efficient. Apart from the water/cement ratio, correlation analysis shows that the electrical resistivity of the GORCCs is highly dependent on the specimen size and measurement method. Feature importance analysis shows that the dispersion of GO has a significant influence on the electrical resistivity. The extreme gradient boosting (XGB) model and the artificial neural network (ANN) model with 3 hidden layers are proven to have better predictions, as evidenced by higher R² and lower root mean square error (RMSE). This work is envisioned to provide an effective and efficient way to identify the complex relationship between the material properties of the GORCCs and the various influencing factors.

1. Introduction

Cement composites have become the world’s most widely used building materials due to their versatility, durability, and cost-effectiveness [1]. They are integrated into buildings, bridges, roads, and other infrastructure projects [2]. However, traditional cement composites with structural functions have fallen short of meeting the contemporary demands for developing construction applications with multifunction [3,4,5]. Researchers have explored different methods to enhance the material properties of cement composites (i.e., compressive strength, flexural strength, electrical properties, thermal properties, etc.) by incorporating various nano-fillers into the cement matrix [6,7,8,9,10,11]. Among the numerous nano-fillers, graphene oxide (GO) [12,13,14], as a graphene derivative, has become a promising reinforcement candidate in the cement matrix due to its improved dispersion, excellent mechanical and physical properties, and low manufacturing cost [13,15,16,17].

Numerous works have been carried out on GO reinforced cement composites (GORCCs). Li et al. [18] added GO to cement and measured the electrical resistivity of the GORCCs by the two-electrode method. They found that the electrical resistivity of the cement pastes initially increased, reaching a peak at 0.02 wt% GO, and then decreased as the GO content further increased to 0.04 wt%. The electrical resistivity of the cement composites with well-dispersed GO was much lower than that of the cement matrix. Sartipi et al. [19] studied the electrical properties of the GORCCs, and the volumetric conductivity of the samples was tested at 7, 14, and 28 days. It was shown that the addition of GO increased the electrical conductivity of the samples. Qureshi et al. [17] found that the electrical resistivity of the cement composites with GO and rGO increased with the increase in hydration time. They concluded that the added nanofillers participated in the cement hydration process, which led to increased C-S-H gel pores in the composite, reduced mean free paths for electron transmission, and increased electrical resistivity. Guo et al. [20] found that the electrical resistivity of the GORCCs varied when the moisture in the cement matrix changed. Their investigation revealed that the electrical resistivity reached its minimum value when the GO content was 0.05 wt% and the water/cement ratio was 0.35. It was observed by Hu et al. [21] that when the GO content was 0.03 wt%, the electrical conductivity of the cement composites was nearly 50% higher than that of the plain sample. They found that an appropriate GO content resulted in a denser and more regular microstructure in the cement, which contributed to the enhanced electrical conductivity. Recently, Yan et al. [22] found that GO nanosheets and rGO could be highly aligned in cement pastes under an external magnetic field. The mechanical and electrical properties were found to be enhanced compared to the cement paste with a random distribution of the fillers. Xiong et al. [23] invented a novel power ultrasound (PUS) method for optimizing the dispersion of GO in cement paste. The synergistic effects of PUS-assisted mixing and PCE resulted in improved dispersion of the GO fillers.

Although numerous experiments have been conducted on the GORCCs, it is challenging to identify an explicit expression between the electrical properties and the many influencing factors such as w/c, content of GO, curing temperature, etc. [24]. In addition, during fabrication, the dispersion and measurement methods may also have potential impacts on the final results [15,18]. Recently, machine learning (ML) has attracted considerable attention in the field of composites due to its powerful data analysis capabilities as well as high efficiency without the need for conducting massive experiments [25,26,27]. Machine learning can consider various characteristics of different experiments in its model by collecting a large amount of data. In addition, machine learning is able to reflect the extent to which individual features affect the final result, as well as the complex nonlinear relationships between the involved features. Huang et al. [28] predicted the compressive and flexural strength of carbon nanotube (CNT) reinforced cement composites via ML methods, i.e., artificial neural network (ANN) and support vector machine (SVM). They found that ANN performed better on small datasets, whereas SVM was more suitable for large datasets. Montazerian et al. [29] investigated the thermal conductivity of graphene reinforced cement composites through a hybrid modeling approach employing ML and computational micromechanics models. Their findings indicated that the ANN demonstrated superior prediction accuracy and generalization ability compared to SVM and other models employed in their study. Bang et al. [30] used different ML techniques to evaluate the piezoelectric properties of CNT reinforced cement composites. They concluded that the type of loading, water/cement ratio (w/c), and CNT content were the influencing factors for the electrical resistance of CNT reinforced cement composites. Kekez et al. [31] investigated the application of ANN to predict the compressive and flexural strength of CNT reinforced cement composites. The regression coefficient of the optimal ANN was 0.94. Li et al. [32] employed linear regression (LR), support vector regression (SVR), random forest (RF), and extreme gradient boosting (XGB) to predict the compressive strength of cement paste, mortar, and concrete. The findings revealed that the optimal length and diameter of CNT were 20 μm and 25 nm, respectively, and the appropriate range of CNT content should be within 0.1 wt%. Alyousef et al. [33] utilized the GEP, ANN, and ANFIS models to predict the electrical properties of graphene reinforced cement composites. They found that the incorporation of 0.1% GNP content resulted in a notable 16.58% increase in strength and a 14% decrease in fractional change in resistivity. In addition, the R-value of the three models employed was 0.974, 0.963, and 0.954, respectively.

