A Deep Learning-Based Approach for a Numerical Investigation of Soil–Water Vertical Infiltration with Physics-Informed Neural Networks

Abstract

The infiltration of water into the soil can lead to slope instability, which is one of the important causes of many geological hazards (such as landslides and debris flows). Therefore, the numerical investigation of the soil–water infiltration process provides the prerequisite for the determination of slope stability, which is of great significance for geological hazard prevention. In this study, we propose a deep learning-based approach for a numerical investigation of soil–water vertical infiltration with physics-informed neural networks and perform a comprehensive evaluation and analysis of the soil–water infiltration process in different soil types. In the proposed approach, the partial differential equation for soil–water infiltration is combined with the neural network based on physics-informed neural networks (PINNs) to obtain numerical analysis of the soil–water infiltration process. The results indicate that (1) compared with the traditional numerical method, the PINN-based method for the numerical investigation of soil–water vertical infiltration proposed in this study has a smaller error and can obtain more accurate numerical results. (2) During vertical infiltration of water in the different soil types, the light loam is the fastest, the heavy-loam the second and the medium loam the slowest. medium-loam soils are less susceptible to water infiltration of the three soil types and are more suitable for the filling of artificial slopes and dams. The proposed approach could be employed for the simulation of soil–water infiltration processes, not only for the discrimination of slope stability under rainfall conditions, but also for the selection of artificial slopes and dams to fill soil to prevent slope instability.

1. Introduction

Due to the existence of various types of geological environments and complex geological conditions, geological hazards are common in China [1]. Frequent geological hazards cause significant economic losses and pose a great threat to peoples’ lives and property [2,3]. The reasons for geological hazards are complex. Rainfall and its infiltration are the main causes of geological hazards, including landslides and debris flows [4,5]. The majority of instability in landslides and geotechnical engineering occurs during or after rainfall due to the scouring effect of rain on slope surfaces and the infiltration of rainwater into the soil, which lead to changes in stress in the geotechnical body and ultimately to instability [6].

Changes in the water content of soil generally lead to changes in the stress on the soil slope. The investigation of changes in the water content of soil is of great significance for the stability of soil slopes. The change in water content in soil can be seen as the infiltration of rainwater into soil, and investigating this infiltration process is an important prerequisite for analyzing the change in stress in geotechnical bodies and thus predicting the occurrence of geological hazards [7]. Furthermore, the investigation of soil–water infiltration processes is an important guide for practical engineering problems. The analysis of different types of soil–water infiltration processes can further determine the infiltration of different soils by water [8,9], thus providing advice on the selection of soil types for artificially filled or artificially mounded slopes and dams and reducing the possibility of landslides. Therefore, the analysis of the rainwater soil infiltration process provides the preconditions for slope stability discrimination and is of great importance for geological hazard detection, early warning, and engineering geological hazard prevention.

The infiltration of rainwater into soil is a complex process that is influenced by many factors such as the physical properties of the soil, the amount of precipitation, and the time of infiltration [10,11]. Much research has been conducted by different scholars on this topic, among which Richards proposed the continuity equation for the movement of unsaturated soil moisture based on Darcy’s law and the law of conservation of mass in 1931 [12]. Subsequently, on this basis of a large number of experimental verifications and considerable theoretical research, various scholars have proposed formulas such as Horton’s formula [13], Philip’s formula [14], and Smith’s formula [15] to calculate the soil–water infiltration process. The physical concepts of these equations are clear, and the results are relatively accurate; however, the traditional method for solving the equations has a very limited scope of application and can only be used to solve simple soil moisture infiltration problems.

To solve the above problems and to investigate the movement of soil moisture under complex conditions, numerical modeling methods have been proposed. For example, Belmans et al. [16] established a simulation model in 1983 to calculate the water dynamics of soils. Tiktak et al. [17] developed a soil hydrodynamic model for forest ecosystems to simulate the movement patterns of soil–water in the unsaturated zone, and it provides a semi-implicit finite difference decomposition for the equations used to model the motion of soil–water. Wang et al. [18] proposed a non-grid Hermite collocation point method based on radial basis functions for solving the non-linear unsaturated soil moisture equation of motion and verified the effectiveness of the algorithm. However, although traditional numerical methods can achieve high efficiency in performing calculation problems and can be used to solve complex large-scale problems, the accuracy is still not quite satisfactory.

