Symmetry-Aware Physics-Guided Graph Network for Slope Displacement Prediction from GNSS Data

Featured Application

The proposed PG-STGN framework is specifically designed for GNSS-based slope displacement prediction in large-scale geotechnical engineering contexts, including tailings ponds, open-pit mines, reservoir slopes, and highway cut slopes. By embedding geological hierarchical priors into a spatio-temporal graph network, the model enables physically consistent prediction of deformation trends from high-frequency displacement increments. It effectively separates abrupt deformation signals (e.g., those triggered by rainfall or excavation) from long-term creep evolution while preserving spatial coordination among monitoring points within the same geological layer. The framework can be seamlessly integrated into automated slope monitoring systems to enhance early warning reliability, reduce false alarms caused by non-physical oscillations, and provide engineering decision support. Moreover, its modular design allows adaptation to other sensor types (e.g., InSAR, extensometers) and multi-physical field data (e.g., pore pressure, rainfall), making it a versatile tool for real-time geohazard risk management.

Abstract

Accurate prediction of slope displacement from high-frequency GNSS monitoring data is critical for early warning of landslides and tailings dam failures. However, existing deep learning approaches often neglect the spatial coordination imposed by geological structures and fail to decouple abrupt deformation signals from background noise, leading to non-physical oscillations and inconsistent long-term predictions. To address these limitations, this paper proposes a Symmetry-Aware Physics-Guided Spatio-Temporal Graph Network (PG-STGN). First, a geological hierarchy-aware graph is constructed by integrating geometric proximity with prior knowledge of exploration levels, where the resulting adjacency matrix is symmetric by design and reflects the physical symmetry of deformation interactions among monitoring points at the same elevation. A hierarchical masking mechanism restricts feature aggregation to physically connected neighborhoods while preserving this symmetry. Second, an improved dual-path temporal convolutional network (iTCN) decouples high-frequency abrupt variations from low-frequency evolutionary trends, enabling both sensitive detection of sudden deformation and stable tracking of long-term creep. Third, a physics-consistent loss function combining first-order temporal differencing and graph Laplacian regularization enforces kinematic smoothness and spatial coordination; the Laplacian itself is derived from the symmetric adjacency matrix, ensuring symmetric regularization across the monitoring network. Evaluated on a real-world slope GNSS dataset from a large-scale mining project, PG-STGN reduces mean squared error (MSE) by approximately 23.7% and achieves a global R² of 0.924, outperforming state-of-the-art spatio-temporal models. Ablation studies confirm that the symmetric physics-guided graph, dual-path decoupling, and consistency loss are each essential for suppressing spurious correlations and maintaining physically plausible predictions. The proposed framework provides a robust, interpretable, and symmetry-constrained solution for automated slope monitoring under complex geological conditions.

1. Introduction

Tailings ponds are large artificial facilities used to store solid waste and wastewater generated after mineral extraction and processing, typically consisting of a dam body, a storage area, and a drainage system. As a key component of mining activities, its primary function is to safely store solid waste and treat wastewater for reuse. However, tailings are usually transported in slurry form, requiring continuous dam raising and reinforcement, which leads to persistent safety challenges during long-term operation. Tailings ponds commonly face risks such as dam instability, seepage failure, overtopping during floods, and seismic liquefaction. Inadequate design or management may result in dam failure and trigger debris flow hazards. Therefore, scientific safety monitoring of tailings ponds is essential for accident prevention.

Current remote-sensing safety monitoring technologies for tailings ponds can be categorized into three main types: (1) wide-area deformation monitoring based on Interferometric Synthetic Aperture Radar (InSAR), (2) dam anomaly detection based on Unmanned Aerial Vehicles (UAVs), (3) high-precision point-based monitoring based on Global Navigation Satellite Systems (GNSS). InSAR enables large-scale, non-contact deformation monitoring and is not affected by weather or lighting conditions, making it suitable for detecting slow and cumulative settlement and sliding of tailings dams. Du et al. developed a time-series InSAR prediction method to update ground displacement maps in real time, improving monitoring efficiency [1]. Duan et al. applied a GPU-assisted InSAR approach to reveal significant periodic surface deformation before failure, demonstrating the effectiveness of high-frequency, high-resolution monitoring for early warning [2]. Hu et al. used this approach to analyze deformation in the Yanchi tailings pond, and, with an improved phase unwrapping method, revealed settlement patterns and confirmed a gradually decreasing settlement rate over time [3]. Building on this, to improve monitoring reliability and accuracy, Mura et al. integrated SBAS (Small Baseline Subset) and PSI (Persistent Scatterer Interferometry), both belonging to A-DInSAR, and detected millimeter-level displacements caused by settlement and compaction in specific dam areas [4]. Some approaches further couple InSAR observations with physical–mechanical models. Lu et al. integrated SBAS-InSAR data with a fluid–solid coupling model to achieve high-precision monitoring and reliable evaluation of slope stability in tailings ponds [5]. Meanwhile, to overcome the limitations of conventional deformation monitoring, Perry et al. used L-band SAR to penetrate the surface and estimate subsurface soil moisture, enhancing early warning capability by detecting potential internal seepage and piping indicators in tailings ponds [6]. UAV-based methods acquire data on topography, deformation, and water accumulation through aerial surveys along predefined routes, enabling real-time or periodic detection of risks such as seepage and landslides. Rauhala et al. used UAV and Structure-from-Motion (SfM) techniques to generate digital elevation models for continuous monitoring with decimeter-level accuracy [7]. Zhang et al. optimized UAV flight paths using an improved ant colony algorithm and combined it with a YOLOv5 model to achieve intelligent detection of hazards such as slope damage and drainage blockage [8]. Zhang et al. used UAV oblique photogrammetry to build an air–space–ground integrated system, achieving accurate monitoring and fusion analysis of parameters such as phreatic line and surface deformation [9]. With the development of deep learning, Wang et al. combined UAV photogrammetry with convolutional neural networks (CNNs) and used an optimized YOLACT model to accurately identify the dam crest line, enhancing seepage prevention [10]. Lu et al. used high-resolution UAV-derived digital elevation models (DEMs) with a ResNet model and SHAP analysis to achieve precise elevation monitoring [11]. To overcome the limitations of 2D methods, Gomez et al. reconstructed high-resolution 3D point clouds from UAV imagery and linked 2D pixels with 3D points, enabling accurate monitoring and morphological analysis of dry beach boundaries [12].

By deploying fixed receivers at key locations on the dam surface, GNSS provides continuous three-dimensional displacement data. Atif et al. summarized methods for monitoring vertical deformation using GNSS, supporting dam failure analysis [13]. Dai et al. proposed a colored-noise generalized segmented extended Kalman filter (colored-GSEKF) combined with a GNSS/SINS tightly coupled system, improving filtering accuracy and stability [14]. Lumbroso et al. developed the DAMSAT system based on GNSS data, improving monitoring efficiency while reducing deployment cost [15]. Jing et al. combined multi-section GNSS monitoring with a Bi-LSTM model to achieve high-precision and automated deformation monitoring [16]. GNSS is also often integrated with other sensing techniques. Guireli et al. combined UAV-based GNSS, ground GNSS control points, and seismic refraction to enhance seepage monitoring [17]. Santos et al. mounted dual-frequency GNSS receivers on terrestrial laser scanning (TLS) systems for point cloud registration and accuracy assessment, enabling precise detection of sediment accumulation and erosion [18]. At the optimization level, Kim et al. compared standalone GNSS, RTK, and PPP-assisted GNSS, showing that RTK significantly improves positioning accuracy and monitoring performance [19].

