Machine learning-based improved method for estimating long-term failure probability with high efficiency of bank slope

Abstract

Affected by water level and rainfall, the failure probability of bank slope is of great time-dependent characteristics. Combining the random field theory with Monte Carlo simulation strategy to calculate long-term failure probability is a time-consuming when considering the spatial variability of geotechnical materials. Therefore, how to efficiently predict the long-term failure probability remains an urgent problem, which is vital to ensure the safety of bank slopes. This study combines remote sensing technology with machine learning methods to conduct comprehensive analysis on the long-term stability and failure probability of bank slope. Firstly, taking an actual bank slope as an example, the failure mechanism and time-dependent characteristics of it is studied with aid of the remote sensing technology. Subsequently, the stability of the bank slope within a year is quantified by the safety factor and seepage field obtained from the numerical model. The failure probability of the bank slope in the same year is calculated by the random field model, and the influence of uncertain parameters on the failure probability is drawn. Three deep learning models, such as multilayer perceptron, convolutional neural networks, and long short-term memory, are adopted to predict the long-term failure probability in 10 years. The results show that the failure probability increases with any of the uncertain parameters, such as the coefficient of variation, correlation coefficient of shear strength, or scale of fluctuation. The utilization of the novel method proposed by this study can efficiently depict the time-dependent characteristic of the long-term failure probability, and it is applicable in predicting the future failure probability of the bank slope. Among three models, the long short-term memory model shows better performance in predicting the time-dependent failure probability. The input data amount and the ratio of the training and test sets have an insignificant effect on the prediction results.

1 Introduction

The stability of coastal slope is vital to the safety of coastal engineering (Li et al., 2024; Li et al., 2025) such as the breakwater and embankments and etc. There are many external affecting factors especially the periodic wave loads and tidal water level variations leading to a time-dependent characteristics for deformation and stability of the coastal slope. The phenomenon is similar in bank slope which has been widely studied. As a result, studying the time-dependent characteristics of bank slope has reference significance on the safety of coastal slope.

With the construction of the Three Gorges Reservoir (TGR), many slopes nearby have become bank slopes. Due to changes in geological conditions and the surrounding environment, bank slopes will further evolve into reservoir bank landslides. For example, one of the greatest losses landslides the Qianjiangping landslide, the volume of the landslide body reaches 24 million m³, and the maximum surge generated by the landslide is 39 m, causing economic losses of approximately 80 million Chinese Yuan (Yin et al., 2015). The bedding slopes are the most common type in TGR. As the causing factors and conditions are different and complex for each landslide, the prevention and control of geological disasters caused by landslides in this area still face big challenges.

During the fluctuations of water level, the rock and soil mass in the sliding zone show an alternating dry and wet state. After being soaked and softened, the shear strength of the rock and soil decreases. This is an important condition for the revival of some ancient landslides or the multistep deformation and failure of reservoir bank landslides (Xu et al., 2020; Wu et al., 2023). By considering the dry-wet cycling effect of the sliding body and sliding zones of the Zhaoshuling landslide due to water level fluctuations, the variation of its stability under the reduction of the shear strength parameter of the slope body was analyzed (Zhang et al., 2022). Coupling the erosion and damage of the slope surface caused by rain, but the sliding zone had not yet reached saturation, Sun et al (Sun et al., 2023). used permeable boxes as the sliding surface and studied the effect of soil types in sliding zone on the revival of landslides by filling the permeable boxes with water. Su et al (Su et al., 2023). analyzed microscopic changes of sliding zone soil after interacting with acidic groundwater, normal groundwater, and distilled water environments and found that all three types of groundwater damaged the microstructure of the sliding zone soil to varying degrees. Among them, acidic groundwater had the most severe degree of damage to the soil structure. Zou et al (Zou et al., 2023). studied the mechanical properties of the sliding zone soil of the Liutang landslide through circum-shear tests and proposed a stability estimation method for landslides. This method was able to quantitatively express the relationship between the shear strength and displacement of the sliding zone soil under different moisture contents and normal stress states.