Recently, deep learning-based approaches have been increasingly used to address infiltration problems in rainwater soils. For example, Oluwaseun et al. [19] predicted unsaturated hydraulic conductivity using multiple linear regression (MLR), an artificial neural network (ANN), and an adaptive neuro-fuzzy inference system (ANFIS). They determined the relationship between moisture content and unsaturated hydraulic conductivity in biochar-amended soil. Cai et al. [20] proposed the deep learning regression network (DNNR) for predicting the soil moisture in the Beijing area, which proved that the deep learning model is feasible and effective. Zhou et al. [21] proposed a convolutional neural network (CNN) to build the functional mapping between the hydraulic conductivity field and the longitudinal macrodispersivity, and it can achieve better results in estimations for fields with less heterogeneity. Lee et al. [22] employed deep learning methods, including deep neural networks (DNNs), long short-term memory (LSTM), gated recurrent units (GRUs), and new DNN-based algorithms (LSTM-DNN and GRU-DNN), to predict porosity and hydraulic conductivity based on electrical resistivity, and a better prediction was achieved.

However, the current research on deep-learning-based methods for rainwater soil infiltration only focuses on the determination of various parameters, and few deep learning models are used to numerically investigate the movement of soil moisture and to predict the process of soil moisture infiltration. This research gap is because most deep learning models are data-driven and are only trained to fit based on large amounts of data but are unable to extract interpretable information from large amounts of data.

The current physics-informed deep learning embeds physical knowledge into deep learning, reduces the dependence of traditional deep learning on large amounts of data, and does not require an abundance of data to fit within the constraints of physical equations to obtain more satisfactory results [23]. At present, physics-informed deep learning has been applied in many fields and has achieved positive results. For example, Mishra et al. [24] proposed a novel machine-learning algorithm based on physics-informed neural networks (PINNs) for simulating inverse problems for radiative transfer; the algorithm is efficient and accurate, alleviates dimensional disasters and reduces computational costs. Shen et al. [25] proposed a physics-informed deep learning approach for bearing fault detection that consists of a simple threshold model and a deep convolutional neural network. The approach is capable of obtaining high detection accuracy at high time-varying speeds, and the incorporated physical knowledge makes the proposed method more interpretable and credible.

Moreover, Zhang et al. [26] proposed an integrated framework for augmenting turbulence models with physically informed machine learning to predict turbulence mode patterns. The effectiveness of the method was demonstrated after several tests. Tartakovsky et al. [27] proposed a deep neural network (DNN) method based on physical information for estimating the hydraulic conductivity of saturated and unsaturated flows controlled by Darcy’s law, ultimately obtaining more accurate results compared to the state-of-the-art maximum a posteriori probability method. In contrast, there has been less research on PINN-based methods in rainwater soil infiltration, and to the best of the authors’ knowledge, there are currently no studies that have used PINN to numerically investigate soil–water vertical infiltration.

In this study, we propose a physics-informed neural network (PINN)-based approach for the numerical investigation of soil–water vertical infiltration. First, we select the one-dimensional equation of motion to model the vertical infiltration of soil–water. The parameters for different soil types, such as hydraulic conductivity and water diffusivity at different moisture contents, are selected to provide a solution to the equation. Thus, the partial differential equation form, initial conditions, and boundary conditions are determined. Then, based on the PINN, the neural network is connected to the partial differential equation to find a solution to the soil–water infiltration equation. Finally, we validate the proposed PINN-based method by comparing it with traditional numerical methods. The higher accuracy and validity of the method in the investigation of the vertical infiltration process of soil–water is demonstrated. The method is also applied to the numerical investigation of one-dimensional water infiltration processes in different soil types, and the soil–water infiltration curves of different soil textures are compared and analyzed to make a preliminary assessment of the infiltration capacity of the soil and to provide prerequisites for the determination of slope stability under rainfall conditions.