Although various ML models have been widely used in the field of composites, limited studies applied ML techniques to estimate the electrical properties of the GORCCs, which are intricately related to many features/influencing factors [34,35,36]. Under such circumstances, it is challenging and time-consuming to conduct massive experiments and establish an explicit expression to describe the nuanced relationships between the electrical properties and the many features. Additionally, the existing investigations neglect the effects of the specimen size as a crucial factor despite its evident influences. Based on the identified research gap, this work is the first attempt to employ ML techniques to investigate the electrical resistivity of the GORCCs considering the sample size as an input feature. Moreover, from the perspective of manufacturing, the dimensions of GO typically conform to a Gaussian distribution. Therefore, this study adopts a Gaussian function to characterize the distribution of the GO dimensions instead of using fixed values. Three different ML models are employed to investigate the complex nonlinear relationship between the electrical resistivity of the GORCCs and the coupling effects of the influencing factors. To train the ML model, this study collects 171 sets of experimental data for the electrical resistivity of the GORCCs. For discontinuous features, such as dispersion methods, cement type, etc., the one-hot encoding method is employed for feature input. Particularly, two parameters, i.e., K_s and K_h, are introduced to represent the dimensions of the specimens. Correlation analysis is conducted to examine the intrinsic relationships between the electrical resistivity of the GORCCs and the individual features, which are quantified through the Pearson correlation coefficient. Four evaluation metrics, i.e., R², mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE), are adopted to evaluate the prediction performance of the ML models.

2. Data Collection and Pre-Processing

Data collection and pre-processing are crucial for the accurate prediction of the ML models. In this study, a total of 171 experimental datasets were collected from the previous literature. In this work, the criteria for data selection include: (1) The source of experimental data should be published papers or technical reports in authoritative journals, and the results must be rigorously tested against specified requirements. (2) The data collected in the study should ensure diversity and be able to adequately reflect the effect of influencing factors on the GORCCs. The study must consider the w/c, type of cement, curing temperature, curing age, method of dispersion, method of testing, sample size, content of GO, and attributes of GO (i.e., diameter, number of layers, and thickness). (3) The data collected in this study can be normalized into a uniform format to facilitate subsequent data processing and the training of the ML models. Details of the 171 samples can be found in Appendix A 

Table A1. Details of the collected data.