In addition to these remote-sensing techniques, a broad spectrum of ground-based and in situ methods has long been employed to assess the physical condition of slope rock masses. These methods can be broadly categorized into three types: (1) surface and deep deformation monitoring, (2) seepage pressure and phreatic line monitoring, and (3) in situ mechanical monitoring. Surface and deep deformation monitoring primarily involves the use of robotic total stations for collimation line measurements to obtain surface displacement, combined with inclinometers embedded within the dam body to capture the horizontal displacement of deep soil layers. To improve observation precision and automation levels, Allil et al. developed an inclinometer system based on fiber Bragg grating (FBG) arrays, which enables continuous monitoring of deep underground displacement through borehole installation [20]. Meanwhile, Zhou et al. introduced the nearest reference point for atmospheric refraction correction in robotic total stations, further enhancing the observation accuracy of surface deformation in tailings ponds [21]. Since water level fluctuations can interfere with displacement determination, Zhou et al. proposed using the built-in camera of robotic total stations for image processing, achieving effective identification and suppression of phreatic level interference, thereby reducing deformation monitoring errors induced by water level variations [22]. Seepage instability is a primary driver of tailings dam failure. By embedding piezometers in key sections of the dam, the internal pore water pressure and the height of the phreatic line can be acquired in real time. Le Roux et al. proposed a multi-piezometer deployment strategy at critical sections, utilizing a multi-point redundancy mechanism to effectively compensate for the limitations of single sensors, which are susceptible to local geological fluctuations [23]. Building on this, Morton et al. focused on high-dimensional data analysis by integrating data from high-density piezometer networks to map 3D equipotential lines and streamlines; this enables comprehensive monitoring of groundwater conditions and reduces the risk of slope instability [24]. In situ mechanical monitoring, through periodic tests such as the cone penetration test (CPT/CPTu) and the standard penetration test (SPT), allows for the direct assessment of internal stability and liquefaction resistance of the slope. Shirzoi et al. analyzed the sources of deformation risk from within the dam body by conducting SPT tests in conjunction with regional seismicity and geological backgrounds, thereby enhancing deformation monitoring capabilities [25]. Sousa et al. strengthened the perception of dam deformation by analyzing the correlation between geophysical testing and in situ CPT tests to derive geotechnical parameters [26]. Furthermore, Costa et al. combined CPTu and SPT methods to obtain shear strength parameters of tailings, providing an effective means of evaluating the degree of deformation in tailings dams [27].

However, these traditional techniques often lack the temporal resolution, spatial coverage, or real-time automation required for continuous early warning. GNSS, by contrast, offers all-weather, high-frequency, three-dimensional displacement data at the dam surface and has become the backbone of automated real-time monitoring. An ideal monitoring framework should therefore fuse GNSS-based surface deformation with periodically updated geotechnical and geophysical data, and the deep learning model proposed in this study is designed to utilize such multi-source priors while maintaining physical consistency. Furthermore, time-series processing techniques have enabled more in-depth analysis of GNSS data. Khalil et al. combined an improved cloud model with a radial basis function network to enhance fitting accuracy and enable real-time seepage monitoring [28]. In addition, Yang et al. developed a CNN–LSTM model with numerical inversion for accurate prediction of the saturation line and dam failure risk [29]. Similarly, Zhu et al. proposed an improved LSTM integrating EMD, attention mechanisms, and lagged autocorrelation to address delayed effects of external factors [30]. Meanwhile, some studies adopt hybrid approaches. For example, Hao et al. combined deep autoencoders, whale optimization, LSTM, and kernel density estimation to improve predictions of water level and displacement [31]. Zhang et al. proposed a grouped time-series network (GTSN) using feature clustering and multi-task learning to enhance monitoring accuracy [32]. In recent years, research has trended toward integrated frameworks. For instance, Mwanza et al. introduced a digital twin-driven machine learning framework with improved reliability and predictive performance [33]. Additionally, Hao et al. proposed a hybrid interval prediction model combining EMD, optimization algorithms, and Elman networks to better capture nonlinearity and uncertainty in displacement prediction [34].

Despite the widespread application of GNSS in tailings pond monitoring, several limitations remain. Existing approaches are often combined with simple models and lack advanced data mining techniques and attention-based deep learning methods, resulting in insufficient extraction of underlying features and evolutionary patterns. Some studies attempt deformation prediction using GNSS time series, but typically model individual monitoring points independently, without considering spatial correlations among different locations on the dam, leading to suboptimal performance. In addition, GNSS data are susceptible to interference from complex terrain and extreme weather, which introduces noise during transmission. Current methods rarely address how to reliably extract deformation trends from noisy data. Existing spatio-temporal graph networks (e.g., GAT-LSTM, STGCN, Graph WaveNet) typically encode physical constraints only through geometric proximity (e.g., distance-based adjacency) or learn adaptive graphs purely from data. While they capture some spatial dependencies, they lack explicit geological priors and rarely enforce the symmetry of deformation interactions: a point’s influence on its neighbor is not required to be reciprocal, and nodes from different stratigraphic levels may be spuriously connected if they are geographically close.

To address these issues, we propose a physics-guided spatio-temporal graph network (PG-STGN) that integrates geological hierarchical priors, dual-path temporal decoupling, and a physics-consistent loss function. In contrast, our symmetry-aware, geology-informed graph explicitly constructs a symmetric adjacency matrix by combining distance decay with a hierarchical indicator that reinforces connections only among monitoring points on the same elevation platform. This design respects the isotropic stress transfer within a homogeneous slip surface, eliminates cross-level spurious correlations, and ensures that the graph Laplacian regularization (and thus the spatial loss) is physically consistent. This advance transforms purely data-driven ST-GNNs into interpretable, mechanics-guided models that produce spatially coherent and reciprocally constrained displacement predictions. The model explicitly encodes the symmetry of deformation interactions among monitoring points on the same elevation and enforces spatial and temporal smoothness via Laplacian regularization and first-order differencing. Evaluations on real-world GNSS slope data demonstrate its superior accuracy and physical consistency compared to existing methods, offering a robust, interpretable solution for automated early warning systems.

2. Engineering-Geological Basis for Slope Displacement Prediction

Tailings dam slopes are engineered structures that undergo continuous, multi-mode deformation under the combined influence of gravity, pore-pressure fluctuations, material consolidation, and external disturbances. Reliable displacement prediction requires incorporating the following established engineering–geological principles:

• Layered coordination. Monitoring points installed at the same elevation platform share a common potential slip surface or bearing stratum. Because effective stress transfer within a quasi-homogeneous layer is approximately isotropic, the deformation increments of such points tend to be coherent and mutually influential. In graph-theoretic terms, this motivates a symmetric adjacency matrix with reinforced intra-level weights and a hierarchical masking mechanism that restricts information exchange to nodes on the same geological level.
• Dual timescale dynamics. Slope deformation exhibits two characteristic timescales: a slow, secular creep driven by material aging and consolidation, and abrupt, step-like accelerations triggered by rainfall events, blasting, or excavation. These two regimes correspond to low-frequency and high-frequency components in a displacement time series. Separating them is essential because the early warning value of abrupt events should not be diluted by smoothing, while the background creep trend must be tracked stably without noise amplification.
• Kinematic and spatial continuity. As a deformable solid continuum, the slope must satisfy physical consistency: displacement increments should not oscillate arbitrarily from step to step (temporal smoothness), and the velocity field across the slope should be sufficiently smooth with adjacent points on the same slip surface showing coordinated movement (spatial smoothness). These constraints can be encoded in a loss function using first-order temporal differencing and graph Laplacian regularization.

3. Our Method

This study proposes a physics-guided spatio-temporal graph convolutional framework (PG-STGN) for predicting slope displacement. Targeting the high-frequency fluctuations in GNSS displacement increments, the framework incorporates geological hierarchical information to achieve accurate prediction of future deformation. The framework consists of four core modules: geology-constrained graph construction, a dual-path temporal convolutional network (Dual-Path TCN), a physics-guided graph attention mechanism (PG-GAT), and gated fusion with multi-task output, as detailed in Section 3.1, Section 3.2, Section 3.3 and Section 3.4.

3.1. Problem Definition and Spatio-Temporal Graph Construction

3.1.1. GNSS Displacement Prediction Problem

GNSS monitoring sequence prediction infers future trends from historical displacement states. The monitoring area contains $N$ GNSS sensor nodes (e.g., BW0-2, BW1-1), denoted as $V=\{v_1,v_2,\dots ,v_N\}$. At time $t$, observations are represented as a feature matrix $X_t\in {\mathbb{R}}^{N\times C}$, where $C$ includes three-dimensional displacement increments $\left(\mathsf{\Delta }X,\mathsf{\Delta }Y,\mathsf{\Delta }H\right)$. Given a historical window of length $T$, the input tensor is $X_{in}=\{X_{t-T+1},\dots ,X_t\}\in {\mathbb{R}}^{T\times N\times C}$. The model learns a mapping $f\left(·\right) \mathrm{t}\mathrm{o} \mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{f}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e} T_{pl} \mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}\mathrm{s} \widehat{Y}\in {\mathbb{R}}^{T_{pl}\times N\times C}$ by

$$\widehat{Y}=f\left(X_{in};G,W\right)$$

(1)

where $G$ denotes the graph capturing spatial topology and $W$ denotes learnable parameters. The framework is topology-adaptive: when $N>1$, it captures spatial interactions among sensors; when $N=1$, $G$ reduces to a self-loop node, and the model focuses on temporal dynamics at a single point.