2. Method

2.1. Overview

In this study, we propose a physics-informed neural network (PINN)-based approach for the numerical investigation of soil–water vertical infiltration. We have also conducted a comprehensive analysis of the soil–water infiltration process for different soil types, thus providing prerequisites for slope stability discrimination and ultimately aimed at improving geological hazard monitoring, early warning, and engineering geological hazard prevention and control.

First, we selected suitable soil moisture equations and their related parameters, which determine the partial differential equations and initial and boundary conditions for soil moisture infiltration. Then, we combined physical knowledge with deep learning to build a deep learning model based on the PINN to solve the soil moisture infiltration equation. Finally, the proposed PINN-based method is validated by comparing it with traditional numerical methods, and the effectiveness of the method is demonstrated. The method has also been applied to the investigation of one-dimensional water infiltration processes in different soil types. By comparatively analyzing the infiltration processes of different soil types, their infiltration capacity is comprehensively evaluated to provide prerequisites for determining the stability of slopes under rainfall conditions. The workflow of the research in this study is illustrated in Figure 1.

2.2. Brief Introduction to the Pinn

The principle of the PINN is to integrate physical knowledge into deep learning networks by training the neural network to construct loss functions from the residual terms of the control equations, which minimizes the loss function and thus achieves approximate solutions to partial differential equations (PDEs) [28,29].

The principle of the PINN solution is illustrated in Figure 2. A neural network alternative model NN(x) with output result u is first constructed and then placed into the PDE information constraint for differentiation and arithmetic operations. The loss function is then defined according to the initial condition, boundary condition, and network, where the loss function terms include the initial condition loss (IC loss), boundary condition loss (BC loss), and PDE loss. Finally, the neural network is trained by minimizing the loss function, and the optimal parameter $\theta {}^{*}$ is finally obtained.

The idea behind the PINN was first proposed in 1994 when Dissanayake et al. used neural network to solve the two-dimensional poisson equation [30]. Later, the PINN was reintroduced in 2017 by Raissi et al. [31,32]. Since then, the PINN has been rapidly developed.

Traditional numerical methods are generally based on physical knowledge and are currently the most widely used methods for finding solutions to various types of partial differential equations (PDEs). However, when the dimensionality of the problem being solved increases, it is prone to dimensional issues that cannot be solved. Furthermore, the value of the solution is only calculated at grid points, while errors are easily introduced at non-grid points. For data-driven deep learning approaches, large amounts of data are needed to train and fit neural networks to obtain the desired results. However, for some areas, deep learning methods can suffer from a lack of data and data noise, among other problems, and the interpretability of data-based fits alone is weak.

The physics-based deep learning approach solves partial differential equations (PDEs) with randomly selected sampling points in the computational domain and ensures that all random sampling points in the computational domain meet the constraints as much as possible by minimizing the loss function, thus greatly reducing the accumulation of errors in the numerical results and computing the desired results [33]. In addition, this approach also improves the generalization of deep learning when facing different problems due to the added constraints of physical conditions.

2.3. Details of the Proposed PINN-Based Method

In this section, we introduce the details of the proposed PINN-based method. We first introduce the governing equation and parameters and then describe the method used to solve the one-dimensional soil–water vertical infiltration equation, which uses the PINN to perform a numerical investigation of soil moisture infiltration.

2.3.1. Governing Equation and Parameters

For unsaturated soil–water infiltration, the Richards equation is generally used, and the one-dimensional Richards equation for volumetric water content is:

$$\frac{\partial \theta }{\partial t}=\frac{\partial }{\partial z}\left[D\left(\theta \right)\frac{\partial \theta }{\partial z}\right]-\frac{\partial K\left(\theta \right)}{\partial z}$$

(1)

where $\theta$ represents the soil–water content, $D\left(\theta \right)$ is the soil moisture diffusivity, $K\left(\theta \right)$ represents the unsaturated hydraulic conductivity, z represents the soil depth and t represents time. The above equation can be translated into:

$$\frac{\partial \theta }{\partial t}=\frac{\partial }{\partial z}\left[D\left(\theta \right)\frac{\partial \theta }{\partial z}\right]-\frac{\partial K\left(\theta \right)}{\partial \theta }\frac{\partial \theta }{\partial z}$$

(2)