No.	Type of Cement	Dispersion Method	Measurement Method	w/c	Curing Temperature	Curing Age	Ks	Kh	Diameter (μm)	Thickness (nm)	Layers	GO Content	Electrical Resistivity	Reference
1	1	1	1	0.40	20	28	0.66	0.3	0.2	2.49	2	0.000	183.95	[15]
2	1	1	1	0.40	20	28	0.66	0.3	0.8	1.43	3	0.005	206.12
3	1	1	1	0.40	20	28	0.66	0.3	2.5	1.56	2	0.010	230.08
4	1	1	1	0.40	20	28	0.66	0.3	0.7	2.39	1	0.015	262.88
5	1	1	1	0.40	20	28	0.66	0.3	1.8	1.08	1	0.020	302.39
6	1	1	1	0.40	20	28	0.66	0.3	0.3	2.69	1	0.025	296.78
7	1	1	1	0.40	20	28	0.66	0.3	0.4	2.74	2	0.030	293.97
8	1	1	1	0.40	20	28	0.66	0.3	0.2	1.38	1	0.035	294.57
9	1	1	1	0.40	20	28	0.66	0.3	2.4	1.75	1	0.040	291.26
10	1	1	1	0.45	23	1	1	1	1.3	2.26	1	0.000	7.58	[55]
11	1	1	1	0.45	23	2	1	1	1.9	1.11	1	0.000	10.41
12	1	1	1	0.45	23	3	1	1	2.1	1.79	2	0.000	13.08
13	1	1	1	0.45	23	5	1	1	1	1.49	3	0.000	15.33
14	1	1	1	0.45	23	7	1	1	1	1.6	1	0.000	17.24
15	1	1	1	0.45	23	10	1	1	0.6	2.51	2	0.000	17.88
16	1	1	1	0.45	23	14	1	1	0.9	2.58	1	0.000	18.68
17	1	1	1	0.45	23	18	1	1	0.9	2.33	1	0.000	19.24
18	1	1	1	0.45	23	20	1	1	1.5	2.32	2	0.000	19.57
19	1	1	1	0.45	23	25	1	1	2.9	2.73	2	0.000	20.22
20	1	1	1	0.45	23	28	1	1	1.4	2.65	2	0.000	20.59
21	1	1	1	0.45	23	1	1	1	0.5	1.21	1	0.010	9.73
22	1	1	1	0.45	23	2	1	1	0.8	1.93	1	0.010	13.06
23	1	1	1	0.45	23	3	1	1	1.3	1.87	1	0.010	15.73
24	1	1	1	0.45	23	5	1	1	0.6	2.83	1	0.010	16.90
25	1	1	1	0.45	23	7	1	1	0.5	2.55	1	0.010	18.10
26	1	1	1	0.45	23	10	1	1	1.5	1.17	1	0.010	18.83
27	1	1	1	0.45	23	14	1	1	0.9	1.61	1	0.010	19.80
28	1	1	1	0.45	23	18	1	1	2.5	2.44	1	0.010	20.79
29	1	1	1	0.45	23	20	1	1	2.1	1.81	3	0.010	21.32
30	1	1	1	0.45	23	25	1	1	1.9	2.84	2	0.010	22.59
31	1	1	1	0.45	23	28	1	1	0.4	1.84	1	0.010	23.29
32	1	1	1	0.45	23	1	1	1	2.6	1.58	1	0.020	7.64
33	1	1	1	0.45	23	2	1	1	1.3	1.71	1	0.020	10.54
34	1	1	1	0.45	23	3	1	1	0.7	1.97	3	0.020	12.91
35	1	1	1	0.45	23	5	1	1	1.3	1.55	1	0.020	14.82
36	1	1	1	0.45	23	7	1	1	2.4	2.17	1	0.020	16.94
37	1	1	1	0.45	23	10	1	1	0.3	1.52	1	0.020	17.93
38	1	1	1	0.45	23	14	1	1	2	2.8	1	0.020	19.14
39	1	1	1	0.45	23	18	1	1	0.9	2.88	2	0.020	19.76
40	1	1	1	0.45	23	20	1	1	0.1	1.28	1	0.020	20.27
41	1	1	1	0.45	23	25	1	1	0.6	2.21	1	0.020	21.09
42	1	1	1	0.45	23	28	1	1	0.2	1.23	3	0.020	21.84
43	1	1	1	0.45	23	1	1	1	1.8	2.84	1	0.040	7.96
144	1	1	1	0.45	23	2	1	1	0.8	1.47	1	0.040	9.32
45	1	1	1	0.45	23	3	1	1	0.6	2.47	3	0.040	10.83
46	1	1	1	0.45	23	5	1	1	2.9	1.46	3	0.040	13.67
47	1	1	1	0.45	23	7	1	1	2.5	2.95	1	0.040	16.30
48	1	1	1	0.45	23	10	1	1	0.6	2.91	3	0.040	17.48
49	1	1	1	0.45	23	14	1	1	0.8	2.06	1	0.040	18.96
50	1	1	1	0.45	23	18	1	1	1.9	2.68	1	0.040	19.36
51	1	1	1	0.45	23	20	1	1	2.7	1.81	1	0.040	19.60
52	1	1	1	0.45	23	25	1	1	0.3	1.74	2	0.040	20.73
53	1	1	1	0.45	23	28	1	1	1.8	2.38	3	0.040	21.64
54	1	1	1	0.45	23	1	1	1	1.9	1.44	2	0.080	7.34
55	1	1	1	0.45	23	2	1	1	1.8	1.21	2	0.080	9.39
56	1	1	1	0.45	23	3	1	1	0.6	1.57	3	0.080	11.23
57	1	1	1	0.45	23	5	1	1	2.2	1.36	2	0.080	13.38
58	1	1	1	0.45	23	7	1	1	0.1	1.97	3	0.080	15.28
59	1	1	1	0.45	23	10	1	1	0.4	1.36	2	0.080	16.90
60	1	1	1	0.45	23	14	1	1	2.1	2.86	2	0.080	18.75
61	1	1	1	0.45	23	18	1	1	1	1.88	2	0.080	19.15
62	1	1	1	0.45	23	20	1	1	0.3	2.27	1	0.080	19.41
63	1	1	1	0.45	23	25	1	1	1.7	1.35	3	0.080	20.28
64	1	1	1	0.45	23	28	1	1	2.3	1.58	1	0.080	20.79
65	1	1	1	0.45	23	1	1	1	1.1	2.78	2	0.160	7.46
66	1	1	1	0.45	23	2	1	1	0.6	1.03	1	0.160	9.42
67	1	1	1	0.45	23	3	1	1	0.8	2.46	1	0.160	10.95
68	1	1	1	0.45	23	5	1	1	1	2.01	2	0.160	13.69
69	1	1	1	0.45	23	7	1	1	0.4	2.39	2	0.160	16.09
70	1	1	1	0.45	23	10	1	1	0.4	2.6	3	0.160	16.98
71	1	1	1	0.45	23	14	1	1	2.8	2.45	2	0.160	18.13
72	1	1	1	0.45	23	18	1	1	1.9	1.73	1	0.160	18.59
73	1	1	1	0.45	23	20	1	1	1.7	2.58	1	0.160	18.84
74	1	1	1	0.45	23	25	1	1	0.7	2.32	1	0.160	19.54
75	1	1	1	0.45	23	28	1	1	2.1	1.31	3	0.160	19.98
76	1	1	1	0.40	20	28	0.66	0.3	0.6	1.91	2	0.007	90.52	[18]
77	1	1	1	0.40	20	28	0.66	0.3	0.3	1.41	2	0.008	85.67
78	1	1	1	0.40	20	28	0.66	0.3	1.4	2.09	3	0.009	78.49
79	1	1	1	0.40	20	28	0.66	0.3	1.8	2.07	2	0.010	69.65
80	1	1	1	0.40	20	28	0.66	0.3	2.4	1.73	3	0.011	61.86
81	1	1	1	0.40	20	28	0.66	0.3	0.1	2.73	3	0.012	53.02
82	1	1	1	0.40	20	28	0.66	0.3	0.4	2.91	2	0.013	43.26
83	1	1	1	0.40	20	28	0.66	0.3	1.2	1.82	2	0.014	41.60
84	1	1	1	0.40	20	28	0.66	0.3	2.7	1.64	1	0.015	39.64
85	1	1	1	0.40	20	28	0.66	0.3	3	1.24	1	0.016	39.02
86	1	1	1	0.40	20	28	0.66	0.3	0.2	1.31	3	0.017	36.08
