1. Introduction
The soil–water characteristic curve (SWCC) is a quantitative relationship that describes the potential energy (matric suction) and water content (volumetric moisture content, saturation, mass moisture content) of soil [
1]. It can reflect the microscopic connections between different phases in unsaturated soil and is used to study the permeability, deformation, strength, and multi-field coupling behavior of unsaturated soil [
2,
3,
4]. It is essential in the research of unsaturated soil theory. Some scholars provide SWCC models by fitting experimental data, such as the well-known FX model [
5] and VG model [
6]. However, the parameters in the model are not fixed and vary with changes in the external environment, which results in a set of model parameters only reflecting the soil–water characteristics under certain specific conditions and which are unable to achieve the predictive function of the model for different environments [
7,
8].
With the development of modern society, the environment in which engineering construction takes place is becoming increasingly complex, involving the coupling problem in multiple fields such as seepage, deformation, temperature, and chemical, which poses severe challenges to geotechnical engineering (geological storage of nuclear waste [
9], design of landfill sites [
10], engineering construction on salinized soil [
11], remediation of contaminated soil [
12]). This also puts high demands on SWCC models that can reflect complex environments. At present, in the research on SWCC models, although some scholars have considered the effects of deformation, temperature, salinity, etc., these studies generally only focus on the influence of a single factor and are limited to a small range of changes, and there are few studies that consider the combined influence of two or three factors [
13,
14]. What is worse, obtaining a soil–water characteristic curve within the full suction range requires a large amount of experimental measurement data, and the entire experimental process requires a long time [
15].
With the widespread application of machine learning, a new approach has been provided for predicting the SWCC in complex environments. By learning from a large amount of existing experimental data, the evolution law of matric suction and water content of unsaturated soil under different conditions can be predicted. Therefore, some scholars have conducted research on the prediction of the SWCC using machine learning methods [
15,
16,
17]. For example, Li and Vanapalli [
17] used the grain-size distribution curve as input information and established a SWCC prediction model using artificial neural networks (ANNs) and multiple adaptive regression splines (MARSs) as analysis methods. Pham et al. [
15] utilized the advantages of multiple machine learning models and the unsaturated soil database (UNSODA) to develop a new pedotransfer function to estimate the SWCC. These methods provide new solutions for the prediction of the SWCC, but there are still some limitations, such as the difficulty in considering the influence of external environments such as stress history, temperature, deformation, salinity, etc.
Currently, there are two main problems with the research on soil–water characterization. On the one hand, it is due to the long SWCC testing period in the full suction range, which leads to insufficient test data at present. On the other hand, most of the studies on SWCC modeling are based on the fitting of empirical formulas, and there are also some studies on simple prediction by machine learning methods, but it is difficult to predict the SWCC directly. Furthermore, these two problems are also more prominent under the condition of considering the influence of multiple factors. If the existing valuable data can be utilized to better predict the soil–water characteristic under the influence of different environmental factors, it can not only compensate for the problem of limited data due to the long testing period and multiple influencing factors, but can also provide a strong support for the research on a multi-field coupling model for unsaturated soil.
This paper adopts a neural network method that can learn from a large amount of existing experimental data and on-site measurement data to more accurately calculate soil–water characteristics under different conditions [
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44,
45]. The input feature quantities can cover factors such as soil particle size, density, plasticity index, environmental temperature, matrix suction, void ratio, salinity, and stress history, and can directly use water content as the predicted output value. The hidden layer includes a certain number of neurons, neural network parameters such as weights and biases, and is optimized using gradient-based algorithms. In addition, further learning can be achieved by increasing the sample size in different regions, thereby improving the adaptability to soil samples under different conditions and having better generalization ability.
2. Construction of Neural Network Prediction Model
2.1. Data Collection and Processing
The physical properties of soil, initial conditions, and external environment are the three main factors affecting the shape of SWCC. Selected physical properties such as soil particle size d10/d30/d60 (types of particle sizes; the mass of soil particles smaller than these sizes account for 10%, 30%, and 60% of the total mass, respectively) and plasticity index Ip are the first components of the input layer. The initial water content w0 and void ratio e0 are the second components of the input layer. The external environmental temperature T and salinity C serve as the third components of the input layer. Simultaneously considering the volumetric water content under different suction conditions, the matrix suction s is also taken as the input value.
Considering that there are almost no experiments in the existing literature that simultaneously consider the effects of deformation, temperature, and salinity on the SWCC, three datasets were established. Dataset 1 considers the effects of salinity and deformation, dataset 2 considers the effects of temperature and deformation, and both datasets consider the other influencing factors mentioned above. The neural network input layers trained on both datasets contain eight variables. Dataset 3 combines datasets 1 and 2 together, with dataset 1 having a data temperature of 25 °C at room temperature and dataset 2 having a data salinity of 0 mol/L. Therefore, the input layer of the neural network trained in dataset 3 contains nine variables.
Before using neural networks for prediction, well-processed datasets can significantly improve the prediction accuracy of neural networks. Optimizing the database not only focuses on the quantity of data samples, but also on their quality, and removing abnormal data can help reduce the workload of optimizing neural network parameters. This paper collected experimental data from the literature on the SWCC published in recent decades, and these have been listed in the References section [
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44,
45]; after sorting, 1000 sets of effective data were selected. The SWCC test types include tests on different types of soil, as well as tests affected by void ratios, temperature, and salinity. The experimental results are obtained through strict testing, which can ensure the accuracy of the results. The specific experimental details are documented in the cited literature and are not elaborated on here. Due to the different methods used in the various experiments, such as the pressure plate method, steam equilibrium method, etc., the range of eigenvalues such as matrix suction varies greatly among the different experiments. The maximum/minimum normalization method is used to normalize the input eigenvalues in the same proportion, and the processed value range is 0–1. At the same time, abnormal data are deleted. Finally, there are 400 sets of data in dataset 1, 400 sets of data in dataset 2, and 800 sets of data in dataset 3.