2.3.2. Linearisation of the Governing Equation

The governing equations were carried out in a series of linearisation processes [34,35]. For this unsaturated one-dimensional soil–water infiltration equation, the initial condition is the water content at the beginning of the solution condition, $\theta (z,0)={\theta }_0$. The boundary condition is that the upper boundary water content after infiltration begins is the upper boundary condition, $\theta (0,t)={\theta }_s$, and the lower boundary condition water content is always $\theta (z,0)={\theta }_0$. Both $D\left(\theta \right)$ and $K\left(\theta \right)$ are determined by fitting the equations to the values of D and K for different water content cases. To facilitate the solution of the equations:

Let $D\left(\theta \right)$ be constant $\overline{D}=D\left(\frac{{\theta }_0+{\theta }_s}{2}\right)$, and let $\frac{\partial K\left(\theta \right)}{\partial \theta }=N=\frac{K\left(\theta s\right)-K\left({\theta }_0\right)}{{\theta }_s-{\theta }_0}$; then, it follows that:

$$\frac{\partial \theta }{\partial t}=\overline{D}\frac{{\partial }^2\theta }{\partial z^2}-N\frac{\partial \theta }{\partial z}$$

(3)

where ${\theta }_0$ is the initial conditional moisture content, and ${\theta }_s$ is the upper boundary condition moisture content.

2.3.3. Solving the Soil–Water Vertical Infiltration Equation Using Pinn

In this study, the one-dimensional soil moisture infiltration equation is solved using PINN, and the principle of the soil moisture infiltration equation solution process is illustrated in Figure 3. The PINN seamlessly integrates information from measurements and partial differential equations (PDEs) by embedding the soil moisture infiltration partial differential equation (SMIPDE) into the loss function of a neural network using automatic differentiation [23]. First, a deep neural network $NN(z,t,\theta )$ is built to obtain the surrogate of the SMIPDE solution u. Then, solution u is substituted into the physical information constraint of the soil moisture infiltration equation. The soil moisture infiltration partial differential equation (SMIPDE) is constructed to obtain the PDE loss, and initial and boundary condition constraints are applied to obtain the initial and boundary condition losses. Finally, the neural network is trained by minimizing the loss until the obtained loss is less than a threshold $\epsilon$. Then, the soil–water content ${\theta }^{*}$ of the obtained optimal result can be output.

The implementation of the soil moisture infiltration partial differential equation is as follows: to provide a more convenient solution, in this study, we use a Python library for PINNs, DeepXDE, to solve the soil moisture equation of motion. DeepXDE can solve positive problems given initial and boundary conditions and inverse problems given some additional measurements [36]. The solutions to the differential equations found by DeepXDE are generally solved using the built-in modules for each specified problem. Moreover, DeepXDE supports complex geometric domains, and it supports four standard boundary conditions: Dirichlet boundary condition, Neumann boundary condition, Robin boundary condition, and Periodic boundary condition. The IC is used to define the initial conditions [36]. DeepXDE contains different neural networks, including feedforward neural networks $(maps.FNN)$ and residual neural networks $(maps.ResNet)$. During DeepXDE model training, different training hyperparameters such as loss types, metrics, optimizers, and learning rate schedules can be selected.

3. Verification of the Proposed PINN-Based Method

In this study, the PINN-based method is validated for the numerical investigation of soil–water vertical infiltration to ensure the reliability of the method when applied to the comparative analysis of water infiltration in different soil types. The PINN-based method was demonstrated to have smaller error than traditional numerical methods and to obtain more accurate results in numerical investigations of soil moisture infiltration. The measured data were obtained from experiments conducted by Murati et al. during their research work on the mathematical modeling of soil–water movement [37].

3.1. Experimental Environment

The hardware and software environment configuration is listed in

Table 1. Environment configurations.

Environment Configurations	Details
OS	Windows 10 Professional
Deep learning framework	TensorFlow
Dependent library	DeepXDE, Numpy etc.
CPU	Intel(R) Core(TM) i5-7200U CPU
CPU core	2
CPU RAM (GB)	8
CPU Frequency (GHz)	2.50

.

3.2. Governing Equation and Parameters

The original equation for the movement of soil–water infiltration used in this study is Equation (3).