87	1	1	1	0.40	20	28	0.66	0.3	2.8	2.37	3	0.018	34.42
88	1	1	1	0.40	20	28	0.66	0.3	2.6	2.82	2	0.019	32.46
89	1	1	1	0.40	20	28	0.66	0.3	1.8	2.23	1	0.020	30.19
90	1	1	1	0.40	20	28	0.66	0.3	2	1.02	1	0.021	28.59
91	1	1	1	0.40	20	28	0.66	0.3	1.6	1.48	2	0.022	25.95
92	1	1	1	0.40	20	28	0.66	0.3	0.8	2.44	3	0.023	24.97
93	1	1	1	0.40	20	28	0.66	0.3	0.9	1.14	1	0.024	24.36
94	1	1	1	0.40	20	28	0.66	0.3	1.3	2.98	2	0.025	21.72
95	1	1	1	0.40	20	28	0.66	0.3	0.9	1	3	0.026	19.75
96	1	1	2	0.43	20	7	1.99	1.41	2.2	1.58	2	0.020	344.46	[56]
97	2	1	2	0.43	20	7	1.99	1.41	0.4	1.75	1	0.040	284.56
98	2	1	2	0.43	20	7	1.99	1.41	0.1	2.54	1	0.080	263.79
99	2	1	2	0.43	20	7	1.99	1.41	2.9	1.04	3	0.160	216.08
100	2	1	2	0.43	20	14	1.99	1.41	0.9	2.74	2	0.020	381.30
101	2	1	2	0.43	20	14	1.99	1.41	2.8	1.32	3	0.040	314.87
102	2	1	2	0.43	20	14	1.99	1.41	2.3	1.42	2	0.080	287.94
103	2	1	2	0.43	20	14	1.99	1.41	0.3	1.75	1	0.160	244.95
104	2	1	2	0.43	20	28	1.99	1.41	0.9	2.83	2	0.020	410.28
105	2	1	2	0.43	20	28	1.99	1.41	3	1.7	1	0.040	341.68
106	2	1	2	0.43	20	28	1.99	1.41	2.1	2.95	2	0.080	314.15
107	2	1	2	0.43	20	28	1.99	1.41	1.4	1.26	1	0.160	299.41
108	1	2	1	0.45	23	1	1	1	0.6	1.41	3	0.020	6.19	[17]
109	1	2	1	0.45	23	3	1	1	1.4	1.09	1	0.020	10.63
110	1	2	1	0.45	23	5	1	1	1.2	2.59	1	0.020	11.73
111	1	2	1	0.45	23	7	1	1	0.5	1.15	3	0.020	12.69
112	1	2	1	0.45	23	10	1	1	2.2	2.39	2	0.020	13.66
113	1	2	1	0.45	23	12	1	1	1.9	1.72	1	0.020	14.22
114	1	2	1	0.45	23	14	1	1	2.7	1.96	3	0.020	14.81
115	1	2	1	0.45	23	17	1	1	1.8	1.37	1	0.020	15.60
116	1	2	1	0.45	23	20	1	1	0.5	1.84	3	0.020	16.48
117	1	2	1	0.45	23	23	1	1	2.1	2.34	3	0.020	17.21
118	1	2	1	0.45	23	25	1	1	2.2	2.54	2	0.020	17.78
119	1	2	1	0.45	23	28	1	1	1.1	2.83	3	0.020	18.56
120	1	2	1	0.45	23	1	1	1	0.1	1.4	2	0.040	6.77
121	1	2	1	0.45	23	3	1	1	2.1	1.53	2	0.040	10.27
122	1	2	1	0.45	23	5	1	1	2.6	1.13	2	0.040	12.49
123	1	2	1	0.45	23	7	1	1	1.4	1.58	1	0.040	14.39
124	1	2	1	0.45	23	10	1	1	1.9	2.32	3	0.040	14.70
125	1	2	1	0.45	23	12	1	1	0.4	2.62	2	0.040	14.93
126	1	2	1	0.45	23	14	1	1	0.6	2.42	1	0.040	15.10
127	1	2	1	0.45	23	17	1	1	1.6	2.81	3	0.040	15.78
128	1	2	1	0.45	23	20	1	1	1.9	1.33	3	0.040	16.44
129	1	2	1	0.45	23	23	1	1	0.3	1.39	2	0.040	17.08
130	1	2	1	0.45	23	25	1	1	0.9	1.66	3	0.040	17.61
131	1	2	1	0.45	23	28	1	1	1.4	1.8	3	0.040	18.16
132	1	2	1	0.45	23	1	1	1	2.5	1.1	1	0.060	6.62
133	1	2	1	0.45	23	3	1	1	0.1	1.58	1	0.060	10.43
134	1	2	1	0.45	23	5	1	1	2.9	2.76	3	0.060	11.58
135	1	2	1	0.45	23	7	1	1	0.7	2.69	1	0.060	12.80
136	1	2	1	0.45	23	10	1	1	0.1	2.21	2	0.060	13.57
137	1	2	1	0.45	23	12	1	1	2.6	2.77	3	0.060	14.07
138	1	2	1	0.45	23	14	1	1	0.4	3	2	0.060	14.59
139	1	2	1	0.45	23	17	1	1	2.1	1.49	1	0.060	15.38
140	1	2	1	0.45	23	20	1	1	1.8	1.78	1	0.060	16.15
141	1	2	1	0.45	23	23	1	1	0.1	1.65	2	0.060	16.86
142	1	2	1	0.45	23	25	1	1	0.8	1.81	1	0.060	17.40
143	1	2	1	0.45	23	28	1	1	0.8	1.66	2	0.060	18.15
144	2	1	2	0.40	22	1	2.56	0.8	0.1	1.03	1	0.040	210.87	[57]
145	2	1	2	0.40	19	2	2.56	0.8	2.1	1.03	1	0.040	282.33
146	2	1	2	0.40	18	3	2.56	0.8	1.4	1.76	2	0.040	386.96
147	2	1	2	0.40	21	4	2.56	0.8	1.1	1.82	3	0.040	386.96
148	2	1	2	0.40	19	5	2.56	0.8	2.1	1.54	3	0.040	378.68
149	2	1	2	0.40	19	6	2.56	0.8	1.9	2.5	2	0.040	382.80
150	2	1	2	0.40	20	7	2.56	0.8	2.9	2.11	2	0.040	382.80
151	2	1	2	0.40	19	8	2.56	0.8	2.1	2.07	2	0.040	382.80
152	2	1	2	0.40	21	9	2.56	0.8	1.7	2.41	1	0.040	386.96
153	2	1	2	0.40	22	10	2.56	0.8	2.6	1.85	3	0.040	382.80
154	2	1	2	0.40	21	11	2.56	0.8	2.7	1.85	1	0.040	391.16
155	2	1	2	0.40	18	12	2.56	0.8	0.6	1.35	2	0.040	395.41
156	2	1	2	0.40	18	13	2.56	0.8	0.8	1.45	2	0.040	398.99
157	2	1	2	0.40	22	14	2.56	0.8	2.3	2.1	3	0.040	403.32
158	2	1	2	0.40	18	15	2.56	0.8	0.3	2.88	1	0.040	412.14
159	2	1	2	0.40	22	16	2.56	0.8	3	2.49	2	0.040	429.57
160	2	1	2	0.40	19	17	2.56	0.8	2.4	1.56	1	0.040	443.73
161	2	1	2	0.40	18	18	2.56	0.8	0.1	2.64	1	0.040	462.50
162	2	1	2	0.40	21	19	2.56	0.8	2.3	2.56	3	0.040	482.06
163	2	1	2	0.40	20	20	2.56	0.8	0.2	2.38	1	0.040	497.95
164	2	1	2	0.40	21	21	2.56	0.8	2.2	2.96	2	0.040	530.35
165	2	1	2	0.40	20	22	2.56	0.8	0.4	1.9	1	0.040	535.15
166	2	1	2	0.40	18	23	2.56	0.8	1.5	1.6	1	0.040	576.17
167	2	1	2	0.40	18	24	2.56	0.8	2.1	2.46	3	0.040	576.17
168	2	1	2	0.40	22	25	2.56	0.8	2.5	1.03	1	0.040	595.16
169	2	1	2	0.40	21	26	2.56	0.8	0.5	2.08	3	0.040	620.34
170	2	1	2	0.40	19	27	2.56	0.8	1.8	1.48	3	0.040	660.71
171	2	1	2	0.40	20	28	2.56	0.8	0.4	1.94	3	0.040	653.60

