Data-Driven Prediction of Stress–Strain Fields Around Interacting Mining Excavations in Jointed Rock: A Comparative Study of Surrogate Models

Abstract

Assessing the stress–strain state around interacting mining excavations using the finite element method (FEM) is computationally expensive for parametric studies. This study evaluates tabular machine-learning surrogate models for the rapid prediction of full stress–strain fields in fractured rock masses treated as an equivalent continuum. A dataset of 1000 parametric FEM simulations using the elastoplastic generalized Hoek–Brown constitutive model was generated to train Random Forest, LightGBM, CatBoost, and Multilayer Perceptron (MLP) models based on geometric features. The results show that the best models achieve R² scores of 0.96–0.97 for stress components and 0.99 for total displacements. LightGBM and CatBoost provide the optimal balance between accuracy and computational cost, offering speed-ups of 15 to 70 times compared to FEM. While Random Forest yields slightly higher accuracy, it is resource-intensive. Conversely, MLP is the fastest but less accurate. These findings demonstrate that data-driven surrogates can effectively replace repeated FEM simulations, enabling efficient parametric analysis and intelligent design optimization for mine workings.

1. Introduction

The assessment of the stress–strain state (SSS) of rock masses characterized by pervasive fracturing in the interaction zone of mining excavations is a central problem in geomechanics [1,2,3], directly affecting the safety and economic efficiency of engineering decisions [4,5]. The finite element method (FEM) remains the standard tool for solving this problem across a wide range of geotechnical applications [6,7,8]. However, in tasks requiring multiple recalculations—such as probabilistic analyses, parameter optimization, and the comparison of alternative design solutions within intelligent mining frameworks—direct use of FEM becomes computationally expensive. Every new configuration requires mesh regeneration and a full re-solution of the problem, and both meshing and computation remain the bottleneck even with automated workflows.

In this context, surrogate modeling based on Soft Computing Methods (SCM) and machine learning (ML) has emerged as a promising alternative capable of rapidly approximating dependencies identified through FEM. As highlighted in a recent state-of-the-art review [9,10], a wide spectrum of algorithms is now available for geotechnical tasks, ranging from neural networks (ANN, CNN, LSTM) and support vector machines (SVM) to powerful ensemble methods like Gradient Boosting and Random Forest. These techniques allow for the processing of heterogeneous geological data and complex non-linear relationships that are difficult to capture with traditional empirical methods, including clustering-based monitoring-data interpretation [11].

Existing studies show successful applications of surrogate models primarily for predicting integral scalar indicators, such as tunnel stability coefficients in different geological conditions, bearing capacity of foundations, slope stability, and retaining structure performance [12,13,14,15,16]. For instance, comparative analyses of soft computing methods for deep excavations have demonstrated that ensemble learning techniques, specifically XGBoost, offer superior predictive capacity over traditional models in estimating maximum wall deflection [17]. Furthermore, transfer learning approaches and knowledge-based deep learning have shown significant potential for tasks involving sequential data, such as the prediction of shield tail clearance using hybrid Transformer-LSTM architectures [18]. Another emerging direction involves Physics-Informed Neural Networks (PINN), which ensure physical consistency even with limited datasets by incorporating governing differential equations directly into the neural network’s loss function [19].

In contrast, predicting full spatial fields of the SSS is a considerably more complex task, generally addressed using architectures designed for spatial data processing. One common approach represents geometry and output fields as images and applies convolutional neural networks (CNNs), including conditional generative adversarial networks (cGANs), which have demonstrated the ability to reproduce stress fields for complex geometries and boundary conditions [20,21,22]. Another approach represents the finite element mesh as a graph and uses graph neural networks (GNNs), such as MeshGraphNets, which can significantly accelerate FEM simulations in solid mechanics by operating directly on unstructured meshes [23,24,25]. Although highly accurate, these models require specialized data representations (images or graphs) and are often computationally demanding to implement in routine engineering practice.

Furthermore, there is a significant research gap in the application of simpler, faster, and more interpretable tabular algorithms (such as Random Forest or Boosting) for full-field prediction. The fundamental limitation of such models is their inherent inability to capture spatial dependencies; unlike CNNs or GNNs, tabular algorithms treat input features as independent variables and cannot natively perceive the underlying geometric topology or the spatial proximity of nodes.

To facilitate AI-assisted design in mining, the present study considers an alternative paradigm: encoding the physics of the problem directly into the feature space by designing a set of engineering (contextual) features that describe the mutual arrangement of nodes and excavations as well as the geometry of the system. This strategy enables the application of classical tabular ML algorithms—primarily tree-based ensembles—that are well suited for structured data and provide high computational performance [26]. In contrast to PINNs, which enforce physical laws via differential equation constraints within the neural network’s loss function, the proposed approach encodes mechanical and spatial context directly into the input representation. This bypasses the high computational overhead [27] and the complex architectural tuning [28] required for PINNs, making the surrogate models significantly easier to implement while leveraging the superior ability of tree-based algorithms to handle non-linear tabular data. However, the applicability and comparative efficiency of such algorithms specifically for reproducing full SSS fields around interacting mining excavations remain insufficiently investigated.

The aim of this study is to compare the effectiveness of surrogate models (Random Forest, LightGBM, CatBoost, MLP) for the rapid prediction of SSS fields around two interacting underground excavations based on FEM simulations.

The objectives of the study are as follows:

• To generate a parametric dataset of 1000 FEM simulations and develop an engineering feature space that reflects the geometry of the system and the spatial position of nodes relative to the excavations.
• To train and compare surrogate models in terms of accuracy ($R^2$, MAE) and computational performance (training time, inference speed, model size, and learning curves).
• To evaluate model interpretability using permutation feature importance and to conduct qualitative validation of SSS fields (including critical zones defined by the limiting shear state (${\tau }_{rel}\ge 1$) against reference FEM results.