From the experimental data, the following can be learned:

The initial condition is the initial value given to the variable at the beginning of the index value calculation, in this paper the initial condition is the moisture content value. The initial condition is $\theta (z,0)=0.09$.

Boundary conditions are the values of the variables at the boundary of the solution region. Boundary conditions are important for the solution of numerical problems. The boundary conditions are: upper boundary condition: $\theta (0,t)=0.42$, lower boundary condition: $\theta (z,t)=0.09$.

The $D\left(\theta \right)=0.0002e^{19.902\theta }$, and the $K\left(\theta \right)=555.22{\theta }^{13.181}$ [37], where $\theta$ indicates moisture content.

The $\overline{D}=D\left(\frac{{\theta }_0+{\theta }_s}{2}\right)=3.1995\times {10}^{-2}$ cm²/min.

The $N=\frac{\partial K\left(\theta \right)}{\partial \theta }=\frac{K\left(\theta s\right)-K\left({\theta }_0\right)}{{\theta }_s-{\theta }_0}=1.8197\times {10}^{-2}$ cm²/min.

From this initial data, the one-dimensional partial differential equation for soil–water infiltration can be determined as follows:

$$\frac{\partial \theta }{\partial t}=3.1995\times {10}^{-2}\frac{{\partial }^2\theta }{\partial z^2}-1.8197\times {10}^{-2}\frac{\partial \theta }{\partial z}$$

(4)

3.3. Numerical Results

The PINN-based numerical investigation method for one-dimensional soil–water infiltration proposed in this study was used to solve the above partial differential equation presented in Section 3.2 and obtain numerical results for soil–water infiltration. The model boundary conditions are illustrated in Figure 4 and the detailed numerical investigation procedure is illustrated in Figure 5. First, information and data concerning soil–water infiltration are input, the model computational domain is determined and meshing is performed. Then, a deep neural network model is built, boundary conditions and initial conditions are established and PDE equations are determined. Finally, the physics-informed neural network is trained to obtain the numerical results of the soil–water infiltration investigation.

The final soil–water infiltration profiles for three different periods, 600 min, 1230 min, and 1890 min, were obtained. In addition, we employed Hydrus-1D to numerically simulate the soil–water infiltration process for comparison with the PINN-based method, as illustrated in Figure 6. From Figure 6, it can be concluded that the soil moisture movement trends of both the traditional numerical method and the PINN-based numerical investigation method are consistent with the measured data, while the PINN-based numerical investigation method is a better fit compared to the traditional numerical method. To make the conclusions more reliable, the predicted water content at different depths and times obtained by the traditional numerical method and the PINN-based numerical investigation method was compared with the measured data and the error analysis for 600 min, 1230 min, and 1890 min were listed at

Table 2. Soil moisture infiltration error analysis.

Time		600 min	1230 min	1890 min
MAE	Traditional numerical method	2.030 × 10${}^{-2}$	2.022 × 10${}^{-2}$	2.658 × 10${}^{-2}$
	PINN-based numerical method	1.150 × 10${}^{-2}$	1.495 × 10${}^{-2}$	1.621 × 10${}^{-2}$
MSE	Traditional numerical method	5.018 × 10${}^{-4}$	7.126 × 10${}^{-4}$	2.659 × 10${}^{-3}$
	PINN-based numerical method	2.185 × 10${}^{-4}$	3.264 × 10${}^{-4}$	4.656 × 10${}^{-4}$
RMSE	Traditional numerical method	2.240 × 10${}^{-2}$	2.669 × 10${}^{-2}$	5.156 × 10${}^{-2}$
	PINN-based numerical method	1.478 × 10${}^{-2}$	1.807 × 10${}^{-2}$	2.158 × 10${}^{-2}$
R2	Traditional numerical method	9.759 × 10${}^{-1}$	9.512 × 10${}^{-1}$	8.269 × 10${}^{-1}$
	PINN-based numerical method	9.895 × 10${}^{-1}$	9.776 × 10${}^{-1}$	9.697 × 10${}^{-1}$

. From Table 2, it can be observed that the error of the prediction results of the method proposed in this study is smaller compared with the traditional numerical method, and the fitting results are closer to the measured data. Therefore, the method proposed in this study is more reliable in the numerical investigation of soil moisture infiltration.