2.2. Model Building
To prevent overfitting of the neural network, Bayesian regularization is adopted to improve the universality of the model. The neural network for optimizing parameters using a Bayesian algorithm, also known as the Bayesian regularization neural network (BRNN), has a longer training time compared to other algorithms, but the prediction accuracy can be improved. It is worth noting that BRNN does not require a validation set, meaning the entire data are randomly divided into two parts, with 80% of the data used for model training and 20% for model testing. When the given training result standard is reached, the training process stops.
After normalizing the dataset, it is randomly divided into training and testing sets to ensure the representativeness of the entire dataset. Among them, there are 320 sets of data in the training set and 80 sets of data in the testing set. As shown in
Figure 1, the BRNN neural network model has nine input variables, namely particle size
d10/
d30/
d60, plasticity index, initial moisture content, void ratio, temperature, salinity, and matrix suction. It is worth mentioning that when only considering two factors among temperature, salinity, and deformation, there are only eight input variables. The output layer has one node, which is the volume moisture content, and the hidden layer has 25 neurons.
The transfer function from the input layer to the hidden layer is the Purelin function
y =
x, and the activation function of the hidden layer is the tansig function
. If the expected output is not obtained at the output layer, it enters the backpropagation stage. The loss function used is the mean square error loss function as follows
Return the error signal along the original connection path and minimize the error signal by modifying the weights of each neuron. This process is optimized using the stochastic gradient descent method. In addition, the model adopts a Bayesian regularization algorithm to prevent overfitting problems.
2.3. Neural Network Training
Use two statistical measures, mean square error (MSE) and fitting degree (R), to evaluate the predictive ability of the neural model, which is the difference between the predicted target value and the true target value. The smaller the MSE value while ensuring that the R value is closer to 1, the stronger the predictive ability of the model. The MSE and R statistics are also influenced by the number of neurons in the hidden layer of the neural network. Increasing the number of neurons can enhance the computational power of the model and improve its predictive performance. However, it is not the case that the more neurons the better, as an increase in the number of neurons can lead to longer training times. For example, during the training process, when the number of hidden layer neurons increased from 10 to 25, the MSE value decreased by 12.56, and the R value increased by 0.13; when the number of hidden layer neurons increased from 25 to 50, the MSE value increased by 9.661, and the R value decreased by 0.082. By continuously adjusting the number of hidden layer nodes in the model, the number of hidden layer neurons is determined to be 25, and the model is optimized until the best predictive model is obtained.
The SWCC prediction model trained on dataset 1 (considering the influence of salinity and void ratio factors) and dataset 2 (considering the influence of temperature and void ratio factors) is shown in
Figure 2 and
Figure 3, respectively. It can be seen that the SWCC prediction model considering the above two different factors generally shows good predictive ability, with MSE values of 12.1128 and 9.8952, and fitting degrees of 0.96711 and 0.93431, respectively. Due to the better data richness of dataset 2, the accuracy of its prediction results is also relatively higher. The training results of the SWCC prediction model for dataset 3, which takes into account the combined effects of temperature, deformation, and salinity, are shown in
Figure 4, and also have good prediction accuracy. Here, no SWCC model prediction training results considering the influence of temperature and salinity factors are provided due to the lack of relevant experimental data.
4. Conclusions and Discussion
This paper focuses on the study of SWCC prediction under the influence of multiple factors by using a neural network machine learning approach, and the main conclusions are as follows:
- (1)
Using neural network machine learning methods to train a large number of SWCC test results under complex environments, a SWCC prediction model that can consider the effects of multiple factors such as temperature, salinity, and deformation was obtained. The model selects the representative particle sizes d10/d30/d60, plasticity index, initial moisture content, void ratio, temperature, salinity, and matrix suction of the soil as input variables, and the volumetric moisture content as the output variable.
- (2)
To improve the prediction accuracy of the model, separate training models were established for the SWCC considering temperature and deformation effects, as well as the SWCC considering salinity and deformation effects, for prediction analysis. By comparing the prediction results of the established model for SWCC characteristics under temperature, salinity, and deformation factors with relevant experimental results, including the effects of two factors and the combined effects of three factors, the validity of the model was verified.
- (3)
The model directly takes environmental variables and the physical properties of the soil as inputs, overcoming the problem of poor prediction accuracy caused by the excessive number of parameters in traditional empirical formulas due to the increase in variables. In addition, this method takes suction conditions as input eigenvalues, which can directly obtain the corresponding water content under certain conditions, making it simpler and more direct.
The research results provide a new method for predicting SWCC under complex environmental influences, and can be combined with experimental data, empirical formulas, and other prediction methods to enhance the accuracy of SWCC prediction. However, there are still certain limitations to the research. The method of using neural networks for SWCC prediction must be based on a large amount of existing data. Due to the limited experimental data for the SWCC under the influence of multiple factors, the prediction accuracy of the prediction model obtained in this paper, by training with not too much experimental data, needs to be further improved.