The remainder of the manuscript is organized as follows. Section 2 describes the numerical model, feature construction, and model training/optimization. Section 3 reports the comparative accuracy and computational performance. Section 4 discusses interpretability, limit-state behavior in critical zones (limiting shear state, ${\tau }_{rel}\ge 1$), and extrapolation limits. Section 5 concludes the study.

2. Materials and Methods

The research methodology (Figure 1) included: parametric FEM modeling to generate a synthetic dataset of simulations; extraction of SSS fields and construction of engineering features; splitting the dataset into training/validation/test subsets with cumulative subsamples for learning curves; hyperparameter optimization using Optuna (version 3.6) and model training (Random Forest implemented in scikit-learn (version 1.8.0), LightGBM (version 4.3), CatBoost (version 1.2), MLP implemented in PyTorch (version 2.2)); followed by a comprehensive evaluation of model accuracy and computational efficiency.

2.1. Numerical Modeling

The dataset used for model training was generated by conducting a series of 1000 parametric simulations in a geotechnical finite element software package. To automate geometry generation, modification of input parameters, and extraction of simulation outputs, an application programming interface (API) controlled by a Python (version 3.10) script was employed.

The computational model (Figure 2) represents a 50 × 50 m rock mass domain containing two parallel horseshoe-shaped underground excavations. The model dimensions were selected to eliminate the influence of boundary effects. The mechanical behavior of the rock mass was simulated using an elastoplastic constitutive model with a Hoek–Brown yield criterion, developed for fractured rock masses [29,30]. In this study, fracturing/jointing is incorporated at the rock-mass scale through the GSI parameter within the generalized Hoek–Brown formulation, without explicit discontinuum joint network modeling. Equivalent Mohr–Coulomb parameters were derived using the procedure proposed by Hoek and Brown [31].

The simulations were performed under plane strain conditions. The boundary conditions were defined as follows: the bottom boundary was fixed in both vertical and horizontal directions (u_x = 0, u_y = 0), while the lateral boundaries restricted horizontal displacement (u_x = 0). The initial in situ (geostatic) stress state was simulated by applying a uniform distributed load to the top boundary, representing the overburden pressure of the strata overlying the model domain at a depth of H = 118 m.

The numerical calculation was implemented in two stages: an initial stage to establish the in situ stress field, followed by a second stage where the excavation was modeled by the deactivation of the finite element clusters within the tunnel contours. The excavations were modeled as unsupported. Hydrostatic pore water pressure was not considered in the analysis.

The computational domain was discretized using 15-node triangular finite elements. To ensure numerical precision in the interaction zone, local mesh refinement was applied around the excavation contours, while standard element sizes were maintained for the rest of the rock mass domain.

Numerical convergence was controlled by an elastoplastic iterative solver with automatic load stepping. The validity of each simulation was confirmed by ensuring that the global error of unbalanced forces remained within standard tolerances and that the target calculation phase was fully completed.

The physical and mechanical properties adopted in the model are presented in

Table 1. Rock mass parameters used in the model.

Parameter	Symbol	Value	Unit
Unit weight	γsat	25.4	kN/m3
	γunsat	23.0	kN/m3
Geological Strength Index	GSI	50	—
Depth of excavation	H	118	m
Hoek–Brown constant for intact rock	mi	15	—
Disturbance factor	D	0.5	—
Dilatancy angle	ψmax	4.0°	deg
Uniaxial compressive strength of intact rock	|σci|	83,356	kN/m2
Equivalent Young’s modulus of rock mass	Erm	4,408,261	kPa
Poisson’s ratio	ν	0.2	—
Model width	—	50	m
Model height	—	50	m

. Input parameters for such constitutive models—including deformability characteristics and shear strength parameters—are typically derived from laboratory tests on rock specimens [32,33]. The computational domain was discretized using 15-node triangular finite elements with local mesh refinement around the excavation contours.

To identify nodes that reached the limiting shear state, a dimensionless relative shear stress parameter ${\tau }_{rel}$ was used:

$${\tau }_{\mathrm{rel}}={\displaystyle \frac{{\tau }_{\mathrm{mob}}}{{\tau }_{\max }}}$$

(1)

where ${\tau }_{\mathrm{mob}}$ is the magnitude of mobilized shear stresses along the local shear plane, and ${\tau }_{\max }$ is the ultimate shear strength computed according to the Mohr–Coulomb criterion: ${\tau }_{max}=c+{\sigma }_n^{\prime }\tan \phi$.

Within this study, the parameters listed in Table 1 were kept constant, while six global parameters governing the geometry and relative position of the excavations were varied: horizontal spacing and vertical offset between excavation centers, as well as the width and height of each excavation. Parameter values were sampled from predefined uniform distributions. The variation ranges are presented in

Table 2. Variable geometric parameters of the FEM model.

Parameter	Symbol	Range	Unit
Horizontal distance between excavations	C	1.0–8.0	m
Vertical offset between excavations	R	−4.0–4.0	m
Width of excavation 1	B1	2.0–5.0	m
Height of excavation 1	D1	2.0–3.0	m
Width of excavation 2	B2	2.0–5.0	m
Height of excavation 2	D2	2.0–3.0	m

.

For each simulation, the following data were extracted from all nodes of the finite element mesh:

Basic features: nodal coordinates (X, Y).

Target variables: six components of the stress–strain state, including effective normal stresses (${\sigma }_{xx}$, ${\sigma }_{yy}$), shear stresses (${\tau }_{xy}$), strain components (${\epsilon }_{xx}$, ${\epsilon }_{yy}$, ${\gamma }_{xy}$), as well as total displacement ($U_{tot}$).