4. Application of the Proposed PINN-Based Method

4.1. Experimental Background

4.1.1. Experimental Parameters

In research from Shao Ming’an et al. on the derivation of soil hydraulic conductivity parameters from soil moisture characteristic curves, the measured hydraulic conductivity and diffusivity of Ansai light-loam, Luochuan medium-loam, and Wugong heavy-loam soils were provided at different moisture contents [38].

Based on the available data regarding the hydraulic conductivity and diffusivity of different types of soils, the values of ${\theta }_0$ and ${\theta }_s$ can be determined as ${\theta }_0=0.188$ and ${\theta }_s=0.337$, respectively, as illustrated in Figure 7. The linear fitting equations for soil hydraulic conductivity and soil moisture diffusivity for the different soil types can be obtained from the measured data as follows, where Equations (5) and (6) is for light-loam soils, Equations (7) and (8) is for medium-loam soils and Equations (9) and (10) is for heavy-loam soils.

$$D\left(\theta \right)=6\times {10}^{-8}{e}^{24.639\theta }, \theta \in (0.063,0.337)$$

(5)

$$K\left(\theta \right)=5.6\times {10}^{-3}{e}^{14.755\theta }, \theta \in (0.063,0.337)$$

(6)

$$D\left(\theta \right)=1\times {10}^{-9}{e}^{22.947\theta }, \theta \in (0.168,0.572)$$

(7)

$$K\left(\theta \right)=4.6216{\theta }^3-1.7655{\theta }^2+0.5168\theta -0.0332, \theta \in (0.168,0.572)$$

(8)

$$D\left(\theta \right)=7\times {10}^{-8}{e}^{16.843\theta }, \theta \in (0.188,0.560)$$

(9)

$$K\left(\theta \right)=-15.656{\theta }^3+16.993{\theta }^2-4.3267\theta +0.4389, \theta \in (0.188,0.560)$$

(10)

Furthermore, the values of $\overline{D}$ and N, which are used to solve the soil–water movement equation for light-loam, medium-loam, and heavy-loam soils, can be determined as illustrated in

Table 3. N and $\overline{D}$ values for different soil types.

Soil Type	N	$\overline{\mathit{D}}$
Light-loamy soil	1.5844 × 10${}^{-3}$	2.6933 × 10${}^{-1}$
Medium-loamy soil	1.1683 × 10${}^{-5}$	6.4401 × 10${}^{-2}$
Heavy-loamy soil	1.2594 × 10${}^{-4}$	1.9088 × 10${}^{-1}$

.

4.1.2. Experimental Equations

The original equation for the movement of soil–water infiltration used in this study is Equation (3).

The initial condition is $\theta (z,0)=0.188$.

The boundary conditions are: upper boundary condition: $\theta (0,t)=0.337$, lower boundary condition: $\theta (z,t)=0.188$.

Therefore, the experimental equation for different soils can be obtained from the data in Section 4.1.1.

4.2. Solving the Soil–Water Infiltration Equation

The flow chart for the solution to the equations obtained using DeepXDE is illustrated in Figure 8. First, we use the geometry module to specify the computational domain, depth in the range (−100, 0), and time in the range (0, 2400); then, we use the TensorFlow grammar to specify the partial differential equation (PDE) and specify the Dirichlet boundary conditions, which are an upper boundary condition of 0.337, a lower boundary condition of 0.188, and an initial condition (IC) of 0.188. The geometry, PDE, and boundary/initial conditions are integrated into $data.TimePDE$. The number of training residuals sampled in the domain is 25,400, the number of training points sampled on the boundary is 800, and the initial conditioned initial residuals are 1600. The maps module is used to build a feed-forward neural network, and then the PDE problem is combined with the constructed neural network to build the model. The optimization hyperparameters, including the optimizers and learning rates, are set via $Model.compile\left(\right)$, where the optimizer is selected as $Adam$ and the learning rate is set to 1 × 10${}^{-3}$, the model is trained via $Model.train\left(\right)$, and finally, the prediction results are obtained with $Model.predict\left(\right)$.