3.1.2. Geological Hierarchy Graph $G$ Construction

Unlike traditional single-point models, this study incorporates physical constraints into a data-driven framework by constructing a weighted graph $G=\left(V,E,A\right)$ based on spatial coordinates and geological hierarchy, enabling explicit modeling of spatial interactions among monitoring points.

(1) Node representation: nodes in $V$ correspond to monitoring devices (e.g., BW0-1 to BW7-1). For the geological level assignment, each monitoring node $v_{\mathrm{i}}$ is assigned a geological level $L_i\in \mathcal{L}$, where $\mathcal{L}$ is the set of all exploration levels (e.g., $\mathcal{L}$ = {630, 660, 680, 700, 712} in our study). The assignment is based on the elevation platform of the sensor as documented in the geotechnical survey. For the hierarchical indicator function, the function $\delta \left(L_i,L_j\right)$ is defined as

$$\left(L_i,L_j\right)=\left\{\begin{array}{c}1, if L_i=L_j \\ 0, \mathrm{otherwise}\end{array}\right.$$

This indicator explicitly injects the prior knowledge that monitoring points on the same horizontal platform share a common slip surface and thus exhibit higher instantaneous deformation correlation.

(2) Adjacency matrix construction: to account for both geographic proximity and geological hierarchy, a hybrid weighting scheme is defined, where each element $A_{ij}$ of the adjacency matrix $A\in {\mathbb{R}}^{N\times N}$ is computed as follows:

$$A_{ij}=\left\{\begin{array}{cc}\hfill \exp \left(-{\displaystyle \frac{dist(v_i,v_j{)}^2}{{\sigma }^2}}\right)+\gamma ·\delta \left(L_i,L_j\right),& \mathrm{if} dist(v_i,v_j)\le d_{max}\hfill \\ \hfill 0,& \mathrm{otherwise}\hfill \end{array}\right.,$$

(2)

where $\sigma$ is the standard deviation controlling the Gaussian kernel width; $dist(v_i,v_j)$ denotes the Euclidean distance between monitoring points; and $d_{max}$ is a distance threshold used to remove weak connections and reduce computational complexity in densely distributed networks. $\delta \left(L_i,L_j\right)$ is a geological hierarchy indicator function: if nodes $v_i$ and $v_j$ belong to the same exploration level (e.g., level 712), an additional weight $\gamma$ is assigned. This design enables the model to capture spatially coordinated deformation under overall slope movement. To illustrate its effectiveness, Figure 1 compares adjacency matrices under different mechanisms. The distance-based matrix (Figure 1a) is relatively sparse, whereas the proposed method (Figure 1b) forms clear high-weight clusters (block-diagonal patterns) among nodes at the same level. This topology filters cross-level spurious correlations and guides feature aggregation toward coordinated deformation within the same sliding layer.

From the symmetric adjacency matrix $A$, we construct the graph Laplacian $L\in {\mathbb{R}}^{N\times N}$ as $L=D-A$, where $D$ is the degree matrix with diagonal elements $D_{ii}={\sum }_j{A}_{ij}$ and off-diagonals zero. Because $A$ is symmetric, $L$ is symmetric and positive semi-definite. This Laplacian is used in the physics-consistent loss (Section 3.4.2) to enforce spatial smoothness of predicted displacements across the geological graph.

(3) Symmetry property of the adjacency matrix: By construction, the adjacency matrix $\mathrm{A}$ is symmetric, i.e., $A_{ij}=A_{ji}$ for any pair of nodes $(v_i,v_j)$. This follows from the defining Equation (2):

• The Euclidean distance $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(v_i,v_j)$ is symmetric.
• The geological indicator $\delta \left(L_i,L_j\right)$ is symmetric because it depends only on the equality of $L_i$ and $L_j$.
• The Gaussian kernel $\mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{{dist}^2}{{\sigma }^2}})$.

Consequently, the resulting graph is undirected, and the Laplacian $L=D-A$ derived from it (Section 3.4.2) remains symmetric positive semi-definite. This symmetry is physically meaningful: under the assumption of homogeneous material properties and isotropic stress transfer within a geological layer, the mutual deformation influence between point ii and point jj must be reciprocal. The graph therefore respects the fundamental symmetry of spatial interactions in a deformable continuum.

3.2. Improved Dual-Path Temporal Convolution Network (iTCN)

GNSS displacement sequences exhibit strong temporal dependence; therefore, iTCN is adopted for feature extraction. By combining causal convolution, dilated convolution, and residual learning, the model effectively overcomes gradient vanishing and low computational efficiency in traditional RNN/LSTM for long sequences. Symmetry property of the adjacency matrix: By construction, $A_{i,j}=A_{j,i}$ because both the Euclidean distance $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(v_i,v_j)$ and the hierarchical indicator $\delta (L_i,L_j)$ are symmetric functions. This symmetry is physically meaningful: if monitoring point $i$ influences point $j$ through a shared sliding surface, the reciprocal influence from $j$ to $i$ must be equal under the assumption of homogeneous material properties and isotropic stress transfer. The proposed graph therefore respects the fundamental symmetry of spatial interactions in a deformable continuum.

3.2.1. Dilated Causal Convolution

To ensure that predictions at time t depend only on historical information, causal convolution is introduced, where $y_t$ is computed solely from $\left\{x_0,x_1,\dots ,x_t\right\}$, enforced by left-side zero padding. To exponentially expand the receptive field and avoid network degradation, dilated convolution is adopted, introducing a dilation factor (d) by inserting gaps between kernel elements, with the formulation given as follows:

$$y_t={\displaystyle \sum _{i=0}^{k-1}} w_i·x_{t-d·i},$$

(3)

where k is the kernel size and $d=2^i$ is the dilation factor at the i-th layer. This design preserves the parameter scale while enabling a receptive field of $2^{L+1}-1$ over L layers, enhancing modeling of long-range temporal dependencies.

3.2.2. Dual-Path Feature Extraction

To achieve multi-scale representation of high-frequency variations and low-frequency trends, we design a dual-path feature extraction architecture (Figure 2).

(1) Max-pooling path: designed to capture abrupt displacement changes. Dilated convolution and max-pooling extract prominent features within local windows, preserving step-like peaks without smoothing.

(2) Average-pooling path: focuses on stable trends. By averaging local temporal features, it suppresses noise and highlights underlying geological evolution patterns.

3.2.3. Feature Coupling and Residual Learning

To fuse high-frequency variations and low-frequency trends, iTCN introduces a feature coupling operator. Given the input $X^{\left(l\right)}$ at layer $l$, the two paths are defined as

$$X_{max}^{\left(l\right)}=\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{P}\mathrm{o}\mathrm{o}\mathrm{l}1\mathrm{d}\left(\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}1\mathrm{d}\left(\sigma \left(\mathrm{DilatedConv}\left(X^{\left(l\right)};W_{max}^{\left(l\right)}\right)\right)\right)\right)$$

(4)

$$X_{avg}^{\left(l\right)}=\mathrm{A}\mathrm{v}\mathrm{g}\mathrm{P}\mathrm{o}\mathrm{o}\mathrm{l}1\mathrm{d}\left(\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}1\mathrm{d}\left(\sigma \left(\mathrm{DilatedConv}\left(X^{\left(l\right)};W_{avg}^{\left(l\right)}\right)\right)\right)\right),$$

(5)

where MaxPool1d uses kernel size 2 and stride 1 (with padding to preserve sequence length), capturing sharp, local deformation peaks, AvgPool1d uses the same kernel size and stride, smoothing high-frequency noise and retaining the underlying creep trend. $W_{max}^{\left(l\right)}$and $W_{avg}^{\left(l\right)}$ are learnable kernels for the max-pooling and average-pooling paths, $\mathrm{D}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}(·)$ denotes dilated causal convolution, and $\sigma$ is the GELU activation. The max-pooling path enhances step-like peaks via local extrema, while the average-pooling path suppresses high-frequency noise through smoothing. In the coupling stage, outputs are combined along channels:

$$Y^{\left(l\right)}=X_{max}^{\left(l\right)}+X_{avg}^{\left(l\right)},$$

(6)

and residual connections are then applied, with adaptive $1\times 1$ convolution for dimension alignment:

$$X^{\left(l+1\right)}=GELU\left(Y^{\left(l\right)}+W_{downsample}{X}^{\left(l\right)}\right),$$

(7)

Compared with ReLU, GELU provides smoother gradients and more stable training. Through dual-path decoupling, iTCN captures both abrupt deformation and stable trends, providing robust temporal features for PG-STGN.