2.2. Feature Space and Its Construction

Since the original nodal coordinates (X, Y) do not provide ML models with explicit information about the geometry of the computational domain, and the global parameters alone are insufficient to capture dependencies explaining the target variables, a set of engineering (contextual) features was developed. These features supply the model with information about the spatial position of each node relative to the excavation contours and the overall system geometry. The full set of input features used for training is summarized in

Table 3. Input features for the machine learning models.

Formulas	Feature	Feature Group
–	X, Y	Nodal coordinates
(2) and (3)	Mean_width, Mean_height	Derived global parameters
(4) and (5)	Aspect1, Aspect2
(6)	Area_ratio
(7)	Dist_norm
(8)	Shift_norm
(9)	Signed_Dist_Norm	Engineering (contextual) features
(10)	Overlap_Index
(11)	Curvature
(12)	Vertical_Projection
(13)	Density_Excavated_Distances

.

The average excavation width ($\mathrm{Mean}\_\mathrm{width}$) and height ($\mathrm{M}\mathrm{ean}\_\mathrm{height}$) were computed as:

$$\mathrm{M}\mathrm{ean}\_\mathrm{width} = {\displaystyle \frac{B_1+B_2}{2}},$$

(2)

$$\mathrm{M}\mathrm{ean}\_\mathrm{height}={\displaystyle \frac{D_1+D_2}{2}},$$

(3)

The aspect ratios (${\mathrm{Aspect}}_1$, ${\mathrm{Aspect}}_2$) and area ratio ($\mathrm{Area}\_\mathrm{ratio}$) were defined as:

$${\mathrm{Aspect}}_1 = {\displaystyle \frac{B_1}{D_1}},$$

(4)

$${\mathrm{Aspect}}_2={\displaystyle \frac{B_2}{D_2}},$$

(5)

$$\mathrm{A}\mathrm{rea}\_\mathrm{ratio}={\displaystyle \frac{B_1{D}_1}{B_2{D}_2}}.$$

(6)

The normalized distance ($\mathrm{Dist}\_\mathrm{norm}$) and shift ($\mathrm{Shift}\_\mathrm{norm}$) were defined as:

$$\mathrm{D}\mathrm{ist}\_\mathrm{norm}={\displaystyle \frac{C}{\mathrm{M}\mathrm{ean}\_\mathrm{width}}},$$

(7)

$$\mathrm{S}\mathrm{hift}\_\mathrm{norm}={\displaystyle \frac{R}{\mathrm{Mean}\_\mathrm{height}}}.$$

(8)

The normalized signed distance ($\mathrm{Signed}\_\mathrm{Dist}\_\mathrm{Norm}$) was given by:

$$\mathrm{Signed}\_\mathrm{Dist}\_\mathrm{Norm}\left(P\right)=\mathrm{sign}\left(P\right)\cdot {\displaystyle \frac{\mathrm{dist}\left(P,B\right)}{\mathrm{M}\mathrm{ean}\_\mathrm{width}}},$$

(9)

where $\mathrm{Signed}\_\mathrm{Dist}\_\mathrm{Norm}\left(P\right)$ is the feature value for node $P$; $B$ is the set of points forming the excavation contour; $\mathrm{M}\mathrm{ean}\_\mathrm{width}$ is the mean excavation width defined in Equation (2); and $\mathrm{sign}\left(P\right)$ is a function returning +1 for rock mass nodes and −1 for excavation nodes.

The overlap index ($\mathrm{Overlap}\_\mathrm{Index}$) was defined as:

$$\mathrm{Overlap}\_\mathrm{Index}\left(P\right)= \exp \left(-{\displaystyle \frac{d_1\left(P\right)}{\mathrm{M}\mathrm{ean}\_\mathrm{width}}}\right)\cdot \exp \left(-{\displaystyle \frac{d_2\left(P\right)}{\mathrm{M}\mathrm{ean}\_\mathrm{width}}}\right),$$

(10)

where $d_1\left(P\right)$, $d_2\left(P\right)$ are the minimum distances from node $P$ to the nearest points on the contours of the first and second excavation, respectively.

The curvature feature ($\mathrm{Curvature}$) was defined as:

$$\mathrm{Curvature}\left(P\right)=\kappa \left(b^{\mathrm{*}}\left(P\right)\right)\cdot \exp \left(-{\displaystyle \frac{\mathrm{dist}\left(P,B\right)}{\sigma }}\right)$$

(11)

where $b^{\mathrm{*}}\left(P\right)=\arg \underset{b\in B\mathrm{dist}}{\min }\left(P,b\right)$ is the point on the excavation contour $B$ closest to node $P$; $\kappa \left(b\right)$ is the local curvature of the contour at point $b$, estimated by approximating a local segment with a circular arc; and $\sigma$ is a parameter controlling the decay rate of curvature influence.

The vertical projection feature ($\mathrm{Vertical}\_\mathrm{Projection}$) was defined as:

$$\mathrm{Vertical}\_\mathrm{Projection}\left(P\right)=\mathrm{clip}\left(\mathrm{sgn}\left(\mathsf{\Delta }y\left(P\right)\right)\cdot f\left({\displaystyle \frac{\left|\mathsf{\Delta }y\left(P\right)\right|}{r\left(P\right)}}\right),-1,1\right),$$

(12)

where $\mathsf{\Delta }y\left(P\right)$ is the vertical offset from node $P$ to the nearest point on the excavation contour; $r\left(P\right)$ is the Euclidean distance to the same point; $f:\left[0,\mathrm{\infty }\right)\to \left[\mathrm{0,1}\right]$ is a monotonically increasing function; and $\mathrm{clip}\left(z,-\mathrm{1,1}\right)$ is a function that limits the final value to the range [−1, 1].