Rainfall is regarded as one of the important external factors causing landslides in the reservoir area (Yazdani et al., 2024; Zhang et al., 2024). Fan et al (Fan et al., 2020). generated a series of time series with different peak rainfall intensities and magnitudes. By comparing the dynamic characteristics of landslides induced by different rainfall patterns, they found that large early rainfall and uniform rainfall were the most probable patterns to cause landslides. Wang et al (Wang et al., 2021). classified the landslides caused by heavy rain into two failure modes. One was the sliding along the shear surface owing to the failure of the soil, and the other was the rapid sliding caused by the erosion of the slope. Yu et al (Yu et al., 2023). studied the failure of bedding slope containing interlayer soil under different rainfall patterns. They found that under uniform rainfall conditions, the rainfall threshold during landsliding would increase with the decrease of rainfall intensity. When there were peaks in the rainfall pattern, the rainfall threshold for landsliding was the smallest before the peak and the largest after the peak. A large number of the above-mentioned studies have shown that rainfall and reservoir water level are the main controlling factors when estimating the stability of reservoir bank landslides. Therefore, estimating the multistep deformation and evolution process of reservoir bank landslides, the effect of rainfall and reservoir water level, so-called the dynamic mechanical environment in this study, needs to be considered.

Reliability analysis quantifies the safety factor of geotechnical engineering from a probabilistic perspective, providing an effective way when considering the uncertainties of geotechnical engineering (Li et al., 2016; Jiang et al., 2022). It has been widely applied and developed in aspects such as uncertainty quantification (Li et al., 2019; Yang HQ. et al., 2021), advanced surrogate models (Li et al., 2023), and reliability design (Vagnon et al., 2020; Yang YJ. et al., 2021; Wang and Owens, 2022). Wu et al (Wu et al., 2017). conducted a reliability analysis and found that the reliability of the landslide gradually decreases as the strength of the sliding zone soil weakens. Zhang et al (Zhang et al., 2020). analyzed the changes in the failure probability of the Bazimen landslide under the dynamic mechanical environment based on monitoring data, and pointed out that the failure probability would increase with the increase of soil variability. Liao et al (Liao et al., 2021)further conducted dry-wet cycle tests to simulate the water level fluctuations on the Majiagou landslide. By analyzing the time-dependent reliability of the landslide for 10 years, they pointed out that the landslide would translate to an unstable state. Guardiani et al (Guardiani et al., 2022). adopted machine learning methods as surrogate models to study the relationship between the safety factor and time, and conducted sensitivity analyses on both the mechanical and hydraulic parameters. However, geotechnical parameters are usually regarded as random variables in the above-mentioned research, and existing research indicates that not considering spatial variability will lead to relatively conservative results (Huang et al., 2021; Li et al., 2022).

The spatial variability of geotechnical materials is always considered in numerical simulation under the basis on random field theory and Monte Carlo simulations. Considering the high cost and time-consuming calculation of MCS (Monte Carlo simulations), scholars have combined data-driven methods to construct surrogate models for achieving reliability analysis quickly and accurately. Wang and Goh (2021) used convolutional neural networks (CNNs) of which the predicting accuracy was very close to the results calculated directly using random fields, but the computational efficiency had been improved. Zhang et al (Zhang et al., 2023b). took the Bazimen landslide as an example and established the XGBoost and LightGBM prediction models to obtain the safety factor. The above-mentioned research verified the applicability of machine learning methods in improving efficiency in estimating the failure probability. However, for the reservoir bank landslides in dynamic mechanical environments, the failure probability of it is bound to show great time-dependent characteristics, and the long-term failure probability is of significant importance in estimating the risk of landslides. A novel method is needed to efficiently predict long-term reservoir bank landslide probability while accounting for soil and rock mass spatial variability.

This study, aiming at improving the prediction efficiency of the long-term failure probability of bank slope under the influence of dynamic mechanical environment, takes the Jiuxianping landslide in TGR as an example and proposes a novel method for the long-term failure probability prediction. To be specific, this paper is organized as follows. First, through the analysis of on-site geological and monitoring data, the failure mechanism and time-dependent displacement characteristics of the Jiuxianping landslide are determined. A numerical model of the landslide is established to assess its seepage field and safety factor in 2020. Subsequently, reliability analysis is performed using a random field model, incorporating the spatial variability of the sliding zone. Also, the effect of the uncertain parameters on failure probability is drawn. Finally, based on the 10-year failure probability from the random field model, a novel method for long-term failure probability prediction through deep learning is proposed and its high efficiency is verified. Furthermore, performance when using different deep learning models are compared and the effect of the data amount is discussed.

