## 1 Introduction  
It is crucial to conduct stability assessments of road and railway slopes throughout their entire lifecycle, including during the construction and operational phases. Factors such as human activity, rainfall, and seismic events disturb the stability of the slopes and can result in slope failure and landslides [1, 2, 3, 4, 5, 6]. For example, in February 2022, a landslide occurred on a slope section of the Zhaoping-Pingle Expressway in Guangxi, China. The event disrupted traffic, buried a passenger vehicle, and resulted in one fatality (Figure 1). Separately, on 11 August 2021, a sudden landslide struck a highway in the Kinnaur mountainous area of Himachal Pradesh, Northern India. This incident buried multiple vehicles, causing at least 28 fatalities and 13 injuries (Figure 2). The frequent occurrences of roadside slope failures necessitate heightened demands for routine inspections, targeted monitoring of geological hazard-prone areas along roadways, and dynamic stability analysis of slopes. At present, methods used to quantitatively analyze slope stability can be roughly divided into the limit equilibrium method and the strength reduction method [7,8]. In this respect, the traditional limit equilibrium method is simple with clear mechanical concepts, and it can be used to evaluate stability problems in geotechnical engineering [9, 10, 11]. However, this method requires advanced knowledge of the geometric characteristics of the slope and the properties of the sliding surface [4,12]. In-depth research on the application of the limit equilibrium method has been conducted, and classic methods, such as the Bishop, Janbu, and Spencer methods have been devised [13, 14, 15]. These methods essentially divide a landslide into a finite number of vertical divisions, thereby searching for a landslide safety factor over a large range. However, owing to the discontinuity of the function and the existence of local minima, it is difficult to determine the critical sliding surface that corresponds to the minimum safety factor [16]. Therefore, many studies have been conducted to devise a method that identifies the most dangerous sliding surface [17, 18, 19, 20]. Although significant progress has been made in the application of the limit equilibrium method, its theoretical basis limits its use; for example, it cannot consider the stress and strain of geotechnical materials, the internal force distribution of a landslide must be appropriately simplified to simplify the control equations, and it cannot be used to study the dynamic processes of landslide failure [21, 22, 23]. The strength reduction method was proposed in 1975, and it provided a theoretical foundation for conducting numerical simulations and calculating the slope safety factor [24]. As such, it has been widely applied in slope stability analysis [21, 22, 25, 26]. [PERSON] used the strength reduction method to conduct a detailed analysis of the stability of a two-layer soil slope with a soft lower layer and a hard upper layer [27], [PERSON] used the double strength reduction method to study the safety factor of layered slopes [28], and [PERSON] used two strength reduction methods to calculate the reduction coefficients of cohesion and internal friction angle and optimize the relationship between them [29]. Based on this,  
Figure 1: Landslide along the Zhaoping-Pingle. Figure 2: Kinnaur high-velocity landslide, Himachal Pradesh. the stability of the mountain slopes was analyzed. [PERSON] used the strength reduction method combined with Flac3D to study the stability of three-dimensional single-layer and three-layer reinforced slopes [30], while [PERSON] used a variational method based on strength reduction to define the failure criterion of non-uniform slopes and determine the safety factor of the slope [31]. However, the slope stability often changes dynamically with time, working conditions, and stage.