The density feature ($\mathrm{Density}\_\mathrm{Excavated}\_\mathrm{Distances}$) was computed as:

$$\mathrm{Density}\_\mathrm{Excavated}\_\mathrm{Distances}\left(P\right)={\left({\displaystyle \frac{1}{\left|E\right|}}\sum _{e\in E} \mathrm{dist}\left(P,e\right)\right)}^{-1},$$

(13)

where $\mathrm{Density}\_\mathrm{Excavated}\_\mathrm{Distances}\left(P\right)$ is the feature value for node $P$; $E$ is the set of nodes belonging to the excavated domain; and $\left|E\right|$ is the cardinality of this set.

To ensure reproducibility, specific parameter values were fixed based on preliminary sensitivity analysis. The full feature-engineering pipeline is provided in the public code repository. The excavation boundary set $B$ in Equations (9), (11), and (12) is represented by a discrete set of boundary-adjacent FEM nodes, and distances $\mathrm{dist}\left(P,B\right)$ are computed as Euclidean nearest-neighbor distances using a k-d tree. For the $\mathrm{Curvature}$ feature (Equation (11)), local curvature $\kappa \left(b\right)$ was estimated using PCA-based local orientation analysis and circle fitting within a neighborhood radius of $R_c=2.0$ m ($N_{min}=12$), with the influence decay parameter set to $\sigma =0.4$ m. For $\mathrm{Overlap}\_\mathrm{Index}$ (Equation (10)), distances $d_1$ and $d_2$ were computed relative to the two connected components corresponding to the two excavations using k-d tree queries. The monotonic mapping $f\left(·\right)$ in $\mathrm{Vertical}\_\mathrm{Projection}$ (Equation (12)) utilizes power-law scaling with $\gamma =1.2$ and edge-aware smoothing (${\tau }_{blend}=0.18$); the complete algorithmic implementation is provided in the repository. For $\mathrm{Density}\_\mathrm{Excavated}\_\mathrm{Distances}$ (Equation (13)), a small stabilizer $\epsilon ={10}^{-6}$ is applied in the inverse-distance computation to avoid division by zero.

Visualizations of the engineering feature fields for a test example are shown in Figure 3.

To ensure methodological rigor and prevent data leakage, the global dataset of nodes was not shuffled prior to splitting. Instead, the partitioning was performed at the simulation file level, treating each FEM calculation as a single independent unit. Consequently, the dataset was divided into training (70%), validation (15%), and testing (15%) subsets based on unique simulation files. This protocol guarantees that the models are evaluated on entirely unseen geometric configurations. Training was subsequently conducted on cumulative subsets of k simulation files, where k ∈{1–10, 16, 24, 32, 40, 56, 72, 80, 160, 240, 320, 400, 480, 560, 640, 700}, to analyze the dependence of predictive accuracy on the volume of physical scenarios. For the MLP, the subset size was capped at 400 simulation files due to GPU memory limits; larger subsets were used for the tree-based models only. Each subsequent subsample included all samples from the previous one, which enabled consistent construction of learning curves.

2.3. Machine Learning Algorithms Considered

The comparative analysis evaluates four regression algorithms frequently established as robust benchmarks for predictive modeling on structured tabular data. Independent surrogate models were developed for each of the seven stress–strain target parameters.

Random Forest (RF): A bagging-based ensemble algorithm implemented via the scikit-learn library. RF was selected for its capacity to mitigate variance through the aggregation of multiple randomized decision trees [34].

LightGBM [35] and CatBoost [36]: Two advanced implementations of the Gradient Boosting Decision Tree (GBDT) framework. LightGBM was utilized for its computational efficiency, derived from histogram-based split searching, while CatBoost was selected for its ordered boosting mechanism, which minimizes prediction bias and effectively captures complex non-linear feature interactions.

Multilayer Perceptron (MLP): A fully connected feed-forward neural network implemented within the PyTorch framework. The MLP was optimized via backpropagation and utilized L2-regularization (weight decay) to prevent overfitting and ensure stable convergence [37].

Detailed architectural specifications and the resulting optimized hyperparameters for all models are summarized in

Table 4. Summary of optimized hyperparameters for surrogate models across all target variables. Values represent the range [min–max] selected by the optimization procedure. Only the main hyperparameters are shown; full configurations are available in the public repository.

Model	Hyperparameter	Optimized Range/Value
Random Forest	Number of trees (n_estimators)	200–566
	Max depth (max_depth)	24–28
	Min samples split	6–12
	Max features	0.75–1.0 (or log2)
LightGBM	Number of estimators	1199–1471
	Learning rate	0.07–0.10
	Num leaves	169–199
	Max depth	19–25
CatBoost	Iterations	1900–3850
	Depth	10
	Learning rate	0.03–0.05
	Bagging temperature	1.8–5.7
MLP (PyTorch)	Hidden layers	3, 4
	Neurons per layer (descending)	(419–509) → (108–256) → (46–128) → (45–64)
	Learning rate	≈8.4 × 10−4–1 × 10−3
	Dropout rate	0.030–0.298

.

Data preprocessing protocols were tailored to the specific inductive biases of each algorithmic family. Tree-based ensembles (RF, LightGBM, CatBoost) were trained on raw input features, as these models are inherently invariant to monotonic transformations. In contrast, the MLP required Z-score standardization (StandardScaler) to facilitate gradient stability. To ensure a rigorous evaluation and prevent data leakage, normalization parameters (mean and standard deviation) were derived strictly from the training subset and subsequently applied to the validation and test sets.

2.4. Training Methodology and Evaluation Criteria

For each algorithm and each target variable, hyperparameter optimization was carried out using the Optun framework [38], which implements Bayesian optimization methods. The optimization objective was the coefficient of determination ($R^2$) on the validation set. A uniform computational budget was imposed in this study: 2 h of hyperparameter search for each algorithm–target pair.

The optimized hyperparameter configurations for all models are summarized in Table 4. Because the optimal settings differ slightly across the seven target variables, the table reports the range [min–max] of values selected during Optuna-based hyperparameter optimization. For transparency and reproducibility, the complete per-target configuration files (JSON)—including all tuned parameters not listed in Table 4—together with the training scripts and datasets, are provided in the public GitHub repository referenced in the Data Availability Statement.