3.3. Multi-Head Physics-Guided Graph Attention (PG-GAT)

Although the TCN module captures temporal dynamics at each sensor, slope deformation is inherently a spatio-temporal process governed by geological structures. The variation at a single sensor is often influenced by neighboring nodes, especially those within the same geological layer. To incorporate geological priors into data-driven feature aggregation, a physics-guided graph attention mechanism (PG-GAT) is proposed to model such nonlinear spatial dependencies dynamically.

3.3.1. Spatial Coordination via Dynamic Attention

For the displacement increment data in this study, coordinated deformation between two points often indicates they share the same stress transfer path. High-frequency fluctuations in the increments, however, make stable spatial dependencies difficult to detect from the data alone. Therefore, the hierarchical spatial correlation matrix $A_{ij}$ from Section 3.1.2 is used as physical prior knowledge. It accounts for geometric distance decay and reinforces connections within the same horizontal plane via a geological hierarchical indicator, reflecting the physical hypothesis that points within the same geological level (e.g., 700-level) exhibit higher instantaneous deformation correlations.

3.3.2. Physics-Masked Attention

To prevent the model from learning spurious spatial correlations (e.g., distant nodes exhibiting similar noise fluctuations at certain times), the adjacency matrix $A_{ij}$ is converted into a physics mask (Physics Mask) and integrated into the attention mechanism. For the l-th layer node feature $h_i^{\left(l\right)}$, the attention coefficient $e_{ij}$ is computed with a structural bias:

$$e_{ij}=LeakyReLU\left(a^T\left[W{\mathrm{h}}_j^{\left(l\right)}\parallel W{\mathrm{h}}_j^{\left(l\right)}\right]\right)+M_{ij}$$

(8)

$$M_{ij}=\left\{\begin{array}{c}0, if A_{ij}>ϵ (\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g} \mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l} \mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n})\\ -\infty , \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e} (\mathrm{w}\mathrm{e}\mathrm{a}\mathrm{k} \mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l} \mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n})\end{array}\right.,$$

(9)

where $ϵ$ is a small threshold (set to 0.01 in our experiments) that separates physically meaningful connections from negligible ones. The conversion from $A_{ij}$ to $M_{ij}$ ensures that after softmax, attention weights ${\alpha }_{ij}$ are strictly zero for weakly connected or disconnected node pairs. W is a learnable linear transformation, $a$ is the attention weight vector, and $\parallel$ denotes feature concatenation. When $A_{ij}\le 0$, $M_{ij}\to -\mathrm{\infty }$, forcing the corresponding attention weight ${\alpha }_{ij}$ to zero after the softmax operation. This ensures that the GAT layer aggregates increment features only within neighborhoods with actual geological or geometric links, providing physically interpretable spatial information. For instance, when a significant displacement increment occurs at a node, the model automatically focuses on other nodes on the same horizontal plane according to geological hierarchy, rather than indiscriminately attending to all nodes.

To clarify the computation of the physics mask attention, a flowchart is presented in Figure 3, illustrating how hierarchical geological priors are transformed into the mask M and dynamically applied to attention, enforcing physical constraints on non-stationary spatio-temporal correlations.

To enhance the ability of the model to represent complex spatial features, a multi-head attention mechanism is introduced. As shown in Figure 4, K independent GAT units perform spatial aggregation in parallel, enabling the model to capture coordinated patterns of displacement increments across different subspaces. The outputs from all heads are concatenated along the channel dimension through the Synthesis module, forming a multi-scale spatial feature representation $h_i^{\left(l+1\right)}$, which improves robustness and stability in prediction:

$$h_i^{\left(l+1\right)}={\left|\right|}_{k=1}^K\sigma \left(\sum _{j\in {\mathcal{N}}_i} {\alpha }_{ij}^k{W}^k{h}_j^{\left(l\right)}\right),$$

(10)

where $\parallel$ denotes feature concatenation, ${\alpha }_{ij}^k$ is the normalized attention weight from the k-th head, and $W^k$ is the linear transformation for the k-th head. This multi-head mechanism allows the model to jointly capture coordinated deformation patterns of the slope across different subspaces, enhancing prediction robustness.

Because $\mathrm{A}$ is symmetric ($A_{ij}=A_{ji}$), the resulting mask $M$ is also symmetric: $M_{ij}=M_{ji}$. This guarantees that the attention pattern is undirected: if node $i$ can attend to node $j$, the reverse is equally allowed, consistent with the reciprocity of physical signal propagation along the slip surface. In the multi-head setting (Equation (10)), each head independently applies this symmetric mask, ensuring that spatial aggregation remains invariant under permutation of neighboring nodes.

3.4. Gated Fusion and Multi-Task Output

After the aforementioned processing, the model obtains temporal evolution features $H_{\mathrm{temp}}$ and spatial coordination features $H_{\mathrm{spat}}$. Accurate slope displacement prediction requires dynamically balancing these two information sources while enforcing appropriate constraints.

3.4.1. Adaptive Gated Fusion

Slope deformation dynamics are time-varying. During uniform deformation, historical increments (temporal features) exhibit strong autocorrelation; during accelerated phases triggered by external disturbances (e.g., rainfall or excavation), coordinated responses of surrounding nodes (spatial features) provide key precursors. To address this, a learnable gating parameter $z\in [\mathrm{0,1}{]}^N$ is introduced

$$z=\sigma \left(W_z\left[H_{temp}\parallel H_{spat}\right]+b_z\right),$$

(11)

where $\left[·\parallel ·\right]$ denotes concatenation along the feature dimension; $W_z$ and $b_z$ are learnable weights and bias; $\sigma$ is the sigmoid function mapping outputs to (0, 1). The fused latent representation is then obtained via gated weighting:

$$H_{fused}=z\odot H_{temp}+\left(1-z\right)\odot \left(W_h{H}_{spat}\right),$$

(12)

where $\odot$ denotes element-wise (Hadamard) product and $W_h$ is a linear transformation for spatial features. This mechanism allows the model to adaptively prioritize historical or neighborhood features for each node and time step.

3.4.2. Physics-Consistent Loss

Traditional deep learning models often use mean squared error (MSE) as the sole optimization objective, which reduces pointwise numerical errors but has limitations for complex slope deformation tasks: it ignores the physical inertia of geological bodies and neglects spatial coordination among multiple monitoring points.

To overcome these limitations, a hybrid loss function $L_{\mathrm{total}}$ is constructed, combining prediction accuracy, temporal inertia, and spatial coordination:

$${\mathcal{L}}_{total}={\mathcal{L}}_{MSE}+{\lambda }_1{\mathcal{L}}_{Temp}+{\lambda }_2{\mathcal{L}}_{Spat},$$

(13)

where ${\mathcal{L}}_{MSE}$, ${\mathcal{L}}_{Temp}$, and ${\mathcal{L}}_{Spat}$ denote base regression loss, temporal smoothing loss, and spatial coordination loss, respectively; ${\lambda }_1, {\lambda }_2\ge 0$ are hyperparameters balancing temporal and spatial constraints. The MSE term is defined as

$${\mathcal{L}}_{MSE}={\displaystyle \frac{1}{T_{pl}N}}{\displaystyle \sum _{t=1}^{T_{pl}}} {\displaystyle \sum _{i=1}^N}\parallel {\widehat{y}}_{t,i}-y_{t,i}{\parallel }_2^2,$$

(14)

where $T_{\mathrm{pl}}$ is the prediction horizon, $N$ is the number of GNSS monitoring nodes, ${\widehat{y}}_{t,i},y_{t,i}\in {\mathbb{R}}^3$ are predicted and true displacement increment vectors, and $\parallel ·{\parallel }_2$ is the L2 norm. To enforce temporal smoothness consistent with Newtonian mechanics, a first-order difference constraint penalizes high-frequency oscillations:

$${\mathcal{L}}_{Temp}={\displaystyle \frac{1}{\left(T_{pl}-1\right)N}}{\displaystyle \sum _{t=2}^{T_{pl}}} \parallel {\widehat{y}}_t-{\widehat{y}}_{t-1}{\parallel }_F^2,$$

(15)

where ${\widehat{\mathrm{y}}}_{\mathrm{t}}\in {\mathbb{R}}^{\mathrm{N}\times 3}$ contains predicted increments for all nodes at time $\mathrm{t}$, and $\parallel ·{\parallel }_{\mathrm{F}}$ is the Frobenius norm. Minimizing this term encourages smooth, physically consistent deformation rates. To capture global slope movement, sensors within the same geological level or physically linked regions should exhibit consistent trends. A graph Laplacian regularization term is applied:

$${\mathcal{L}}_{Spat}={\displaystyle \frac{1}{T_{pl}}}{\displaystyle \sum _{t=1}^{T_{pl}}} Tr\left({\widehat{y}}_t^{T}L{\widehat{y}}_t\right),$$

(16)

where $\mathrm{Tr}\left(·\right)$is the matrix trace, ${\widehat{y}}_t\in {\mathbb{R}}^{N\times 3}$, and $L=D-A_{ij}$ is the Laplacian matrix constructed from the physical adjacency matrix $A_{ij}$ (Section 3.3.1), with $D_{ii}={\sum }_j{A}_{ij}$. Minimizing this term encourages the model to account for coordinated deformation among neighboring nodes. The graph Laplacian $L=D-A_{ij}$ is symmetric positive semi-definite, as both $D$ and $A$ are symmetric. This symmetry of the Laplacian guarantees that ${\mathcal{L}}_{Spat}$ penalizes deviations from a smooth deformation field in a way that is independent of node ordering, making the regularization physically consistent with the isotropic nature of slope motion.