2 Stability evaluation of Jiuxianping landslide

2.1 Overview of landslide

The Jiuxianping landslide is a large wading landslide with a total volume of approximately 6902.5×10⁴ m³. The distance between it and dam is 239 km. The center point coordinates are 108°4657.35” east longitude and 30°5627.63” north latitude. The average thickness of the landslide body is 40 m, and the area is about 174×10⁴ m². The bedding landslide with large-scale is shown in Figure 1. It is in a continuous deformation state, and once the landslide becomes unstable, it will cause incalculable losses.

Figure 1

Figure 2 plots the overall view of the Jiuxianping landslide, including the monitoring layout and actual view of the landslide. As plotted in Figure 2, the rear edge of the Jiuxianping landslide shows obvious signs of sliding. The middle of the slope is relatively straight, while the front part is characterized by steep cliffs and slopes. The overall failure mechanism of the Jiuxianping landslide is sliding-bending. To be specific, the shallow, fragmented rock masses are sliding locally while the rear accumulation body is sliding along the bedrock surface in a pushing manner. Because the Jiuxianping landslide is a bedding slope, the dip angle of the rock strata is greater than the gradient of the slope. Under the circumstance that the dip angle of the sliding surface is greater than the friction angle of that surface, landslide occurs However, there still exists a resistance for the rock mass to slide, and this makes the rock mass in the middle and lower parts bend and deform until it leads to collapse and failure. Therefore, the development of the landslide can be divided into three stages: the slightly bended stage, the strongly bended and uplifted stage, and the slide surface formed stage. Owing to the release of energy developed in the first 2 stages, the landslide accelerates with additional thrust.

Figure 2

2.2 Measured data

This section exhibits the measured data from the global navigation satellite system (GNSS) for which the monitoring layout is shown in Figure 2A. Figures 3 and 4 plot the cumulative displacement and deformation rate with time, as well as the corresponding water level and rainfall from June 2016 to November 2021.

Figure 3

Figure 4

The Jiuxianping landslide’s deformation primarily occurs at the middle and leading-edge sections. The displacement trends at multiple monitoring points in the middle and leading-edge areas are consistent, suggesting the entire landslide is sliding downward. From 2016 to 2021, the cumulative horizontal displacement of the middle and leading edge of the Jiuxianping landslide reached 237.5 mm. The change of displacement was the greatest in 2017, with an increment of approximately 66.3 mm. The cumulative horizontal displacement at the rear edge was relatively small (i.e., 55.9 mm), and the increment within 2021 was approximately 5.7 mm.

The cumulative displacement curve shows a stepped pattern, reflecting intermittent sharp increases followed by slower deformation periods. Further observation of the deformation rate of the middle and leading-edge positions reveals that the displacement shows a great time-dependent characteristic. The displacement shows regular changes. Take the data of 2021 as an example. From January to April, the deformation rate of each monitoring point was relatively low at 1mm/month. From May to October, the deformation rate increased sharply, with the deformation rate being around 2.0~8.2 mm/month. From November to December, the deformation rate decreased to 1 mm/month.

Compared with the corresponding reservoir water level and rainfall data, these two factors are the most important affecting landslide deformation. Displacement peaks annually in June (with deformation rate reaching 1.9 mm/day) during flood season water level fluctuations (see Figures 3, 4). In other months, the reservoir water level was increased or at a high water level, and the landslide did not show obvious deformation. Water level decline creates an outward hydrodynamic pressure (higher inner/outer head difference) at the landslide front, reducing slope stability.

With the influence of rainfall in the flood season, the landslide displacement shows a stepwise deformation. On the one hand, rainwater seeps into the slope, increasing the gravity of the sliding body and thus the sliding force. On the other hand, the softening effect of landslide soil lowers the resistance of the slope and leads to the instability of the landslide. However, the impact of rainfall individually is smaller than water level. Even at 0.1 m/d rainfall inducing 0.25 mm/d landslide displacement, the movement remains smaller than during reservoir drawdown.

2.3 Numerical model

To quantitatively evaluate the stability of the landslide, the numerical model (Zheng et al., 2025a; Zheng et al., 2025b) of the Jiuxianping landslide is established in Slide 2. Based on the Morgenstern-Price method, the safety factor F_s is calculated, and the mechanism of interaction between reservoir water level and rainfall is investigated in combination with the theories of unsaturated seepage and unsaturated shear strength. The typical cross-section of the Jiuxianping landslide is selected within a hydrological year. When conducting the steady-state seepage calculation, the constant-head boundary is prescribed at the leading edge with 175 m and the below area, and the water head on the left side is set as 360 m according to the monitoring data. For transient seepage analysis, a variable-head boundary (145~175 m) matching the Yangtze River’s actual water level is prescribed at the leading edge. The remaining areas of the slope are set as rainfall flux boundaries, consistent with the monitored daily rainfall. The numerical model is divided by using three-node triangular meshes, with a total of 3,273 meshes. Since the sliding zone has been identified, the sliding surface in the simulation is completely specified.