In addition, the following assumption was adopted: hyperparameter optimization was performed once on the full training set, and the resulting optimal configuration was then used to train models on all smaller subsamples. This strategy was chosen to standardize the comparison conditions and reduce the total computational time. However, it should be noted that each subsample may in principle have its own locally optimal set of hyperparameters.

The experiments were conducted on a workstation running Windows 11 Pro with an AMD Ryzen 5–class CPU (6 cores, 12 threads), 32 GB DDR4 RAM, an NVIDIA GeForce RTX 4070 GPU (12 GB VRAM) and a 512 GB NVMe SSD. The GPU was used for training and inference of the MLP (PyTorch) and LightGBM models, while Random Forest and CatBoost were executed on the CPU. The MLP models were optimized using the Adam or AdamW optimizer (selected by Optuna), with full-batch updates. This choice was supported by preliminary experiments showing that mini-batch training (stochastic updates) increased gradient noise and led to less stable convergence and lower validation $R^2$ under the same training budget. Because the selected training subsets (up to 400 simulation cases for the MLP, limited by the available GPU memory) fit in GPU memory, full-batch optimization was adopted to obtain a deterministic objective during hyperparameter tuning and to maximize training stability and predictive accuracy.

All computations were performed in 32-bit precision (single precision), as increasing the numerical precision did not improve the results but led to higher demands on computational resources.

In this study, the uncertainty associated with surrogate predictions is treated as epistemic uncertainty, i.e., the approximation error arising from finite training coverage of the parameter/feature space. Since the training data are generated by deterministic FEM simulations, we evaluate this uncertainty empirically using two complementary views: (i) global generalization metrics on the independent test set (e.g., $R^2$, MAE), and (ii) spatial diagnostics via absolute error maps $\left|{\mathrm{Y}}_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}-{\mathrm{Y}}_{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}}\right|$, which highlight zones where approximation errors increase (typically near excavation boundaries and other high-gradient regions). The performance of the surrogate models was assessed and compared using five criteria:

• Global predictive accuracy: The primary metric was the coefficient of determination ($R^2$) computed on the held-out test set. The dependence of $R^2$ on the size of the training data was analyzed to construct learning curves and to identify the saturation point of accuracy [34].
• Accuracy in critical zones: To evaluate the ability of the models to predict material behavior near the limit state, an additional analysis was carried out. For a series of simulations with fixed excavation geometry but varying distance between the excavations, the model predictions were compared with reference values of ${\sigma }_{xx}$, ${\sigma }_{yy}$ and ${\tau }_{xy}$ at nodes that had entered the limiting shear state. These points were defined as nodes where the mobilized shear stress reached the ultimate shear strength according to the Mohr–Coulomb criteria (${\tau }_{\mathrm{rel}}=1$). The coordinates of these limiting-shear-state points were extracted directly from the reference FEM solutions and used as a targeted test set to evaluate whether the surrogate models could accurately recover stress fields in high-gradient, non-linear regions. The evaluation metric was the mean absolute error (MAE).
• Computational efficiency: Both training and inference (prediction) speed were assessed:
  • Training time required to fit the model on training sets of different sizes;
  • Inference time required to generate predictions of the stress–strain fields for a single simulation case from the test set. For the MLP and LightGBM models, these computations were performed on the GPU, whereas RF and CatBoost used the CPU.
• Model compactness: The physical size of the trained model file on disk (in megabytes) was measured as an indicator of storage and deployment requirements.
• Interpretability: The contribution of each input feature to the final prediction was quantified using the Permutation Feature Importance (PFI) method [34]. The method consists of measuring the drop in $R^2$ after random permutation of the values of a given feature on the test set.

3. Results

Figure 4 shows the dependence of $R^2$ on the training set size for each of the seven predicted stress–strain parameters, with separate curves for RF, LightGBM, CatBoost and MLP. The corresponding maximum $R^2$ values are summarized in

Table 5. Maximum coefficient of determination ($R^2$) values for each model and target parameter.

Model	Stresses			Strains			Utot
	σxx	σyy	σxy	εxx	εyy	γxy
LightGBM	0.912	0.967	0.914	0.601	0.865	0.702	0.998
MLP	0.778	0.920	0.893	0.536	0.826	0.690	0.984
RandomForest	0.892	0.960	0.919	0.594	0.865	0.716	0.998
CatBoost	0.910	0.967	0.914	0.579	0.859	0.685	0.998

.

The analysis demonstrates the high predictive capability of the developed surrogate models for the stress components (${\sigma }_{xx}$, ${\sigma }_{yy}$, ${\sigma }_{xy}$) and the total displacement $U_{tot}$. For these quantities, the tree-based models exhibit a consistent increase in $R^2$ as the size of the training set grows, with the accuracy approaching saturation at approximately 300–400 FEM simulations. When trained on the full dataset, high final $R^2$ values are achieved, in particular 0.967 for ${\sigma }_{yy}$ (LightGBM, CatBoost) and 0.998 for $U_{tot}$ (LightGBM, Random Forest, CatBoost).

A comparison of the algorithms shows that LightGBM, Random Forest and CatBoost provide similarly high performance for the stress components and $U_{tot}$. LightGBM yields the best results for ${\sigma }_{xx}$ and ${\epsilon }_{xx}$, Random Forest performs best for ${\tau }_{xy}$ and ${\gamma }_{xy}$, while for ${\sigma }_{yy}$, ${\epsilon }_{yy}$ and $U_{tot}$ the differences between the three tree-based models are negligible. The MLP model is consistently less accurate than the tree-based algorithms in all cases.

The lowest $R^2$ values are obtained for the strain components, particularly for ${\epsilon }_{xx}$, where the maximum $R^2$ is 0.601.