.

Several traditional data pre-processing methods were used in this study. For example, for features such as cement type, dispersion method, measurement method, etc., one-hot encoding was used, where all types of features are converted to binary 0 and 1 for feature input. For cement types, 1 and 2 represent Type I 42.5 Ordinary Portland Cement and Type II 42.5 Ordinary Portland Cement, respectively. These two types of cement are the most commonly used in practical construction and in research as well. The selected curing temperature range was from 18 °C to 23 °C, which represents the typical laboratory conditions for curing cement composites. For the dispersion method, 1 represents mixing after ultrasonication, and 2 represents magnetic mixing followed by ultrasonication. For the measurement method, 1 and 2 represent the four-electrode method and the two-electrode method, respectively. The dimensions of the specimen were normalized according to 50 mm (length) × 50 mm (width) × 50 mm (height), where two dimensionless parameters, K_s and K_h, were introduced. K_s is the ratio of the bottom area of the specimen to the area of 50 mm × 50 mm, and K_h is the ratio of the height of the specimen to the height of the standard specimen, i.e., 50 mm. For the dimensions of GO, from the perspective of manufacturing, the dimensions of GO adhere to a Gaussian distribution. In this study, Gaussian random numbers within the upper and lower limits of the interval were employed to characterize the dimensions of GO, including diameter, thickness, and number of layers. In the data collected in this work, the dispersant used in most studies was PCE. Therefore, the dispersant was not considered as a separate feature in this work.

Table 1. Description of the collected data.

Features	Counts	Mean	Minimum	Maximum	Median	Standard Deviation
w/c	137	0.43	0.4	0.5	0.45	0.02
Curing temperature (°C)	137	21.72	18	23	23	1.64
Curing age (days)	137	16.18	1	28	15	9.77
Ks	137	1.29	0.66	2.56	1	0.65
Kh	137	0.88	0.3	1.41	1	0.29
Diameter (μm)	137	1.56	0.1	3	1.6	0.90
Thickness (nm)	137	1.95	1	3	2.07	0.57
Layers	137	1.95	1	3	2	0.84
Content of GO (wt%)	137	0.05	0	0.16	0.04	0.04
Cement type 1	102	-	0	1	-	-
Cement type 2	35	-	0	1	-	-
Dispersion method 1	113	-	0	1	-	-
Dispersion method 2	24	-	0	1	-	-
Measurement method 1	102	-	0	1	-	-
Measurement method 2 Electrical resistivity (Ω·m)	35 171	- 123.78	0 6.19	1 660.71	- 19.6	- 176.26

presents a description of the collected data. The 171 collected data points were partitioned into 0.8 and 0.2, resulting in 136 training sets and 35 test sets, respectively.

3. Methodology

Different ML models have been developed and used in various fields [37]. Among the models, ANN has demonstrated an excellent ability to deal with complex nonlinear problems [38]. RF also exhibits capabilities in handling nonlinear problems [39,40] and mitigates overfitting issues by eliminating datasets featuring noise and outliers [41]. Similar to the RF model, another robust ensemble learning algorithm is XGB. Compared to RF, XGB employs a gradient boosting algorithm, which trains the decision tree iteratively. For each iteration, the model is adjusted based on the results of the previous iteration. XGB introduces a regularization term in the objective function, which is used to optimize the residual using gradient boosting [42,43]. The three models mentioned above are employed for the prediction of the electrical resistivity of the GORCCs in this work.

3.1. Random Forest

Random forest (RF) is an ensemble learning algorithm composed of decision trees. A decision tree is a model in which each node represents a feature, each branch denotes a possible value of this feature, and each leaf represents an output variable. RF combines several different decision trees, each of which generates a result. Then RF generates the final result by “voting”. For regression problems, this “voting” is the averaging of the values. The architecture of the RF model is shown in Figure 1. The loss function for the regression tree is

$$L\mathrm{oss}={\displaystyle \sum _{i\in C_1(j,s)}{\left(a_i-y_1\right)}^2}+{\displaystyle \sum _{i\in C_2(j,s)}{\left(a_i-y_2\right)}^2}$$

(1)

where a_i represents the actual value, y₁ and y₂ are the predicted values, C₁ and C₂ denote the two split subtrees, and j and s stand for splitting variable and point, respectively.

The randomness of RF is mainly reflected in two aspects [44], i.e., randomness of data sampling and feature selection. RF creates different training datasets by the bootstrapping method. This is accomplished by randomly selecting samples from the original dataset with replacement. The introduction of randomness helps reduce the variance of the model, preventing a particular feature from dominating the entire model due to the increased variability of each tree.

3.2. Extreme Gradient Boosting

Similar to the RF algorithm, extreme gradient boosting (XGB) is an ensemble learning algorithm [45], which demonstrates a strong learning ability when dealing with complex nonlinear problems. Additionally, this ensemble learning approach enables the model to be robust when subjected to noise and outliers, which improves the stability of the model.