In the typical cross-section, strata are formed by silted clay containing crushed stones, fractured rock mass, sliding zone, and bedrock, as shown in Figure 5. The geotechnical parameters for the typical strata are listed by conducting the laboratory tests in Table 1. Mohr-Coulomb criteria are adopted in the model. Based on the Slide 2 seepage analysis module, the van-Genuchten empirical model is used to obtain the soil-water characteristic curves and permeability functions.

Figure 5

2.4 Stability evaluation

In 2020, the TGR faced its largest flood peak (i.e., 75,000 m³/s) during the No. 5 flood event in July-August which is the highest since the reservoir’s operation began. Therefore, stability evaluation is carried out based on the condition in 2020. Figures 6, 7 show the variation of seepage field and F_s in 2020.

Figure 6

Figure 7

As shown in the Figure 6, the water level experienced great decline and increase in the duration from January to April and from June to September. It is easy to find that F_s shows a positive correlation with reservoir water level. Both the variation trend and the varying rate are consistent between F_s and the reservoir water level. Furthermore, affected by rainfall, F_s reaches the minimum value 1.189 in June. Compared to the reservoir water level, rainfall has a more irregular and subtle effect on landslide behavior.

Combined with the seepage field shown in Figure 7, it can be observed that the infiltration line at the leading edge of the landslide becomes upward convex shape and upward concave shape (e.g., on April 5^th and October 29^th) when water level declined and increased. A faster-changing water level makes the phenomenon more noticeable. This occurs because the rate of change in the reservoir water level exceeds the slope’s permeability coefficient, resulting in a lag effect of the infiltration line in slope. Also, this is the failure mechanism of landslide affected by reservoir water level. When the infiltration line becomes upward convex shape, the outward seepage-induced water pressure reduces slope stability, triggering landslide movement. On the contrary, the slope is stable.

After October 24^th, the water level increased to 174.04m and then began to operate at a high level. On October 29^th, F_s of the landslide reached its peak (i.e., 1.274). Following this, the reservoir water level did not change significantly, but F_s slightly decreased. This phenomenon results from the reduction and eventual dissipation of the hydraulic head gradient across the slope interface. That means the infiltration pressure towards the slope is gradually disappearing. F_s shows a slightly decreasing trend due to the floating effect brought by the reservoir water.

3 Reliability analysis considering the variability of sliding zones

3.1 Establishment of the RFM

Stochastic analysis is conducted in this section to further obtain the failure probability of the landslide. To consider the spatial variability of geotechnical materials, the Local average subdivision method is adopted to expand the random field. Based on the local average theory, the accuracy is achieved through the continuous subdivision of elements. The Markovian autocorrelation function is adopted shown as Equation 1.

where and represent the horizontal and vertical scale of fluctuation, respectively; and represent the horizontal and vertical distances between two points. Stochastic analysis is carried out by combining the Monte-Carlo Simulation (MCS) strategy with Latin-hypercube-sampling (LHS).

To reflect the uncertainty of the slide zone, the mean μ and standard deviation σ of the shear strength parameters are obtained according to the onsite borehole data. Statistical analysis shows the sliding zones’ shear strength follows a lognormal distribution. Referring to the literature of similar bedding landslides in the TGR (Li et al., 2015; Zhang et al., 2023a), l_x is in a range of 20–80 m, while l_y is in a range of 2–6 m. As a result, the spatial variability parameters are determined in Table 2.

Figure 8 shows one of the implementations of the random field. In the software, the grid is automatically divided by the software based on the input variability parameters and the Markov autocorrelation function. The grid size is at least half of the l_x or l_y, and each material layer has a sufficient number of grids to correctly represent the spatial variability. A total of 5,000 implementations is conducted in every stochastic case. As shown in Figure 9, both the μ and σ of F_s exhibit stabilizing tendencies when the implementation time is 5000, indicating that the probability analysis results can meet the accuracy requirements.