Figure 5 presents the dependence of model training time on the number of training samples. The plots reveal fundamental differences in the scalability of the considered algorithms. The maximum training times are summarized in

Table 6. Maximum training times (s) for each model and target parameter.

Model	Stresses			Strains			Utot
	σxx	σyy	σxy	εxx	εyy	γxy
LightGBM	164	166	157	131	162	171	168
MLP	1787	2189	2097	1252	1985	2604	1893
RandomForest	7790	7961	8111	1445	2402	3872	6477
CatBoost	249	251	249	121	191	243	271

.

For the gradient-boosting models, the increase in computational cost is the most predictable and efficient. The training time of LightGBM on the full dataset is on the order of ≈130–170 s, while CatBoost requires ≈120–270 s depending on the target parameter. Here, training time is defined as the wall-clock time required to fit the model on training sets of different sizes.

Random Forest proves to be the most computationally demanding algorithm: for some target parameters, the maximum training time reaches ≈7.8–8.1 × 10³ s, i.e., roughly an order of magnitude higher than for the gradient-boosting models.

The MLP model trains faster than the other algorithms for small and medium training set sizes; however, after a certain threshold (around 160–240 files) the training time increases by an order of magnitude and reaches ≈1.2–2.6 × 10³ s at the maximum dataset size. This behavior is likely related to the limitations of the available hardware resources and specifics of the training implementation.

In addition to accuracy and training time, an important practical aspect is the compactness of the trained models, i.e., the physical size they occupy on disk. Figure 6 shows the dependence of model size on the number of training samples, and the corresponding maximum values are reported in

Table 7. Maximum file sizes of trained models (MB) for each model and target parameter.

Model	Stresses			Strains			Utot
	σxx	σyy	σxy	εxx	εyy	γxy
LightGBM	23.3	24.3	23.5	18.0	23.3	25.3	23.7
MLP	0.6	0.4	0.6	0.4	0.3	0.6	0.5
RandomForest	24,716.1	21,406.5	45,946.1	31,379.1	12,435.2	37,147.7	29,298.6
CatBoost	140.9	143.9	139.9	67.4	107.3	134.0	161.7

. Model compactness is measured as the size of the serialized model file on disk (in megabytes), which serves as a proxy for storage and deployment requirements.

The MLP model is the most compact and is practically insensitive to the size of the training set: the file sizes remain in the range of ≈0.3–0.6 MB for all target parameters.

The gradient-boosting models differ substantially in compactness. The size of LightGBM models stabilizes at ≈23–25 MB, whereas for CatBoost the model size after an initial growth reaches ≈70–160 MB, depending on the target parameter.

In contrast, the Random Forest models exhibit a pronounced and almost linear increase in size with the number of training samples, reaching several gigabytes on the full dataset. The pronounced growth in Random Forest file size is largely driven by the optimized RF configuration: the hyperparameter search favored very deep trees (max_depth = 24–28; Table 4), which increases the number of split nodes and, consequently, the serialized model size (Table 7).

A key indicator governing the applicability of surrogate models for rapid assessment tasks is the inference speed, i.e., the time required to generate predictions for new data. Inference time is measured as the time required to generate predictions of the stress–strain fields for a single simulation case (all nodes) from the test set. Figure 7 compares the prediction times of models trained on datasets of different sizes with the reference time of a single FEM simulation. The maximum inference times are summarized in

Table 8. Maximum inference times (s) for each model and target parameter.

Model	Stresses			Strains			Utot
	σxx	σyy	σxy	εxx	εyy	γxy
LightGBM	0.292	0.264	0.251	0.140	0.163	0.138	0.229
MLP	0.004	0.007	0.004	0.004	0.004	0.004	0.004
RandomForest	0.429	0.225	15.967	1.744	0.111	8.474	1.425
CatBoost	0.134	0.135	0.150	0.071	0.109	0.139	0.136

.

The results indicate a substantial (one to two orders of magnitude) speed-up in prediction for most models compared with direct numerical simulation (4.86 s).

The highest and most stable prediction speed is achieved by the MLP: its inference time does not depend on the training set size and is ≈0.004–0.007 s, corresponding to a speed-up of about 700–1200 times relative to FEM.

The gradient-boosting models are also efficient. For LightGBM, the maximum inference time lies in the range ≈0.14–0.29 s, providing a speed-up of approximately 15–30 times. For CatBoost, the values are ≈0.07–0.15 s, which corresponds to a speed-up of about 30–70 times.

For Random Forest, a pronounced dependence of inference time on model size and complexity is observed. For some target parameters, the prediction time remains at 0.1–0.4 s (up to ≈40-fold speed-up), but for the shear components (${\tau }_{xy}$, ${\gamma }_{xy}$) it increases to 8–16 s, making the inference comparable to or even slower than a direct FEM simulation and severely limiting the practical applicability of this algorithm.

To assess the ability of the models to correctly predict material behavior near the limit state, an additional analysis was performed. The prediction accuracy was evaluated at nodes that reached the limiting shear state (${\tau }_{rel}\ge 1$), i.e., where the mobilized shear stress attained the ultimate Mohr–Coulomb shear strength, as identified from the reference FEM solution. The metric used was the mean absolute error (MAE), averaged over the three stress components.

Figure 8 shows the dependence of MAE on the distance between the excavations C. The training data covered the range $C\in \left[\mathrm{1,8}\right]$ m; therefore, the points at $C < 1$ m and $C > 8$ m characterize the behavior of the models under extrapolation.

Within the training range ($C\in \left[\mathrm{1,8}\right]$ m), the gradient-boosting models provide the best accuracy: the average MAE is 197.3 kPa for LightGBM and 217.1 kPa for CatBoost. Random Forest yields a slightly higher error level (245.52 kPa), and the worst result is obtained with MLP (358.2 kPa).