Compared to RF, the XGB model employs the idea of gradient boosting. The model is deployed by gradually introducing new trees, each of which attempts to correct the bias (the model’s prediction error relative to the true value) of the previous tree [46]. The final model is a weighted sum of these trees (for regression problems) or a voting result (for classification problems). The flowchart of the XGB model is illustrated in Figure 2. The training objective of XGB is to minimize an objective function consisting of a loss function and a regularization term. The general form of this objective function is given as

$$Objective={\displaystyle {\sum }_{i=1}^{n}L\left(y_i,{\widehat{y}}_i\right)}+{\displaystyle {\sum }_{k=1}^K\Omega \left(f_k\right)}$$

(2)

where n is the number of samples, $L\left(y_i,{\widehat{y}}_i\right)$ is the loss function measuring the prediction error of the model for sample i, K is the number of trees, f_k is the kth tree, and Ω(f_k) is the regularization term of the tree.

To enhance the performance of XGB and ensure the robustness of the model, XGB introduces the concept of learning rate, which is used to control the weight of each tree. This concept enables the model to be more stable by reducing the impact of each tree in every iteration.

3.3. Artificial Neural Network

Artificial neural network (ANN) is a computational model inspired by the biological neural network, which emulates the information processing mechanism of the human brain [47]. ANN consists of three distinct layers, including input, hidden, and output. The structure of ANN is shown in Figure 3. Neurons are interconnected through assigned weights, reflecting the degree of influence of neurons in the previous layer on the ones in the next layer [48].

Weight coefficients and activation functions are essential for the ANN model [49,50]. The function of the weight coefficients is to facilitate the propagation of neural signals, and the propagation process can be expressed by

$$f_i={\displaystyle {\sum }_{i=1}^n{w}_{ij}·x_i+b}$$

(3)

where f_i represents the weighted sum of the outputs of the ith layer in the forward propagation, w_ij represents the corresponding weighting coefficients of the neurons between the previous jth layer and the ith layer, x_i represents the inputs of the neurons in layer i, and b represents the bias. The activation function introduces nonlinear characteristics that empower ANN to learn and represent more intricate relationships. The commonly used functions include Sigmoid, Tanh, ReLU (Rectified Linear Unit), etc. In this work, the ReLU function is employed. The expression for the activation function is shown in Equation (4).

$$O_i=h\left(f_i\right)$$

(4)

where O_i represents the output value of layer i, and h represents the activation function.

3.4. Evaluation Metrics

This work evaluates the performance of the model using four parameters, which are R², MAE, RMSE, and MAPE. The definitions of these four parameters are

$$R^2=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}}{{{\displaystyle {\sum }_{i=1}^n\left(y_i-{\overline{y}}_i\right)}}^2}}$$

(5)

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n\left|y_i-{\widehat{y}}_i\right|}$$

(6)

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

(7)

and

$$MAPE={\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n\left|\frac{y_i-{\widehat{y}}_i}{y_i}\right|}\times 100\%$$

(8)

where n represents the number of samples, y_i represents the actual value, ${\widehat{y}}_i$ represents the predicted value, and ${\overline{y}}_i$ represents the average of the actual values.

3.5. K-Fold Cross-Validation

In this study, k-fold cross-validation is utilized to assess the performance of a model on a limited dataset and to improve the generalization of the model. The steps of the k-fold cross-validation method are as follows:

(1)

Divide the original dataset into k folds of mutually exclusive subsets with equal size.

(2)

For each round, one fold is selected as the test set and the remaining folds are used as the training set. The model is trained, and its performance is evaluated by the test set. Repeat this process until each fold has been used as a test set.

(3)

The results of each round of the model evaluation are summarized, and the final output is the average of the results of the k rounds.

It has been proven that 10-fold cross-validation is an effective approach, which balances performance and time consumption [51]. Therefore, this study applies the 10-fold cross-validation method to improve the performance of the model and avoid the overfitting issue. A general schematic diagram of the 10-fold cross-validation method is shown in Figure 4.

4. Results and Discussion

Based on the collected data, the electrical resistivity of the GORCCs was predicted and analyzed by employing RF, XGB, and ANN, respectively. As the prediction performance of the ANN models is intricately related to the configuration of the hidden layer, four ANN models with different hidden layers were applied in this work.

4.1. Correlation Analysis

In this study, correlation analysis was conducted during feature engineering. Figure 5 demonstrates the Pearson correlation coefficient between continuous features and the electrical resistivity of the GORCCs. A strong negative correlation was observed between the electrical resistivity and the w/c, with a value of −0.7. The experimental samples were maintained in an environment with humidity exceeding 90% and kept in water at about 23 ℃ after being unmolded. During the cement curing process, the pores within the cement were filled with capillary water, adsorbed water, etc. The porosity in the cement increased with the increase of the w/c, resulting in greater absorption of water with dissolved ions (i.e., hydroxide and calcium ions) in these pores. Ionic conduction dominated at this stage, leading to an increase in the electrical conductivity of the cement specimens. Another strong negative correlation was observed between curing temperature and the electrical resistivity, with a value of −0.68, while a positive correlation was observed between curing age and the electrical resistivity, with a value of 0.17. These correlations emphasize the importance of the curing environment, which has a significant influence on the electrical resistivity of the GORCCs. The strong positive correlation of K_s, with a value of 0.86, indicates the important effect of specimen size. The positive correlation between specimen size and the electrical resistivity can be attributed to the fact that smaller specimens had a higher probability of forming effective conductive networks because of their reduced volume, enhancing the electrical conductivity. In contrast, larger specimens had a lower probability of forming such conductive networks throughout the entire sample, leading to higher electrical resistivity. In addition, larger specimens might be subjected to non-uniform curing conditions from outside to inside, which might induce different microstructure, resulting in an increase in the resistivity of the sample. Therefore, a positive correlation was shown between specimen size and the resistivity. The Pearson correlation between the dimension of GO and the electrical resistivity of the GORCCs is weak. Effective dispersion is crucial for forming conductive networks within the composite. Therefore, the uniform distribution of GO nanosheets in the specimen had a greater impact on the final results compared to the size of GO. The negative correlation between the GO content and the electrical resistivity shows that the incorporation of GO reduces the electrical resistivity of the GORCCs. When well dispersed within the cement matrix, GO sheets form interconnected conductive pathways, facilitating electron transport across the composite and thereby reducing overall electrical resistivity. Additionally, the incorporation of GO improves the microstructure of the cement composite by filling voids and reducing porosity. A denser microstructure with fewer defects enhances the material’s electrical conductivity. Figure 6 further reveals the correlation between the electrical resistivity and the w/c, K_s, and K_h, respectively. The electrical resistivity tends to decrease with the increase of the w/c. Figure 6b,c suggest that the electrical resistivity increases with the increase of K_s and K_h. In addition to the positive correlation between specimen size and the resistivity, another possible reason is that the presence of cracks increases as the specimen size increases. According to previous studies, cracks may interrupt the conductive paths formed by the conductive nanoparticles, leading to increased electrical resistivity [52].