Figure 8

Figure 9

3.2 Reliability analysis

Referring to the “Technical requirements for engineering geological investigation of geological hazard prevention in TGR (Three Gorges Reservoir Area Geological Hazard Prevention Headquarters, 2014), simulation cases with F_s less than 1.2 are considered unstable. Also, the reliability analysis is conducted according to the time stage clarified in the Table 2.

Figure 10 demonstrates the variation of failure probability (P_f) during each time stage and the whole of 2020. According to Figure 10A, P_f in stage 1 is relatively small, ranging from 0 to 5.9%. As the reservoir water level declines more obviously in stage 2 (see Figure 10B), P_f increases sharply, with a maximum reaching 83.32%. Seeing from Figure 10C, as the water level went through a quick increase and decline, so does the variation of P_f. From June 10^th to July 21^st, the water level increased, and P_f decreased sharply from 81.96% to 0.22%. On August 14^th, as the water level declined to its lowest, P_f increased to 41.82%. During August and September, P_f decreased to 0 and then increased to 44.72% with a water level increase first and decline. According to Figures 10D, E, the water level increased from 155.28m to 172.34m at stage 4, and the water level stayed at a high level at stage 5. From September 23^rd to September 26^th, the reservoir water level declined slightly, P_f decreased from 45.06% to 0. After that, P_f stays 0.

Figure 10

Figure 10F demonstrates the variation of P_f in the whole year. Compared with the deterministic analysis in section 2.4, the variation pattern of P_f demonstrates remarkable consistency with deterministic evaluation results. However, the response of P_f is more intense. In other words, the reliability analysis can provide early warning of reservoir bank landslides.

3.3 Parametric analysis

3.3.1 Coefficient of variation

To study how the variation in c and φ of the sliding zone affects P_f, the heaviest rainfall day in the 2020 flood season (July 11th, with 0.1m of rainfall) is selected as the study case. The variation of P_f with COV_c and COV_φ is plotted in Figure 11. P_f increases from 7.76% to 8.04% when COV_c increases from 0.05 to 0.4. P_f increases from 0 to 35.96% when COV_φ increases from 0.05 to 0.4, with a relatively large variation range. When COV_c and COV_φ increase, P_f increases simultaneously. However, compared with the influence of COV_φ, COV_c has a relatively small effect on P_f.

Figure 11

To further investigate the effect of COV_φ, the probability density curve and cumulative distribution curve of F_s for different COV_φ are drawn in Figure 12. As shown in Figure 12, the shape of the curves changes a lot when COV_φ varies, and this is the main reason that P_f is sensitive to COV_φ. When COV_φ is as small as 0.05, the probability density curve shape is steep and presented as a concentrated distribution. With the increase of COV_φ, the peak value decreases and gradually shifts towards the left, and the curve shape becomes short and wide, indicating a more unstable and uncertain condition for the landslide. As for the cumulative distribution of F_s, these curves intersect at one point at which F_s is about 1.23. When COV_φ increases, the slope of the curve decreases when Fs is smaller than 1.23 but increases when F_s is larger than 1.23. That is to say, the frequency for smaller F_s also increases, showing a more unstable condition of landslide.

Figure 12

3.3.2 Correlation coefficient

Figure 13 shows the effect of the correlation coefficient of c and φ on P_f. When the correlation coefficient changes from -0.7 to 0.5, P_f increases from 7.32% to 9.22%. This indicates that, compared with the negatively correlated, positively correlated c and φ are more unfavorable for the stability of the landslide. However, P_f is not that sensitive to the correlation coefficient, as the variation range is not that large. The probability density curve and cumulative distribution curve of the F_s with different correlation coefficients are drawn in Figure 14. With different correlation coefficient, the shape of the probability density curves remains almost the same, and the peak value increases slightly. The cumulative distribution curves almost coincide with each other.

Figure 13

Figure 14

3.3.2 Scale of fluctuation

To compare the vertical and horizontal scale of fluctuation together, define the coefficient for scale of fluctuation as the ratio of l_x or l_y between the case and the basis case mentioned in the Table 3. Figure 15 shows P_f with l_x or l_y. P_f increases with both l_x and l_y, but to different extents. As l_x increases from 10m to 60m, P_f increases from 5.42% to 11.32%. As l_y increases from 1m to 6m, P_f increases from 4.26% to 10.1%. When the l_x or l_y is large enough, the stochastic field representation reduces to a random variable formulation. This also indicates that the random variable model can obtain a conservative result of P_f. The probability density curve and cumulative distribution curve of the F_s with different l_x are drawn in Figure 16. As l_x increases, the peak of its probability density curve gradually decreases, and the distribution becomes more extensive. The overall shape evolves from a tall and narrow type to a short and wide type, and the slope of the cumulative distribution function curve gradually becomes flatter. The failure probability of the landslide will also increase accordingly.