The behavior of the models outside the training range shows the expected degradation in accuracy. When extrapolating to small distances ($C < 1$ m), where stress concentrations and non-linear effects are most pronounced, a sharp increase in error is observed for all models; a similar trend is seen for $C > 8$ m.

To quantify the relative contribution of the input features to the model predictions, the Permutation Feature Importance (PFI) method was applied. The method consists of measuring the drop in $R^2$ after randomly permuting the values of a given feature on the test set; this drop is directly proportional to the importance of the feature. The results are shown in Figure 9, where the numerical values correspond to the normalized feature importance (in %).

The heatmaps reveal a high degree of consistency among the tree-based models (RF, LightGBM, CatBoost). For these algorithms, the main share of importance (on average about 70–75%) is attributed to the engineered features ($V\mathrm{ertical}\_\mathrm{Projection}$, $\mathrm{Signed}\_\mathrm{Dist}\_\mathrm{Norm}$, $\mathrm{Curvature}$, $\mathrm{Overlap}\_\mathrm{Index}$, etc.), whereas the raw coordinates X and Y account for the remaining ≈25–30%. For the normal components ${\sigma }_{xx}$, ${\sigma }_{yy}$, ${\epsilon }_{xx}$, ${\epsilon }_{yy}$, the contribution of the engineered features reaches 80–90%, while for the shear components and total displacement (${\tau }_{xy}$, ${\gamma }_{xy}$, $U_{tot}$) their share is about 55–60%. At the same time, global parameters such as $\mathrm{Mean}\_\mathrm{width}$, $\mathrm{Mean}\_\mathrm{height}$, $\mathrm{Aspect}1/2$, $\mathrm{Dist}\_\mathrm{norm}$, $\mathrm{Shift}\_\mathrm{norm}$ and $\mathrm{Area}\_\mathrm{ratio}$ show zero importance for all models.

In the updated setting, the MLP model preserves the general tendency of dominance of the engineered features, but redistributes importance differently. For most normal components (${\sigma }_{xx}$, ${\sigma }_{yy}$, ${\epsilon }_{xx}$, ${\epsilon }_{yy}$) and for the total displacement $U_{\mathrm{t}\mathrm{o}\mathrm{t}}$, the key predictor becomes $\mathrm{Overlap}\_\mathrm{Index}$, whose contribution reaches ≈45–55% of total importance, while the shares of $\mathrm{Signed}\_\mathrm{Dist}\_\mathrm{Norm}$ and $\mathrm{Vertical}\_\mathrm{Projection}$ are noticeably lower than in the tree-based models (typically 3–15%). For the shear components (${\tau }_{xy}$, ${\gamma }_{xy}$), the distribution is more balanced: X, $\mathrm{Vertical}\_\mathrm{Projection}$ and $\mathrm{Overlap}\_\mathrm{Index}$ all retain a substantial influence.

To qualitatively assess the ability of the surrogate models to approximate the spatial distribution of the stress–strain fields, a visual comparison was performed for a randomly selected test case. As an illustration, Figure 10 shows triptychs for ${\sigma }_{xx}$ for each algorithm. Each triptych includes the reference field from the FEM simulation, the field predicted by the model, and the map of absolute errors $\left|{\mathrm{Y}}_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}-{\mathrm{Y}}_{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}}\right|$. To improve readability and suppress the influence of rare outliers, the color scale on the error maps is clipped at the 99th percentile of the error distribution.

The obtained absolute error maps are generally consistent with the $R^2$ values (Table 4). The largest discrepancies for all models are localized in the immediate vicinity of the excavation boundaries, whereas in the bulk of the rock mass the errors remain relatively small. For LightGBM and Random Forest, the errors are mainly concentrated near the contours, with Random Forest providing a slightly more accurate reproduction of the fields; CatBoost is characterized by artifacts in the form of radially diverging “rays” emanating from the excavations. In the case of MLP, a fundamentally different pattern is observed: for ${\sigma }_{xx}$, the model fails to reproduce the detailed structure of the target field, and the predicted distributions are noticeably smoother than those of the tree-based models, which leads to an underestimation of local stress extrema around the excavations.

4. Discussion

The comparison shows that, in terms of the coefficient of determination $R^2$, the best performance is achieved by the gradient-boosting models (LightGBM, CatBoost) and Random Forest [39]. At the same time, the qualitative analysis of the fields revealed differences in the error patterns: for gradient boosting (CatBoost in particular), numerical artifacts were observed (radial “rays”, non-physical elongated zones of concentration), whereas Random Forest more frequently reproduced field patterns closer to the reference FEM solutions. The MLP exhibited smoothing, accompanied by a loss of local extrema.

From the viewpoint of computational characteristics, fundamental trade-offs were identified. LightGBM and CatBoost train rapidly on the full dataset (≈130–170 s and ≈120–170 s, respectively), as their effective complexity plateaus through sequential regularization, with model sizes of ≈18–25 MB (LightGBM) and ≈70–160 MB (CatBoost), and stable inference times of ≈0.14–0.29 s and ≈0.07–0.15 s. Random Forest proved to be the most resource-demanding: to capture steep stress gradients, hyperparameter optimization necessitated exceptionally deep trees which, under the bagging paradigm, leads to a near-linear growth in split nodes. Consequently, training time reaches ≈7.8–8.1 × 10³ s, model size grows to tens of gigabytes, and inference time for some target variables increases to 8–16 s, making these configurations comparable to or slower than a direct FEM run (4.86 s), which negates the advantage of using surrogates for real-time mining monitoring tasks. The MLP provides the lowest inference latency (≈0.004–0.007 s, a speed-up of about 700–1200 times relative to FEM) by utilizing optimized matrix operations on the GPU, with very compact models (≈0.5 MB) due to its fixed architectural weight count, but it is less accurate than the tree-based models.