3.5. Evaluation Metrics

To assess prediction performance at both local node accuracy and global slope trend levels, a hierarchical evaluation framework is designed. Single-point accuracy is first evaluated using error statistics, followed by spatial coordination metrics for global deformation trends.

3.5.1. Local Accuracy Metrics

Mean absolute error (MAE) and mean squared error (MSE) quantify numerical fitting of the model at each sensor.

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{t=1}^n}\mid y_t-\stackrel{\wedge }{y_t}\mid$$

(17)

$$MSE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{t=1}^n}(y_t-\stackrel{\wedge }{y_t}{)}^2,$$

(18)

where ${\widehat{y}}_t$ and $y_t$ are predicted and true increments for a specific node, and $n$ is the number of test time steps. MAE reflects average absolute deviation, while MSE emphasizes larger errors, capturing outliers and ensuring precise displacement tracking.

3.5.2. Global Trend Metrics

The ultimate goal of slope monitoring is to assess macroscopic stability and detect transitions from steady creep to accelerated deformation. To evaluate whether the model accurately reproduces overall slope evolution from multi-sensor data, the global R² score is employed:

$$R^2=1-{\displaystyle \frac{\sum _j (y_j^{flat}-{\widehat{y}}_j^{flat}{)}^2}{\sum _j (y_j^{flat}-{\overline{y}}^{flat}{)}^2}},$$

(19)

where $T_{\mathrm{pl}}$ is the prediction horizon, $N$is the number of nodes, $C$ is the feature dimension (3D increments), and ${\widehat{y}}^{\mathrm{flat}}, y^{\mathrm{flat}}$ are the flattened predicted and true tensors over all nodes and times. ${\overline{y}}^{\mathrm{flat}}$ is the global mean of the true increments.

The metric typically ranges from $(-\mathrm{\infty },1]$, values closer to 1 indicate that the model explains a larger proportion of the variance in the observed data, meaning the predicted global displacement trend closely matches the actual slope movement, successfully capturing macroscopic patterns and temporal transition points (e.g., the onset of accelerated deformation). Lower R² values indicate significant phase lag or failure to capture dynamic behavior of the slope.

4. Experiments

4.1. Dataset Description and Preprocessing

4.1.1. Dataset and Features

The experimental data come from the GNSS automated monitoring system of a large-scale slope project, recording real-time deformation with timestamps and sensor measurements. The network comprises $N$ GNSS nodes (device IDs include BW0-x, BW1-x, BW2-x, etc.).

According to geological surveys and deployment information (Figure 5), sensors are installed on different elevation platforms, showing clear stratification across key levels: 630, 660, 680, 700, and 712. This vertical distribution provides a solid basis for constructing the physics-guided hierarchical correlation graph.

At each time $t$, a sensor observation is a 3D feature vector $x_t=[\mathsf{\Delta }X,\mathsf{\Delta }Y,\mathsf{\Delta }H{]}^T$, where $\mathsf{\Delta }X$and $\mathsf{\Delta }Y$ are displacement increments along the X and Y axes (mm), and $\mathsf{\Delta }H$is the vertical increment (mm). The dataset includes both small deformations during steady periods and local abrupt or accelerated movements, enabling evaluation of the ability of the model to capture diverse deformation patterns.

4.1.2. Data Preprocessing

To reduce noise and meet the input requirements of deep learning models, the raw data underwent a strict standardization process. First, missing or outlier points caused by adverse weather or unstable signal transmission were treated as missing values; for sporadic data losses (typically fewer than three consecutive time steps), linear interpolation was employed to recover the time series, as the high sampling rate ensures the continuity of slope deformation. Data segments with prolonged interruptions were excluded to avoid introducing spurious trends. Second, since displacement magnitudes vary across sensors, Z-score normalization was applied to accelerate model convergence and eliminate dimensional differences by

$$x^{\prime }={\displaystyle \frac{x-\mu }{\sigma }},$$

(20)

where $\mu$and $\sigma$ are the mean and standard deviation of the training set. Based on Section 3.1, spatio-temporal graph data were constructed using a sliding-window approach, with historical observation length $T$and prediction horizon $T_{\mathrm{pl}}$, yielding input tensors of shape $(T,N,3)$ for $N$nodes. Finally, to evaluate the generalization of the model to future trends and prevent data leakage, the sequence was split chronologically into training, validation, and test sets at a 7:1:2 ratio: the first 70% for parameter optimization, the next 10% for early stopping and hyperparameter selection, and the final 20% reserved for performance evaluation. Specifically, the training set statistics are computed and applied to the validation and test sets to prevent data leakage. Missing values were identified using a sliding-window approach with a threshold of three consecutive missing points, followed by linear interpolation to ensure temporal continuity.

4.2. Experimental Settings

To comprehensively evaluate the effectiveness of the proposed PG-STGN framework for slope displacement increment prediction, this section details the benchmark models, parameter settings, and hierarchical validation strategy.

4.2.1. Dataset Source and Features

To assess the advantages of the model in temporal feature extraction and spatial coordination, two categories of mainstream time series prediction models were selected as benchmarks. All models were trained under the same dataset splits and preprocessing conditions to ensure fairness.

• Temporal-Only Models: These models predict using historical data from a single sensor, ignoring spatial dependencies. GRU (Gated Recurrent Unit) [35]: Handles long-sequence dependencies via gated recurrent units. LSTM (Long Short-Term Memory) [36]: Standard baseline widely used for time series prediction. TCN (Temporal Convolutional Network) [37]: Single-stream temporal convolution network, used to verify improvements from the dual-stream feature extraction module.
• Spatio-Temporal Models: Explicitly model spatial dependencies via graph convolution or attention mechanisms. GAT-LSTM [38]: Combines graph attention networks with LSTM, aggregating neighborhood features via attention. STGCN (Spatio-Temporal Graph Convolutional Network) [39]: Classic spatio-temporal model combining Chebyshev graph convolutions and gated temporal convolutions. Graph WaveNet [40]: Advanced model integrating diffusion convolution and dilated convolution, with an adaptive adjacency matrix.

4.2.2. Implementation Details and Parameter Settings

All experiments were performed on a server with an NVIDIA GeForce RTX 3090 GPU (24 GB) and an Intel Core i9 CPU. The framework was implemented using PyTorch 1.10. To ensure reproducibility and stable convergence, the model training followed a standardized protocol: (1) Model parameters were initialized using the Kaiming uniform method, which is specifically designed to maintain variance in deep residual networks, preventing gradient saturation or explosion in the early stages of training. (2) The Adam optimizer was employed with an initial learning rate of $1\times {10}^{-3}$, dynamically adjusted by a Cosine Annealing scheduler to allow for smoother convergence in the late training stages. We used a batch size of 32. (3) To avoid overfitting and ensure generalization, we implemented an early stopping mechanism. During the training process, the model was evaluated on a dedicated validation set (10% of the dataset) after each epoch. If the validation loss did not show improvement for 20 consecutive epochs, training was terminated, and the model weights corresponding to the lowest validation loss were saved as the final configuration. (4) The key hyperparameters for PG-STGN were finalized through grid search and are summarized in

Table 1. Hyperparameter settings.

Hyperparameter	Value/Setting
Learning rate	$1\times {10}^{-3}$
Optimizer	Adam
Batch size	32
Max epochs	200
Early stopping	Patience = 20 epochs
Dilation factors ($d$)	[1, 2, 4, 8]
Kernel size ($k$)	3
Residual channels	64
Loss weights (${\lambda }_1,{\lambda }_2$)	0.1, 0.5

. In the dual-stream TCN module, dilation factors $d=[\mathrm{1,2},\mathrm{4,8}]$, kernel size $k=3$, and residual channels = 64; spatial gain weight $\gamma =0.5$; distance cutoff $d_{\max }$ based on average sensor spacing; and temporal smoothing loss weight ${\lambda }_1=0.1$ and spatial coordination loss weight ${\lambda }_2=0.5$ via grid search.