Figure 15

Figure 16

4 Time-dependent failure probability prediction

Due to the high time cost when using MCS, surrogate models are widely used in reliability analysis. Nevertheless, their predictive accuracy requires further enhancement, and these models can not reflect the time-dependent characteristic of the predicted scope. As has been analyzed above, the displacement, stability, and P_f of the Jiuxianping landslide show very intense time-dependent characteristics affected by reservoir water level and rainfall. Therefore, this study proposes a novel method based on deep learning that has been successfully applied in time series prediction in stochastic analysis, and attempts to predict the temporal evolution of P_f. Through the time series prediction, P_f at future time points and long-term reliability evaluation can be further obtained.

4.1 Methodology

4.1.1 Multilayer perceptron

Single-layer neural networks require linear separability, but adding more neurons between the input and output layers can solve nonlinear separation problems. The Multilayer Perceptron (MLP), a simple yet powerful deep learning model, acts as a mapping function. MLP can approximate arbitrary nonlinear functions, making them highly effective for time series prediction.

4.1.2 Convolutional neural networks

CNN uses convolutional layers to extract features via sliding kernels, ensuring translation invariance—detecting features regardless of their position. Pooling layers (e.g., max/average pooling) down sample these features, expanding the kernel’s receptive field to capture broader patterns. Finally, a fully connected layer integrates all features for classification or recognition. Though designed for images, CNN can also outperform recurrent neural networks (RNN) in certain sequence tasks while being computationally cheaper.

4.1.3 Long short-term memory

Unlike standard RNNs, LSTM introduces a cell state C_t and three gating mechanisms to control information flow. The forget gate determines the content to be forgotten from C_t-1 using a sigmoid-activated output f_t. The input gate determines which new information to store in the cell state via sigmoid (i_t) and tanh _t layers. The output gate regulates the next hidden state h_t by filtering the updated cell state C_t through a sigmoid o_t and tanh layer. By selectively retaining or discarding information at each step, LSTM effectively captures long-range temporal dependencies, and it is a key advantage over traditional RNN in tasks like time-series prediction.

4.2 Preparation of data

P _f of the landslide during 2011–2020 is calculated according to the numerical method mentioned in section 3, and the results are demonstrated in Figure 17. The deep learning models are established using the TensorFlow framework. The dataset is partitioned into training and test sets with a 7:3 ratio. The data involving 2011–2017 and 2018–2020 are regarded as the training set and test set, with a data amount of 2558 and 1095, respectively. Taking the current day as the starting point, the model is trained using the data of the previous five days to predict the failure probability of the next day.

Figure 17

Three evaluation indices, such as the determination coefficient (R²) defined in Equation 2, mean absolute error (MAE) defined in Equation 3, and root mean square error (RMSE) defined in Equation 4, are calculated for the deep learning models.

where N represents the total amount of the data; is the value of original data; is the predicted result; is the average of the original data.

4.3 Long-term failure probability

The structure of the MLP model consists of four fully connected layers. The structure of the CNN model consists of a total of three conv1D layers, which include a maxpooling1D layer and a global-average-pooling1D layer in the middle, and ends with a fully connected layer. In the LSTM model structure, four layers of LSTM are established, and the final layer is a fully connected layer. Figures 18, 19 shows the prediction results of three deep learning models compared with the training set and test set, respectively. Failure probability and its variation trend predicted by MLP, CNN, and LSTM models are generally consistent with the calculated value based on MCS in 2 figures. This indicates that time-series prediction based on deep learning models is of high accuracy. Moreover, these models are of high efficiency. While Slide 2 software takes about 35 minutes to compute the Jiuxianping landslide’s one-day failure probability, a deep learning model can predict decade-long P_f in just one minute. This drastically reduces computational demands for time-dependent reliability analysis and offers an effective solution for long-term landslide risk assessment. Other stability analysis methods for slope, such as the strength reduction method, are also applicable. This verifies the practical applicability of the novel method proposed in this study.