4.3. Comparative Analysis of Vertical Soil–Water Infiltration Processes in Different Soil Types

4.3.1. Various Soil–Water Infiltration Processes

According to the solution to the soil–water infiltration equation, the water infiltration processes of light-loam, medium-loam, and heavy-loam soils are obtained, and the three-dimensional point diagram of the infiltration process was generated and is illustrated in Figure 9a–c. The three-dimensional point diagram of the infiltration process is generated by combining the three soil types as illustrated in Figure 9d. The three-dimensional point map allows the water content of the soil to be determined at any time and at any depth, thus allowing a dynamic observation of the water infiltration in each soil type. The three-dimensional point diagram indicates that water infiltrates deeper and deeper with time in all three types of soil.

In Figure 10, the water infiltration with time is visualized for the three types of soil. It can be seen that light loam, medium loam, and heavy loam all continue to infiltrate vertically with time, among which the light loam and heavy loam both have a deeper vertical infiltration depth with time, and after the same amount of time, the vertical infiltration depth is larger and the infiltration rate is faster. Although the medium-loam also adheres to the law of vertical infiltration with time, the infiltration rate is slower.

The numerical predictions results of soil–water infiltration for light-loam, medium-loam, and heavy-loam soils are illustrated in Figure 11, Figure 12 and Figure 13, which also indicates that the depth of infiltration increases with time for different soil types. Based on the comparison of the numerical predictions for the different soil types, it can be found that the depth of infiltration increases fastest with time in light-loamy soil followed by heavy-loamy soil, and it is slowest in medium-loamy soil.

4.3.2. Comparison of Soil–Water Infiltration in Different Soil Types at the Same Time

The comparative results for the water infiltration processes in light-loam, medium-loam, and heavy-loam soils at 10 min, 20 min, 30 min, and 40 min, respectively, are illustrated in Figure 14, which shows that soil–water infiltrates deeper as time increases. At 10 min, 20 min, 30 min, and 40 min, the light-loam reaches the greatest depth of infiltration of the three, the heavy loam is second, and the medium-loam is shallower. After 20 min, the infiltration depths of the light and heavy-loam soils gradually increased compared to that of the medium-loam soil, and by 40 min, the infiltration depths of the light- and heavy-loam soils had already increased compared to that of the medium-loam soil. These results indicate that the light loam has the highest infiltration capacity, the heavy loam has the second-highest infiltration capacity, and the medium-loam has the weakest infiltration capacity due to the low hydraulic conductivity and diffusivity of the medium-loam. When constructing artificial dams or filling artificial slopes, it is advisable to use medium-loam soils with a low infiltration rate. For artificial dams or slopes composed of light or medium-loamy soils, a corresponding slope safety factor must be determined, and certain control measures should be taken to prevent slope instability.

5. Discussion

In this study, we propose a PINN-based approach for the numerical investigation of soil–water vertical infiltration, and we perform a comprehensive evaluation and analysis of the soil–water infiltration process in different soil types. The approach proposed in this study provides smaller error in numerical results than traditional numerical methods and improves the accuracy of numerical investigations of soil–water infiltration problems. The solution to the soil–water infiltration equation based on the PINN in this study is the first attempt to use this network in the investigation of soil–water infiltration, which fills the previously described gap in the field and is conducive to further research of deep learning for soil infiltration problems in the future.

The PINN-based approach achieves better numerical results than traditional numerical methods for the investigation of one-dimensional soil–water vertical infiltration in this study. In traditional numerical methods, the computational domain is divided into a certain number of grids/elements, the partial differential equations (PDEs) to be solved are discretion, the discreteness partial differential equations (PDEs) are solved in each grid/element, and the approximate solutions of the equations in each grid/element are obtained, and the solutions of all the grids/elements are then aggregated to obtain the final results. In contrast, the PINN-based deep learning methods, for example, the numerical investigation of soil moisture infiltration, involves randomly selecting sampling points in the computational domain and ensuring that all random sampling points in the computational domain meet the constraints as far as possible by minimizing the loss function. The errors in traditional numerical methods arise mainly from local errors arising in the local approximation in the grid cells and global errors arising from the global approximation when the solutions of each grid cell are aggregated into the final numerical result for the whole computational domain. The PINN-based methods are based on the error that arises when minimizing the loss function to satisfy the constraints at the sampling points as far as possible. As a result of the different sources of error, the PINN-based method may, under certain conditions, have a smaller error than the traditional numerical method, resulting in more accurate numerical results.