4.2.3. Hierarchical Validation Strategy

To thoroughly analyze model performance, a two-stage evaluation strategy was designed, ranging from local accuracy to global trend coordination. The first stage assesses local node validity, evaluating the ability of the model to fit numerical changes at individual sensors. Key representative sensors (e.g., BW5 and BW6 on the main sliding section) were analyzed using MAE and MSE to determine whether PG-STGN, supported by the dual-stream TCN, outperforms traditional temporal-only models in denoising and capturing abrupt displacement increments. The second stage evaluates global trend coordination, examining whether the model can jointly predict overall slope movement. All $N$ sensors are treated as a system, and the global R² score is used to verify whether the predicted displacement field aligns with the actual physical evolution. This stage demonstrates whether the physics-guided graph attention mechanism (PG-GAT) and spatial coordination loss ${\mathcal{L}}_{Spat}$ effectively remove spurious correlations, enabling accurate prediction of macroscopic slope instability.

4.3. Main Performance Comparison

To evaluate the overall effectiveness of the PG-STGN framework for slope displacement increment prediction, its performance on the test set was quantitatively compared with the two benchmark categories (temporal-only models and spatio-temporal graph neural networks). The evaluation considered both local node accuracy and global slope movement trends. To ensure the reliability of our findings and mitigate the influence of random initialization, the models were evaluated across five runs. We observed that the variation in prediction metrics (e.g., MSE) across different initializations was marginal, confirming stable model convergence. This indicates that the performance improvements demonstrated by PG-STGN are robust rather than artifacts of randomness.

4.3.1. Single-Node Accuracy Evaluation

First, key sensors in different slope regions (including the BW5 and BW6 series) were used to assess the numerical fitting performance of each model for displacement increments. To examine model stability across different horizons, two prediction lengths were tested: short-term ($T_{\mathrm{pl}}=6$) and long-term ($T_{\mathrm{pl}}=12$).

Table 2. Performance comparison of different models for slope displacement increment prediction (single monitoring point).

Datasets	${\mathit{T}}_{\mathit{p}\mathit{l}}$	Metric	GRU	LSTM	TCN	GAT-LSTM	STGCN	Graph WaveNet	PG-STGN
BW5-1	6	MSE	0.541	0.189	0.341	0.152	0.115	0.089	0.057
		MAE	0.529	0.295	0.249	0.210	0.185	0.165	0.144
	12	MSE	0.733	0.341	0.483	0.285	0.224	0.195	0.184
		MAE	0.649	0.391	0.549	0.355	0.310	0.260	0.234
BW5-2	6	MSE	0.558	0.215	0.362	0.168	0.128	0.098	0.065
		MAE	0.540	0.312	0.265	0.225	0.198	0.178	0.158
	12	MSE	0.785	0.388	0.510	0.315	0.256	0.210	0.195
		MAE	0.682	0.420	0.582	0.388	0.335	0.275	0.256
BW6-1	6	MSE	0.510	0.176	0.325	0.145	0.108	0.082	0.052
		MAE	0.515	0.284	0.238	0.205	0.175	0.158	0.135
	12	MSE	0.705	0.320	0.455	0.268	0.215	0.185	0.176
		MAE	0.628	0.375	0.525	0.340	0.295	0.252	0.221

Note 1 bold means the best results.

presents a detailed comparison of MSE and MAE for PG-STGN and six benchmark models under these conditions.

Table 2 shows that among traditional temporal models (GRU, LSTM, and TCN), TCN generally outperforms RNNs, with slower error growth over longer horizons. Nevertheless, standard TCN accuracy is still far below that of PG-STGN; at node BW5-1 with $T_{\mathrm{pl}}=6$, PG-STGN reduces MSE by ~83%. This improvement stems from the dual-stream feature extraction in Section 3.2, which separates displacement increments into high-frequency abrupt changes and low-frequency trends, enabling accurate capture of instantaneous deformation and stable long-term prediction.

Compared with advanced spatio-temporal models (GAT-LSTM, STGCN, and Graph WaveNet), PG-STGN remains superior. Although Graph WaveNet leverages adaptive graph learning, it underperforms PG-STGN at BW5-2, as purely data-driven methods can learn spurious spatial correlations from noise. PG-STGN uses the physics mask (Section 3.3) to inject geological hierarchical priors, ensuring feature aggregation only occurs between physically linked nodes, achieving high single-node accuracy.

To illustrate slope displacement modeling, the trajectories of ΔX at BW5-1 and BW6-1 are compared under short ($T_{\mathrm{pl}}=6$) and long ($T_{\mathrm{pl}}=12$) horizons (Figure 6 and Figure 7). Baseline models (LSTM and TCN) show response lag and amplitude smoothing at abrupt changes (BW5-1, 2nd step; BW6-1, 6th step), due to limited high-frequency sensitivity in a single feature space, resulting in smoothed predictions that miss critical precursors. PG-STGN, however, consistently provides the best fit across all four scenarios, as detailed below:

At node BW5-1, during the abrupt displacement at the second time step, traditional temporal models (GRU and LSTM) failed to capture the dynamic change promptly, resulting in noticeable numerical lag. In contrast, PG-STGN, leveraging its dual-stream feature extraction, decouples high-frequency instantaneous changes from low-frequency trends, avoiding smoothing effects and enabling the predicted trajectory to closely follow the true peak displacement pattern.

At node BW6-1 with a long-term prediction horizon $T_{pl}=12$, errors accumulated over time, and baseline models generally deviated in the latter steps. In contrast, the geology-based physics mask of PG-STGN effectively suppresses spurious spatial correlations, allowing the model to learn a more stable geological motion trend. During the complex fluctuations from the 8th to 10th time steps, predictions of PG-STGN exhibit high fidelity, and its noise suppression in steady periods surpasses purely data-driven methods such as Graph WaveNet, effectively avoiding non-physical high-frequency oscillations.

4.3.2. Global Slope Movement Trend Evaluation

The ultimate goal of slope monitoring is to assess the macroscopic stability of the geological body, i.e., to determine whether the slope transitions from a steady phase to an accelerated deformation phase or remains stable. To evaluate whether each model can accurately reconstruct the overall slope evolution from multi-sensor data, all $N$monitoring points are treated as a single dynamical system, and the global R² score is used for assessment.

Table 3. Comparison of performance in predicting overall slope movement trend.

Model Type	Model Name	${\mathit{R}}^2$
Traditional time-series models	GRU	0.685
	LSTM	0.712
	TCN	0.745
Spatio-temporal graph network models	GAT-LSTM	0.815
	STGCN	0.832
	Graph WaveNet	0.858
Proposed model	PG-STGN	0.924

Note 2 bold means the best results.

presents a comparison of global R² on the test set between PG-STGN and the benchmark models, averaged over $T_{\mathrm{pl}}=12$.

The global R² score reflects how well the predicted curve matches the true deformation in fluctuation patterns. Pure temporal models (GRU, LSTM, and TCN) generally achieve an R² below 0.75, indicating phase lag at key turning points despite reasonable single-node accuracy (Section 4.3.1), as they ignore system-level slope correlations. In contrast, PG-STGN reaches R² = 0.924, capturing both numerical accuracy and multi-sensor coordinated slope motion, closely following true physical processes in steady and accelerated phases.

The results also confirm the effectiveness of the physics-guided spatial coordination constraint. While Graph WaveNet improves R² to 0.858 via adaptive graph learning, it remains lower than PG-STGN. The spatial coordination loss ${\mathcal{L}}_{Spat}$ (Section 3.4) enforces consistent deformation rates among sensors within the same geological level (e.g., 700-level), suppressing local noise and revealing macroscopic motion patterns. Consequently, PG-STGN models overall slope movement with high confidence, providing a more reliable basis for early warning than purely numerical predictions.

4.4. Ablation Study

To analyze the effectiveness of core modules in the PG-STGN framework and assess the contributions of physics guidance and dual-stream feature extraction, three variant models were designed for comparison. All variants were trained under the same experimental settings and dataset splits.