Figure 18

Figure 19

However, prediction results differ slightly when using different models. The MLP and CNN models may significantly overestimate or underestimate P_f when its corresponding calculated value is at the maximum or minimum. By contrast, the results predicted by the LSTM model deviate slightly from the extreme points, but the time-series curve predicted by the LSTM has smaller fluctuations, indicating the robustness of the LSTM model. The theoretical foundation of the MLP lies in its ability to approximate any continuous function through global, dense matrix operations. However, because it treats input features as an unordered set, the model lacks inherent awareness of temporal structure. As illustrated in the figure, while the MLP achieves reasonably accurate predictions overall, it struggles to reliably predict Pf at the next time step based on preceding parameter values—particularly at extreme points. In contrast, the CNN captures local features via convolutional filters. Its incorporation of pooling operations enhances robustness to minor local variations in the input, which explains why the CNN exhibits slightly better predictive performance in local regions compared to the MLP. The LSTM network, with its recurrent connections and gating mechanisms, is specifically designed to model temporal dependencies. As a result, it shows high adaptability and delivers precise predictions for tasks estimating failure probability in reservoir slopes with strong temporal correlations, where historical parameter values significantly influence future states.

Table 3 lists the calculated R², MAE, and RMSE of 3 models. The R² values of all 3 models exceed 0.99 on the training set, indicating excellent goodness of fit. However, on the test set, R² of the LSTM model is the highest (i.e., 0.9823), while R² of CNN and MLP is 0.9784 and 0.9741, respectively. MAE represents the average absolute error, and is insensitive to outliers. Therefore, it has a stronger explanatory power for the regularity of time-dependent data. Among the three deep learning models, MAE value of LSTM is the smallest, followed by CNN. This indicates that these two models can overall predict the time-dependent characteristics the failure probability of bank slope. RMSE represents the overall error level, which amplifies larger errors and is highly sensitive to outliers. Among the three deep learning models, RMSE value of LSTM is still the smallest. This indicates that the LSTM model can better predict outliers, such as when the annual rainfall is particularly high and extreme situations may occur. In conclusion, it can be seen that the LSTM model performs well in predicting the long-term failure probability in this study.

5 Discussion

5.1 Total amount of data

This section discusses the performance of three deep learning method when data amount is changed. Firstly, the influence of the total amount of data on the prediction results is discussed. The structures of the three models remain unchanged. Meanwhile, in order to ensure the consistency of the changing trend of the predicted data, the data amount from 2011 to 2020 is reduced by half, that is, the data is from 2016 to 2020. The ratio stays the same. The data involving 2016–2019 and 2019–2020 are regarded as the training set and test set, with a data amount of 1274 and 546, respectively.

Figure 20 shows the prediction results when the data amount is reduced by half. It can be seen that the three models can still capture the changing trend of the time-dependent failure probability. Among them, the LSTM model still shows the best performance in estimating the failure probability value and varying trend. That is to say, the total amount of data has a limited effect on the time-series prediction.

Figure 20

Table 4 shows that reducing the input data amount does not degrade the prediction performance of any model. For example, as the MLP model generalization ability is relatively weak, its prediction may be greatly affected. However, the evaluation indices of the MLP model are not changed that greatly. R² of the MLP model in the test set decreased from 0.9741 to 0.9579. MAE of the MLP model changed from 2.25 to 2.98, and the RMSE increased from 4.48 to 5.64. The error of the LSTM model is the smallest relative to alternative approaches, and it captures the time-dependent characteristic better. Therefore, the LSTM model is still superior to the others after reducing the data amount. It is reasonable to deduce that the 3 models can still predict the future failure probability of the Jiuxianping landslide accurately. This validates that the novel method can effectively predict long-term landslide failure probability even with limited input data.

5.2 Training and test set

The ratio of the training set and the test set (named ratio hereafter for simplicity) is discussed in this section. The ratio is set as 8:2 and 6:4, respectively. Figures 21, 22 show the prediction results of the three models at a ratio of 8:2 and 6:4. It can be seen from the figures that the three models can still capture the changing trend of the time-dependent failure probability, as the prediction results are consistent with the calculated results. Compared with the CNN and MLP models, the failure probability predicted by LSTM rarely has sudden change points. It can be concluded that the prediction results of the LSTM model are relatively stable, showing strong robustness. It is verified that these models can still capture the changing trend of the time-dependent P_f when the ratio varies. However, there are still problems with accurately estimating the extreme points for all three models.