Due to the complexity of solving the soil–water infiltration equation, previous research has mostly focused on solutions to the infiltration equation, but there is little comprehensive analysis and research on the infiltration process of rainwater in different soil types. In this study, the PINN-based method is employed for numerical investigation of water infiltration in different soil types, not only to simultaneously obtain the water infiltration profiles of different soil types but also to compare and analyze the infiltration capacity of different soil types. The dynamic process of water infiltration over time for a particular soil type can also be studied, and the soil–water infiltration profile at any given time can be extracted, which allows for a comprehensive analysis of soil–water infiltration processes in different soil types. For geological hazards that have already occurred, the infiltration of soil moisture at any point in the process can be inverted to provide experience for future geological hazard prevention and control. In addition, it is possible to predict the state of soil moisture infiltration at different times before rainfall to further determine the stability of slopes and to monitor and warn of geological hazards in advance. Finally, for artificially filled slopes and dams, suitable filling soils can be selected according to the different types of soil moisture infiltration processes to prevent the instability of slopes and dams.

However, in this study, the values for hydraulic conductivity and diffusivity were simplified when solving the soil–water movement equation to make the equations more convenient, which affected the accuracy of the final calculation results to a certain extent. The data collected in this study are limited and only include hydraulic conductivity and diffusivity values for some water content cases involving light-loam, medium-loam, and heavy-loam soils, making the range of the upper and lower boundary conditions of water content in the soil–water infiltration investigation small, and the final research results have some limitations.

In future work, more experimental studies will be conducted to determine the moisture content of more types of soil and to measure the hydraulic conductivity and diffusivity to enrich the experimental data. We will also obtain more representative results for numerical investigations. In addition, the method used for numerical investigation of soil–water vertical infiltration proposed in this study will be further optimized, e.g., soil–water characteristic curves can be linked to Richards’ equation, such as Gardner’s model [39,40] and van Genuchten’s model [41], so that the values of various parameters of the soil–water infiltration process can be determined more accurately, and the linkage between soil–water characteristic curves and soil–water infiltration equation can be established in deep neural networks to obtain more accurate numerical results. Moreover, further analysis of the solution to the two-dimensional soil moisture equation of motion will be carried out. Fieldwork will also be conducted to achieve an efficient and comprehensive analysis and evaluation of different soil percolation processes in different areas and to obtain equation solution results that are more suitable for better monitoring and early warning of geological hazards.

6. Conclusions

In this study, a PINN-based approach for the numerical investigation of soil–water vertical infiltration is proposed, and a comprehensive evaluation and analysis of the soil–water infiltration process in different soil types is performed. Compared with the traditional numerical method, the method proposed in this study combines physical constraints with data-driven models, the simulation error of the soil moisture infiltration process is smaller, better prediction results can be achieved, and the soil moisture movement process analysis can be carried out more effectively. A dynamic understanding of the soil moisture infiltration process is achieved, which is conducive to the real-time forecasting of geological disaster monitoring and early warning.

The results show that (1) the numerical investigation method of soil–water vertical infiltration proposed in this study has smaller error than the traditional numerical method and can obtain more accurate results. (2) The analysis of soil–water infiltration processes for different soil types indicates that light-loam, medium-loam and heavy-loam soils all infiltrate vertically with increasing infiltration time, but the degree of infiltration is different. The light loam has the fastest infiltration rate, followed by the heavy loam. In contrast, the medium loam has a slower infiltration rate compared to the other types, which is due to the lower diffusivity and hydraulic conductivity, and this indicates that medium-loam soils are less susceptible to water infiltration than the other two types of soil and are more suitable for filling of artificial slopes or dams.

In the future, we will collect more soil-moisture-infiltration-related data for the PINN-based soil moisture infiltration problem to obtain more accurate numerical prediction results. In addition, a link between the soil–water characteristic curve and the soil–water infiltration equation will be established in the deep neural networks to obtain more accurate numerical results of the soil–water infiltration process. The solution to the 2D soil moisture infiltration problem will be carried out and further extended to the analysis of slope stability and better monitoring and early warning of geological hazards.