The permutation feature importance analysis showed that the tree ensembles rely primarily on engineering features, validating the proposed physics-informed approach. For the vertical components (${\sigma }_{xx}$, ${\epsilon }_{yy}$), $\mathrm{Vertical}\_\mathrm{Projection}$ dominates (in total ≈ 50–65%); for ${\sigma }_{yy}$, ${\epsilon }_{yy}$, $\mathrm{Curvature}$ and $\mathrm{Signed}\_\mathrm{Dist}\_\mathrm{Norm}$ are particularly important; for ${\tau }_{xy}$, ${\gamma }_{xy}$, the contribution of X is large; and for $U_{\mathrm{t}\mathrm{o}\mathrm{t}}$, Y and $\mathrm{Vertical}\_\mathrm{Projection}$ play a leading role. The high importance of geometric and positional features is typical for stability assessment problems and is consistent with previous studies [14,16,40].

The dependence of accuracy on the data volume exhibits saturation for most targets at ≈300–400 simulations (30–40% of the dataset); further increases in $N$ yield only limited improvements in R². For the strains, especially ${\epsilon }_{xx}$, the achieved values are lower. Strains are spatial derivatives of displacements and therefore amplify approximation errors, making them more sensitive to the smoothing inherent to surrogate regression. Moreover, once yielding occurs, stresses are constrained by the yield surface and evolve more smoothly, while plastic strains can accumulate and develop sharper gradients in localized zones, which are harder to approximate. However, for the intended purpose of rapid screening, the accuracy is sufficient: the models correctly identify the location and relative magnitude of deformation zones, allowing engineers to filter out high-risk scenarios before detailed FEM verification. The analysis of accuracy in limiting-shear-state points (${\tau }_{rel}\ge 1$) confirmed that prediction errors increase significantly when the distance between excavations C goes beyond the training range ($C\in \left[\mathrm{1,8}\right]$ m). This highlights the limited ability of the models to approximate limit-state behavior outside the training domain—a critical consideration for safety assessments in mining design when solving extrapolation problems. Consequently, given that all considered data-driven models exhibited a significant drop in accuracy outside the training range, their application in safety-critical design must be strictly limited to the interpolation domain. Comparative analysis of architectures with superior extrapolation capabilities remains a subject for future research. Additionally, the study was deliberately limited to geometric factors. While geological uncertainty (e.g., GSI, rock strength, and excavation depth) is critical, its inclusion would exponentially increase dimensionality; extending the framework to include these stochastic parameters is a key direction for future research.

From a practical point of view, for rapid assessment and multi-scenario analysis, CatBoost and LightGBM are optimal in terms of the combination of accuracy, training time and model size, whereas MLP provides the lowest inference latency provided that adequate regularization is applied. Although Random Forest can reproduce the stress–strain fields more faithfully than gradient boosting, its use is not justified for large datasets.

The practical advantages of the surrogate models are expected to be most significant for three-dimensional layouts, where a single FEM simulation may require hours or days. In such cases, surrogates can replace FEM in tasks involving a large number of forward evaluations, while FEM is retained for detailed verification. Typical examples include probabilistic analyses of pillar stability, optimization of mining excavation layouts, and routine screening of design alternatives within Digital Twin frameworks. For these applications, increasing the dimensionality of the problem primarily affects the cost of generating the training dataset, whereas the prediction time of the LightGBM, CatBoost, and MLP models remains essentially constant. Although the current study focuses on the geometric interaction of unsupported excavations, the proposed feature engineering framework is adaptable. Future iterations can incorporate ground support systems (e.g., rock bolts, shotcrete) by introducing additional input descriptors representing support stiffness and density.

5. Conclusions

This study has demonstrated and systematically evaluated the applicability of classical tabular machine-learning algorithms for constructing surrogate models capable of predicting, with high accuracy and speed, the full stress–strain fields in a rock mass around a system of two interacting mining excavations.

It has been confirmed that tree ensembles (Random Forest, LightGBM, CatBoost) provide high predictive accuracy, reaching $R^2>0.967$ for the stress components and $R^2>0.998$ for the total displacements. The most important factor for model accuracy is the use of engineering features, whose importance is consistent with the physical pattern of stress distribution in fractured rock masses treated as an equivalent continuum.

The comparative analysis revealed fundamental trade-offs between the algorithms suitable for intelligent design:

• The MLP offers the lowest inference latency (≈0.004–0.007 s, a speed-up of 700 to 1200 times relative to FEM) with compact models, but produces smoother fields with partial loss of local features.
• LightGBM and CatBoost provide the most balanced combination of accuracy and computational cost ($R^2$close to Random Forest, speed-up of 15 to 70 times relative to FEM), making them ideal for iterative design optimization.
• Random Forest yields fields closest to reference FEM solutions but is too resource-demanding for large-scale deployment.

It has been established that model accuracy saturates at ≈300–400 parametric simulations, allowing for the optimization of computational costs during data preparation. The analysis of the mean absolute error (MAE) at limiting-shear-state points (${\tau }_{rel}\ge 1$) showed that within the training range ($C\in \left[\mathrm{1,8}\right]$ m), LightGBM attains the lowest error values. Outside this range, the error increases systematically, reflecting the limitations of data-driven models in extrapolation. Given the observed smoothing of local stress extrema, the proposed surrogate models are intended for preliminary design screening and topology optimization. They do not replace rigorous FEM analysis for the final verification of safety-critical parameters.

Overall, the proposed approach represents an effective tool for rapid stress–strain assessment at the design stage, substantially reducing computational costs. These properties make the proposed surrogates suitable as core components of probabilistic and optimization-based design workflows for underground mining. Large ensembles of scenarios can be evaluated using surrogate models, while detailed FEM analyses are reserved for critical cases. This division of roles allows engineers to account for parameter uncertainty and explore alternative layouts systematically, supporting the transition towards AI-assisted intelligent mining.