4.2. Electrical Resistivity by RF and XGB

Both RF and XGB belong to the ensemble learning algorithms.

Table 2. Results of RF and XGB for the electrical resistivity of the GORCCs.

Index	Models	MAE	RMSE	MAPE
1	RF	3.95	10.33	5.75%
2	XGB	0.76	3.18	1.04%

presents the performance of the models used in this work. It can be seen that the XGB model demonstrates superior prediction performance, with MAE, RMSE, and MAPE being 0.76 Ω·m, 3.18 Ω·m, and 1.04%, respectively. In contrast, the RF model shows inferior prediction performance, with MAE, RMSE, and MAPE being 3.95 Ω·m, 10.33 Ω·m, and 5.75%, respectively.

Figure 7 illustrates the relationship between the actual and predicted values for the RF and XGB models. The distribution of data points for both models shows close proximity to the line y = x, suggesting excellent performances of the models. However, Figure 7a shows that there is a data point noticeably deviating from the line y = x. The actual and predicted values of this point are 24.97 Ω·m and 36.78 Ω·m, respectively. In contrast, in Figure 7b, the prediction for all data points demonstrates closer proximity to the actual values, with reduced average and maximum error values compared to the predictions obtained from RF. The results indicate that the XGB model performs better than the RF model in predicting the electrical resistivity of the GORCCs.

Figure 8 displays the distribution of data points representing actual and predicted values for both the RF and XGB models. The data points with errors smaller than 10 Ω·m exceed 90% for both models. The maximum, minimum, and average errors for the RF model are 39.81 Ω·m, 0.002 Ω·m, and 3.95 Ω·m, respectively. In contrast, these three corresponding errors for the XGB model are 18.75 Ω·m, 0.002 Ω·m, and 0.76 Ω·m, respectively, which indicates that the maximum and average errors are significantly reduced compared to the ones of the RF model. The value of R² is 0.99 for both models. As shown in Equation (5), when the predicted value is relatively close to the actual value, and the actual value significantly deviates from the average of all actual values, the value of R² approaches 1.

Figure 9 shows a feature importance analysis of the predicted results by the RF and the XGB models. The metrics in the figures are used to quantify the degree of contribution of each feature to the final result. A higher value indicates a more significant effect on the electrical resistivity of the GORCCs. Figure 9a,b show feature importance using the RF technique and the F-score of the XGB model, respectively. It is evident in Figure 9a that the measurement method, specimen size (Ks), and cement type have significant influence on the electrical resistivity, with the values being 0.232, 0.217, and 0.213, respectively. This suggests that it is of great importance to consider the effect of the selection of measurement method, specimen size, and cement type on the electrical resistivity of the GORCC samples. In addition, both curing age and the content of GO have a moderate impact on the electrical resistivity. In Figure 9b, the thickness of GO emerges as the most influential factor on the electrical resistivity, as indicated by the high value of the F-score value. The second most influential factor is the measurement method, whose F-score reaches 1045. It is found that the four-probe method has higher accuracy compared to the two-electrode method. This can be explained by the fact that the accuracy of the two-electrode method is affected by the contact resistance between the electrodes and the specimen. In contrast, the four-probe method can compensate for the inaccuracy caused by the contact resistance by using separate pairs of electrodes for current injection and voltage measurement. The content of GO, with an F-score exceeding 1000, also exhibits apparent effects on the electrical resistivity. When well dispersed within the cement matrix, GO sheets form conductive pathways that facilitate electron transport and reduce the electrical resistivity by filling voids and reducing porosity. In conclusion, both the RF and XGB models place emphasis on the measurement methods, curing age, and content of GO, which indicates that more attention should be paid to these aspects when designing experiments.

4.3. Electrical Resistivity by ANN

ANN provides an efficient and accurate computational method for solving complex nonlinear problems [53]. However, the structure of ANN has a substantial impact on the performance and prediction of the model. In order to determine the optimal structure of ANN, this study adopted ANN models with different numbers of hidden layers to predict the electrical resistivity of the GORCCs. The four distinct ANN models are as follows: the first features a single hidden layer with 32 neurons; the second comprises two hidden layers with 64 and 32 neurons, respectively; the third involves three hidden layers with 64, 32, and 16 neurons, respectively; and the fourth incorporates four hidden layers with 64, 32, 16, and 8 neurons, respectively. Among them, the ReLU function was utilized as the activation function, which is expressed as

$$f(x)=\max \left(0,x\right)$$

(9)

where x is the input value. The ReLU function takes the value of x in the region x > 0 and 0 in the region x ≤ 0.

Table 3. Predictions of the ANN models with four different hidden layers.

Index	Models	MAE	RMSE	MAPE
1	ANN:15-32-1	14.88	27.73	43.9%
2	ANN:15-64-32-1	3.84	10.25	5.21%
3	ANN:15-64-32-16-1	1.11	3.07	2.36%
4	ANN:15-64-32-16-8-1	4.15	12.66	4.76%

demonstrates the evaluation metrics of the employed ANN models. It is evident that as the number of hidden layers increases, the performance of the model initially improves and then declines. This observation suggests that an increase in the number of hidden layers does not necessarily lead to better performance. The ANN model with 3 hidden layers exhibits the best performance in predicting the electrical resistivity of the GORCCs, with MAE, RMSE, and MAPE being 1.11 Ω·m, 3.07 Ω·m, and 2.36%, respectively.