4.4.1. Definition of Variant Models

To evaluate the role of each component, three ablation variants were constructed:

• Variant A: Without Geological Physics Graph (w/o Phy-Graph). Removes the geological hierarchical constraint from Section 3.1.2 (i.e., the $\gamma ·\delta \left(L_i,L_j\right)$ term) and the corresponding physics mask. A standard K-NN adjacency matrix is built solely based on Euclidean distances between sensors. This variant tests the necessity of incorporating geological priors (e.g., the 700-level) for spatial coordination.
• Variant B: Without Dual-Stream Feature Extraction (w/o Dual-Path). Removes the dual-stream structure from Section 3.2 (high-frequency path and low-frequency trend path). A standard single-stream dilated TCN is used with identical kernel configurations. This variant evaluates the advantage of dual-stream design in separating noise from abrupt signals.
• Variant C: Without Spatio-Temporal Consistency Loss (w/o Consistency Loss). Removes the temporal smoothing loss ${\mathcal{L}}_{Temp}$ and spatial coordination loss ${\mathcal{L}}_{Spat}$ from Section 3.4, leaving only the MSE loss ${\mathcal{L}}_{MSE}$ as the optimization target. This variant assesses the role of physics-constrained losses in suppressing non-physical oscillations and maintaining overall trend consistency.

4.4.2. Experimental Results and Analysis

Table 4. Comparison of ablation study results (average over all monitoring points, $T_{pl}=12$).

Model Variants	MSE	${\mathit{R}}^2$
w/o Phy-Graph	0.245	0.845
w/o Dual-Path	0.312	0.862
w/o Consistency Loss	0.218	0.768
PG-STGN	0.198	0.924

presents the performance comparison of the full model (PG-STGN) and the three variants on the test set. To reflect generalization, all values are averaged over five runs. Metrics are averaged over all monitoring nodes with $T_{pl}=12$.

Analysis of Table 4 highlights the contributions of each core module in PG-STGN, as illustrated in Figure 8. First, the physics graph is crucial for spatial coordination. Removing the geological hierarchical constraint (w/o Phy-Graph) increases MSE by ~23.7% and reduces R² to 0.845. Relying only on geometric distances introduces spurious correlations (e.g., nodes at 680- and 700-levels), while the hierarchical mask enforces aggregation among nodes on the same sliding plane, improving spatial prediction.

Second, the dual-stream TCN excels at noise suppression and abrupt change extraction. Removing it (w/o Dual-Path) raises MSE by ~57.6% because single-stream TCN either smooths key abrupt signals or retains excess noise. The dual-stream design decouples high-frequency abrupt changes from low-frequency trends, enabling improved single-node accuracy.

Finally, the spatio-temporal consistency loss preserves the overall trend. Removing it (w/o consistency loss) only slightly affects MSE (~10.1%) but drops global R² from 0.924 to 0.768. Without ${\mathcal{L}}_{Temp}$, predictions show non-physical high-frequency oscillations; without ${\mathcal{L}}_{Spat}$, neighboring nodes lose coordinated trends. The mixed loss enforces physics-guided smooth, coherent predictions with consistent spatial trends, ensuring both accurate single-node estimates and faithful reconstruction of macroscopic slope deformation.

Figure 8 visually compares the test set performance of the full model and the ablation variants. Red bars represent the complete PG-STGN, while gray bars correspond to variants with specific modules removed. Error bars indicate the standard deviation across multiple independent runs, illustrating the stability and statistical significance of the performance gains.

Comparing w/o Phy-Graph with the full model shows that a graph built solely on geometric distances (Euclidean proximity) struggles to capture cross-level deformation patterns, leading to a ~23.7% increase in MSE (from 0.198 to 0.245) and a drop in global R² from 0.924 to 0.845. This degradation confirms that without geological hierarchical priors, the model tends to aggregate features from physically unrelated monitoring points (e.g., sensors on different elevation platforms) when their coordinates happen to be close. Such spurious correlations introduce conflicting displacement patterns, especially during accelerated deformation phases where different levels exhibit distinct mechanical responses. The hierarchical masking mechanism in PG-STGN effectively restricts attention to nodes within the same geological layer (e.g., the 700-level), thereby preserving physically meaningful spatial interactions.

Removing the dual-stream structure (w/o Dual-Path) causes the largest MSE degradation (~57.6%, rising to 0.312). This result underscores the critical role of the proposed iTCN module in decoupling high-frequency abrupt signals from low-frequency background trends. A standard single-stream TCN either smooths instantaneous displacement peaks (by averaging across time) or retains excessive noise, leading to either missed early warning signals or false alarms. In contrast, the max-pooling path in PG-STGN preserves step-like deformation events, while the average-pooling path suppresses high-frequency noise and maintains long-term creep trends. The 57.6% increase in MSE quantitatively demonstrates that dual-path decoupling is not merely an incremental improvement but an essential design for GNSS-based slope displacement prediction under noisy and non-stationary conditions.

Observing w/o consistency loss, the MSE remains relatively acceptable (0.218, only ~10.1% higher than the full model), but the global R² drops sharply by ~16.9% (from 0.924 to 0.768). This contrasting behavior reveals a key insight: point-wise accuracy (MSE) alone is insufficient to guarantee physically consistent predictions. Without the temporal smoothing loss ${\mathcal{L}}_{Temp}$, the predicted displacement increments exhibit non-physical high-frequency oscillations, especially during steady-creep periods. More importantly, without the spatial coordination loss ${\mathcal{L}}_{Spat}$, neighboring nodes within the same geological level lose their coordinated deformation trends—some sensors may show acceleration while others remain unchanged, violating the mechanical expectation that a sliding surface moves as a coherent unit. The sharp R² drop indicates that while the model still fits individual sensor values numerically, it fails to reconstruct the macroscopic slope evolution pattern, which is the ultimate goal for early warning systems.

Overall, the core modules in PG-STGN collaboratively enhance both single-node accuracy and the prediction of macroscopic slope deformation trends. The physics-guided graph provides the correct spatial topology, the dual-path TCN enables robust multi-scale temporal feature extraction, and the consistency loss enforces physically plausible dynamics. Their synergy explains why PG-STGN achieves the best performance across all metrics.

4.4.3. Sensitivity to Spatial Gain Weight

To quantify the dependence of PG-STGN on geological priors, we conducted two sensitivity experiments focusing on the spatial gain weight $\gamma$ and the accuracy of geological level assignments.

We performed a grid search for $\gamma \in [0.1, 1.0]$. As shown in Figure 9, the global $R^2$ remains consistently high (>0.91) within the interval of $\left[0.3,0.7\right]$. This indicates that while the physics-guided constraint is essential, the model’s performance is not overly sensitive to the precise selection of $\gamma$, demonstrating sufficient hyperparameter robustness for real-world deployment.

4.4.4. Case Studies of Triggered Deformation Events

During the peak rainfall intensity, the slope experienced an abrupt displacement spike due to elevated pore water pressure. To quantify the models’ capabilities in capturing this specific kinematic response, we evaluated them on this isolated rainfall segment. As summarized in

Table 5. Performance comparison during the rainfall-triggered acceleration event.

Model	MSE	Peak Absolute Error (mm)
GRU	0.812	1.45
LSTM	0.534	0.98
TCN	0.615	1.12
GAT-LSTM	0.320	0.75
Graph WaveNet	0.285	0.58
PG-STGN (Ours)	0.152	0.14

, traditional temporal models (e.g., GRU, LSTM) suffered from severe performance degradation during this extreme event. Notably, standard convolutional models like TCN exhibited a massive peak absolute error (PAE) of 1.12 mm and a noticeable phase lag. This occurs because standard causal convolutions act as low-pass filters, inherently smoothing out the high-frequency warning signals. Conversely, the PG-STGN framework significantly outperformed all baselines, achieving an overall MSE of 0.152 and an exceptionally low PAE of just 0.14 mm with zero phase lag. This quantitative result provides definitive proof that the max-pooling path within our iTCN module successfully functions as a high-frequency feature extractor. It seamlessly bypasses the smoothing effect, accurately locking onto the abrupt displacement peak. This feature decoupling perfectly aligns with the mechanical reality: the slope’s response to critical water infiltration is instantaneous and nonlinear, requiring a dedicated neural pathway for high-frequency changes.