Figure 21

Figure 22

From the perspective of the evaluation indices as shown in the Table 5, the accuracy of the three models in capturing the variation of time-dependent failure probabilities has slightly decreased, but the overall prediction effect remains at a comparable level when the ratio is 7:3. For instance, compared with ratio of 7:3, in the test set, R² of the LSTM model decreases to 0.9811 and 0.9823 respectively, with a reduction of only 0.0012 and 0.0009. The increase in MAE is only 0.04% and 0.08%, and RMSE increased by 0.21% and 0.26%. The fluctuation in the evaluation index is owing to the uncertainty of the training effect of deep learning every time. In short, the ratio changing from 7:3 to 8:2 or 6:4 does not improve the accuracy in capturing the time-dependent trend.

6 Conclusions

This study, aiming at the reservoir bank landslides under the influence of dynamic mechanical environment, takes the Jiuxianping landslide in the TGR as an example and proposes a novel prediction method for the long-term failure probability. This method, based on machine learning algorithms, can significantly reduce the calculation time when considering the spatial variability of rock and soil masses. Also, it can effectively depict the combined effect of reservoir water level and rainfall, and increase prediction accuracy at each time point the failure probability of reservoir bank landslides within ten years. The main conclusions drawn from the full text are as follows:

• (1) Firstly, by analyzing the on-site geological and monitoring data, the failure mechanism of the Jiuxianping landslide and the time-dependent characteristics of displacement are obtained. The overall failure mechanism of the Jiuxianping landslide is sliding-bending mode, and the landslide goes through three stages: the slightly bended stage, the strongly bended and uplifted stage, and the slide surface formed stage Furthermore, significant deformation occurs during May to October each year, and the deformation positions are distributed in the middle and the leading edge, with almost the same displacement amounts, indicating that it is an overall slip. The displacement is not significant in other months, and the displacement develops as a stepped pattern. The deformation of landslides is mainly affected by the reservoir water level and rainfall. Compared with the influence of the water level decline, the influence of rainfall on the deformation of landslides is relatively small.

• (2) The seepage field and safety factor of 2020 year is analyzed through the established landslide numerical model. The consistency of variation trend is found between the safety factor and reservoir water level. As the decline rate of the reservoir water level is greater than the permeability coefficient of the slope, it makes the infiltration line at the leading-edge of the landslide turn to an upward convex or upward concave shape when the water level increases or declines. Water level decline creates an outward hydrodynamic pressure, leading to the decrease in safety factor. That’s the main reason for safety factor decrease with water level of reservoir. On the contrary, water level increase makes the slope more stable. The upward convex or upward concave shape of the infiltration line is more obvious when the variation rate of water level is higher. Also, compared with rainfall, the safety factor is more sensitive to the reservoir water level.

• (3) Reliability analysis is conducted on the Jiuxianping landslide in 2020 based on the random field model, considering spatial variability of cohesion and friction angle of the sliding zone. Results show that the failure probability increases with the decreasing of reservoir water level and this is consistency with the stability evaluation. Parametric study results show that the failure probability increase with any of the parameters such as COV, correlation coefficient of cohesion and friction angle or scale of fluctuation. However, among them, the effect of the correlation coefficient is the smallest. Moreover, the effect of COV of friction angle is bigger than that of cohesion.

• (4) By calculating 10-year P_f using the random field model, this study proposes a novel method for predicting the long-term failure probability based on 3 deep learning models. 3 models can accurately describe the changing trend of the time-dependent failure probability of the Jiuxianping landslide, and can predict the future failure probability of the landslide. The calculation time decreases from 35 minutes of the daily failure probability to only 1 minute for ten years’ failure probability calculation. It has been proved that the novel method is effective. The prediction results of the LSTM model is superior to that of the other models. In the case of reducing the amount of the input data and changing the ratio of the training set to the test set, the three models can still reasonably predict the time-dependent failure probability. However, the prediction ability of the three models has not been improved.

This study also has some limitations. This study only considered the spatial variability of the shear strength of the sliding zone. In fact, some other property such as hydraulic parameters also shows spatially variable characteristic, and they vary with time under the dry-wet cycles. Therefore, the research results are applicable to the situations where the shear properties of the sliding zone are significantly different from sliding body. In future research, it is also possible to consider the spatial variability of multiple parameters throughout the entire landslide and to study the long-term dynamic changes considering properties change with time.

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