Figure 10 illustrates the scatter distributions of actual and predicted values for the four ANN models on the test set. It can be observed from Figure 10a that a number of data points deviate from the line y = x, which indicates the poor performance of the ANN 15-32-1 model. The maximum error value reaches 97.44 Ω·m, which is larger than the average error value of 14.88 Ω·m. In comparison, the accuracy of the predictions of the remaining three ANN models has improved significantly. Among them, the ANN 15-64-32-16-1 model demonstrates the best performance, with 80% of data points having an error value below 1 Ω·m. It is evident from Figure 10c that all the data points align closely with the line y = x, which means that the predicted values are relatively close to the actual values. The maximum error is 17.33 Ω·m, with the average error being 1.11 Ω·m only.

Figure 11 demonstrates the errors in predictions from the four ANN models. It is found that the maximum error in Figure 11a is the largest, surpassing 90 Ω·m. Furthermore, there are 8 data points with an error exceeding 10 Ω·m, which indicates the worst performance among the four models. In Figure 11b,d, the number of data points with error values exceeding 10 Ω·m declines, although the value of the maximum error data point is still around 50 Ω·m. In contrast, the errors in Figure 11c are significantly reduced. There is only one data point with an error exceeding 10 Ω·m, while approximately 91.4% of the data points exhibit an error of less than 2 Ω·m. This indicates a significantly improved prediction, highlighting the superior performance of the ANN model with three hidden layers.

In addition, the function of the ANN model with three hidden layers can be expressed as

$$O=F\left(\left({\displaystyle \sum _1^{16}{\overline{w}}_{q-i}^3}\cdot F\left(\left({\displaystyle \sum _1^{32}{\overline{w}}_{r-q}^2}\cdot F\left(\left({\displaystyle \sum _1^{64}{\overline{w}}_{p-r}^1}\cdot {\overline{i}}_p\right)+{{\overline{b}}_r}^1\right)\right)+{{\overline{b}}_q}^2\right)\right)+{{\overline{b}}_i}^3\right)$$

(10)

where ${\overline{w}}_{m-n}^l$ represents the weights connecting the mth neurons in the lth layer and the nth neurons in the (l + 1)th layer, ${\overline{b}}_n^l$ and ${\overline{i}}_n$ are the added bias and the net input in the lth layer for the nth neuron in the (l + 1)th layer, and F represents the activation function, i.e., the ReLU function.

4.4. Comparison of Performance Evaluation for Different Models

Figure 12 demonstrates the modified Taylor diagram of different ML models as involved in this work, where ANN-1 to ANN-4 represent the ANN models with 1, 2, 3, and 4 hidden layers, respectively. The Taylor diagram provides a visual representation of the relationship between actual and predicted values, facilitating the comparison of the evaluation metrics. The radial line in the Taylor diagram represents the correlation coefficient, indicating the relationship between the experimental results and the predictions. A higher correlation coefficient means better performance of the model. The horizontal and vertical axes of the Taylor diagram correspond to the normalized standard deviation, representing the ratio of the standard deviation of the predicted value to the standard deviation of the actual value. From the diagram, it is found that the ANN model with one hidden layer demonstrates the worst prediction performance, featuring lower correlation coefficients and higher normalized RMSE values. As observed in the enlarged figure, the XGB and ANN-3 models exhibit superior performance in predicting the electrical resistivity. The normalized standard deviation for both models is close to 1, indicating an equivalent dispersion between the actual value and the predicted data. Furthermore, the XGB and ANN-3 models demonstrate higher correlation coefficients and lower normalized RMSE. The performance of the ANN-4 model exhibits a clear overall decline. This further suggests that more hidden layers are not always favorable for better performance. Although increasing the number of hidden layers and neurons normally may increase the learning ability of the network, allowing it to better fit complex data patterns, an overly complex network structure may lead to overfitting [54], resulting in poor performance of the ANN model.

5. Conclusions

In order to investigate the complex nonlinear relationship between the coupling of multiple factors and the electrical resistivity of the GORCCs, three different ML algorithms are employed in this study, including RF, XGB, and ANN. A total set of 171 data points is collected and utilized after data cleaning. In this work, the dimensions of GO, such as diameter, thickness, and number of layers of GO, are considered as features following a Gaussian distribution. Particularly, apart from considering the w/c, cement type, etc. as input features, this work is the first attempt to incorporate the effects of specimen size on the electrical resistivity of the GORCCs. Among them, cement type, dispersion method, and measurement method are processed by the one-hot encoding method for feature engineering. Furthermore, in order to describe the correlation between individual features and the electrical resistivity of the GORCCs, the Pearson correlation coefficient matrix is calculated in this study. This work conducts the error analysis of the ANN models with different hidden layers. All the above-mentioned models are evaluated by four metrics, including R², MAE, RMSE, and MAPE. The results are visualized in the form of a Taylor diagram. The major conclusions of this work can be summarized as follows:

(1)

The three ML models are evidenced to be capable of predicting the resistivity of the GORCCs, which suggests that ML is promising for practical applications in the field of identifying the nonlinear and complex relationship between the material properties and various influencing factors.

(2)

The structure of the ANN has a significant impact on the prediction performance. It is found that the ANN model with hidden layers may increase the learning ability of the network, allowing it to better fit complex data patterns. However, more hidden layers may lead to overfitting and are not always favorable for improving the prediction performance. The ANN model with the structure of 15-64-32-16-1 is proven to have the best performance in this work. This can provide useful guidelines when conducting ML modeling for predicting the material properties.

(3)

The strong correlation between specimen size and the resistivity suggests that the consideration of the specimen size is important for accurate capture of the material properties. In addition, it is found that the electrical resistivity of the GORCCs is also highly dependent on the measurement method. This observation is of great importance when selecting the measurement method to obtain accurate and objective experimental data.