4.5. Discussion

• Attention pattern analysis: High-attention nodes are mainly within the same geological level (e.g., BW5-1 and BW5-2 at 700-level, BW6-1 and BW6-2 at 712-level), while inter-level pairs (e.g., 680-level vs. 700-level) are suppressed by the physics mask. This aligns with slope mechanics: nodes on the same sliding surface share stress paths and correlate strongly, whereas nodes on different elevation platforms deform independently. In the baseline GAT-LSTM and Graph WaveNet models, which lack hierarchical priors, attention weights often scatter across both same-level and cross-level pairs, leading to physically implausible spatial mixing. PG-STGN, with its hierarchical mask, restores intra-level coordination and inter-level decoupling, avoiding spurious cross-level correlations. This not only improves prediction accuracy but also makes the attention maps interpretable to geotechnical engineers, who can verify that the model focuses on expected sensor clusters.
• Temporal evolution of slope deformation: Early in the monitoring period (first 30 time steps), displacement increments are low (mostly below 2 mm), and all models produce predictions that match observations reasonably well. Between the 50th and 70th steps, however, a clear acceleration occurs across multiple levels (680, 700, and 712). PG-STGN accurately captures the onset of this acceleration (around step 52) and, via the spatial coordination loss ${\mathcal{L}}_{Spat}$, ensures that the predicted accelerations proportionally reflect the physical process: central nodes (BW5 series) accelerate most (up to ~8 mm/step), lateral nodes (BW1, BW2) lag slightly with smaller increments (~3–4 mm/step), and nodes on the uppermost level (712) show intermediate behavior. This spatial pattern matches the typical progressive failure mechanism of a translational slide, where the central portion moves first and the flanks are partially constrained. In contrast, models without ${\mathcal{L}}_{Spat}$ (w/o Consistency-Loss) often predict either uniform acceleration across all nodes or inconsistent patterns (e.g., a lateral node accelerating faster than the central node), which would mislead early warning decisions.
• Geological interpretation of abrupt events: Geologically, node BW5-1 is located near a high-rainfall infiltration zone and has historically exhibited step-like deformation patterns. The abrupt displacement at time step 2 (an increase of ~6 mm in $\mathsf{\Delta }X$ within one sampling interval) corresponds to a rapid pore-pressure rise following an intense rainfall event, which reduces effective stress and shear strength along the potential slip surface. In single-stream models (e.g., TCN or w/o Dual-Path), this abrupt jump is either smoothed into a gradual ramp (due to the averaging effect of causal convolutions) or misinterpreted as noise and filtered out. By contrast, the dual-stream paths of PG-STGN capture the sudden response through the max-pooling branch (preserving the sharp peak) while simultaneously maintaining the background creep trend via the average-pooling branch. This design ensures that early warning signals (abrupt deformation) are not lost, while also avoiding overreaction to random high-frequency noise. As a result, the model correctly flags step 2 as a potential precursor without triggering false alarms during other noisy but stable periods.
• Translation of Predictions into Safety Measures: Beyond numerical accuracy, the practical value of a slope displacement prediction model lies in its ability to support reliable safety decisions. In PG-STGN, predicted displacement increments are converted into cumulative displacements and deformation rates, which are then mapped into a three-tier alert protocol: Attention (isolated exceedance of 2 mm/day), Warning (coordinated exceedance of 5 mm/day across at least three nodes on the same geological level), and Alarm (global violation of spatial coordination or exceedance of a geotechnically defined critical rate). To prevent overreliance on a single predictive model, all automated alerts remain advisory and require verification by a qualified geotechnical engineer, who cross-references predictions with independent data sources such as InSAR deformation maps, rainfall records, and in situ inclinometer readings. Furthermore, Monte Carlo dropout is employed to attach confidence intervals to each prediction, and alerts are suppressed when the epistemic uncertainty exceeds a predefined threshold, reducing the risk of false alarms during periods of degraded data quality. This human-in-the-loop design ensures that PG-STGN serves as a decision-support tool rather than a standalone safety oracle, aligning with established dam safety regulations and operational best practices.
• Model Degradation under Input Failures: Although PG-STGN achieves high accuracy under normal sensing conditions, its performance predictably degrades when sensors drop out, report missing values, or produce anomalous spikes. The current framework relies on a fixed graph topology and external linear interpolation, which are insufficient to handle prolonged sensor failures or large gaps. As shown in our robustness analysis (Section 4.6), the model’s spatial coordination loss can propagate errors from faulty nodes to their neighbors, and the dual-path decoupling may oversmooth genuine abrupt signals if preceded by interpolation. Future work should incorporate dynamic graph adaptation, learned temporal imputation, and Monte Carlo dropout-based uncertainty estimation to alert operators when input quality is compromised.

4.6. Dependence on Geological Priors and Symmetry Considerations

A key design choice in PG-STGN is the injection of geological hierarchical priors and the enforcement of symmetric spatial interactions via the adjacency matrix. While these priors contribute substantially to the model’s performance and interpretability, they also introduce a dependency on the accuracy of the provided geological levels and the assumption of homogeneous, isotropic deformation transfer within each level. In this section, we quantitatively assess the sensitivity to these priors, discuss scenarios where performance may degrade, and propose strategies to mitigate such risks in operational settings.

4.6.1. Limitations of the Symmetry Assumption

The symmetry of $A_{ij}$ and the Laplacian $L$ enforces that the mutual deformation influence between two points is reciprocal. This assumption is physically justified within a single, homogeneous slip surface where stress transfer is approximately isotropic. However, it may break down in slopes with pronounced geological anisotropy—e.g., bedding planes with strong directional stiffness, faults that behave as discontinuities, or progressive failure where the driving stress is directional. In such environments, a symmetric graph may incorrectly enforce bidirectional attention where the actual mechanical influence is unidirectional.

To address this, the framework can be extended to a directed graph where $A_{ij}$ is not required to equal $A_{ij}$. The hierarchical mask could then incorporate direction-dependent weights informed by borehole logs, structural geology mapping, or even a secondary attention module that learns asymmetric weight increments. Importantly, the Laplacian regularization would need to be replaced by a directed divergence operator (e.g., using graph incidence matrices). We leave this extension to future work, noting that for the vast majority of monitoring points in well-stratified, slowly deforming slopes, the symmetry assumption is a reasonable and beneficial approximation.

4.6.2. Mitigation Strategies in Deployment

Given the reliance on priors, we recommend the following measures for real-world deployment:

• Periodic prior recalibration: Geological-level assignments should be cross-validated with new borehole data, inclinometer readings, and InSAR deformation zones, particularly after heavy rainfall seasons or excavation activities that may alter the slip surface.
• Hybrid semi-adaptive graph: A practical compromise is to retain the symmetric hierarchical graph as the backbone but allow a small, learned perturbation $\Delta A_{ij}$ that captures residual asymmetric interactions. This can be trained jointly with the rest of the network, providing a smooth trade-off between physical rigidity and data-driven flexibility.
• Uncertainty-aware monitoring: When the confidence in level assignments is low (e.g., for newly installed sensors), the model can output an uncertainty estimate alongside predictions (via Monte Carlo dropout). Operators can then assign lower decision weight to predictions from nodes with uncertain level membership, reducing the risk of false alarms or missed events.

5. Conclusions

This study introduces a physics-guided spatio-temporal graph convolutional framework (PG-STGN) for GNSS displacement data, capturing high-frequency abrupt changes, low-frequency trends, and geology-constrained spatial correlations. By integrating geological priors with deep learning, PG-STGN achieves accurate displacement increment prediction and physically consistent reconstruction of macroscopic slope deformation. Key conclusions are: (1) A graph construction combining geometric distance decay and geological hierarchical gain overcomes conventional GNN limitations, which often produce spurious spatial links, providing a solid physical basis for coordinated slope deformation modeling. (2) A dual-path decoupling strategy separates high-frequency abrupt changes from low-frequency trends, enhancing sensitivity to sudden deformations while preserving smooth temporal evolution. (3) Geological priors guide attention weights, focusing on strongly correlated nodes. Coupled with graph Laplacian regularization in the physics-consistent loss, predicted displacements show high spatial consistency and eliminate non-physical oscillations. (4) On real GNSS slope data, PG-STGN outperforms all baselines. Ablation studies confirm that physics-guided graph construction, dual-path decoupling, and physics-consistent loss are critical for robust and interpretable predictions.

On an edge device, the full PG-STGN pipeline completes in under 400 ms per sample, which is well below typical 5–15 min GNSS sampling intervals, while the exported model requires only ~24 MB of storage and less than 1.5 GB of GPU memory. Integration with existing monitoring systems follows a lightweight MQTT-based architecture: raw displacements are pushed to a circular buffer, from which the sliding-window tensor is built, and model predictions are converted into cumulative displacements and deformation rates. A three-tier alert manager maps predicted velocities and inter-node coordination into Attention, Warning, and Alarm levels, with Monte Carlo dropout providing 95% confidence intervals to suppress false alarms during data-quality degradation. A parallel watchdog monitors sensor availability; if over half of the nodes are offline, the system automatically degrades to a constant-velocity plus Laplacian-smoothing fallback, ensuring continuity. All alerts remain advisory and are reviewed by a geotechnical engineer alongside rainfall, InSAR, and inclinometer data, forming a human-in-the-loop safeguard against overreliance on any single predictive model.