Comparative Evaluation of Feed-Forward Neural Networks for Predicting Uniaxial Compressive Strength of Seybaplaya Carbonate Rock Cores

Abstract

Accurate estimation of the uniaxial compressive strength (UCS) of carbonate rocks underpins safe design and stability assessment in karst-influenced geotechnical projects. This work presents a comprehensive evaluation of four feed-forward artificial neural network (ANN) architectures—radial basis function (RBF), Bayesian regularized (BR), scaled conjugate gradient (SCG), and Levenberg–Marquardt (LM)—to predict UCS from three readily measured variables: water content, interconnected porosity, and real density. Fifty core specimens from the Seybaplaya quarry in Campeche, Mexico, were split into training and testing subsets under uniform preprocessing. Each model’s predictive performance was assessed over 30 independent runs using mean absolute error, root mean squared error, and coefficient of determination, with statistical differences tested via nonparametric hypothesis testing. The RBF network achieved the highest median R² and significantly outperformed the other variants, while the BR model demonstrated robust generalization. SCG and LM converged faster and efficiently but with slightly lower accuracy. Sensitivity analysis identified interconnected porosity as the primary predictor of UCS. These results establish RBF-based ANNs with appropriate regularization and feature importance assessment as a novel, practical, and reliable framework for UCS prediction in heterogeneous carbonate formations.

1. Introduction

Accurate prediction of the uniaxial compressive strength (UCS) of rock masses is essential for the safe and cost-effective design of a wide range of geotechnical and civil infrastructure, including foundations, tunnels, and slope stability assessments. Traditional empirical and regression-based models—linking UCS to parameters such as porosity, density, or ultrasonic pulse velocity—provide valid first-order estimates but frequently underperform when applied to heterogeneous, partially saturated, or karst-affected carbonate formations. This performance gap stems from the rigid functional forms and limited textural variability assumed in these models, which fail to capture complex pore structure–mechanical strength relationships observed in field settings [1,2,3]. In the Seybaplaya bank rocks of Campeche, Mexico—a formation characterized by mixed bioclastic limestones, coquinas, and breccias with wide-ranging water content, interconnected porosity, and skeletal density—such limitations become particularly acute, leading to potential under- or overestimation of rock strength and consequent engineering risk.

Recent advances in machine learning offer promising alternatives: artificial neural networks (ANNs) can automatically learn complex, nonlinear interactions among multiple geomechanical inputs without requiring explicit correlation assumptions. Multiple studies have demonstrated that ANNs, when trained with features such as water content, porosity, and density, achieve markedly improved predictive accuracy for UCS across diverse lithologies and saturation states [4,5]. Further, hybrid frameworks—ranging from adaptive neuro-fuzzy inference systems (ANFISs) to particle swarm optimization (PSO)-tuned networks—underscore the flexibility and robustness of data-driven approaches in rock mechanics contexts.

Research Gap. Despite these successes, there remains a notable lack of systematic, side-by-side evaluations comparing the effects of different ANN training algorithms and regularization schemes on predictive performance for karst-influenced carbonate rocks. Specifically, the impact of convergence speed, overfitting control, and noise resilience across algorithmic variants—radial basis function (RBF), Bayesian regularization (BR), scaled conjugate gradient (SCG), and Levenberg–Marquardt (LM)—has not been fully elucidated for formations similar to Seybaplaya. Without such comparative insight, practitioners lack clear guidance on selecting the most appropriate network architecture for reliable UCS estimation under field-analog conditions.

Contributions of This Study. To address this gap, the present work delivers the following key contributions:

• Direct Algorithmic Comparison. We implement and rigorously compare four feed-forward ANN architectures (RBF, BR, SCG, LM) on a uniform dataset of 50 Seybaplaya carbonate core specimens, leveraging identical preprocessing pipelines, data split protocols, and performance metrics.
• Statistical Validation. Through 30 independent training–testing runs per model and hypothesis testing via the Friedman test with Benjamini–Hochberg correction, we objectively determine the statistical significance of inter-algorithm performance differences.
• Feature Importance Analysis. Employing a partial derivatives sensitivity analysis, we quantify the relative influence of water content, porosity, and density on UCS predictions, providing actionable insight into variable prioritization for field measurements.
• Practical Recommendations. We synthesize our findings into clear recommendations for selecting ANN training strategies that balance predictive accuracy, convergence efficiency, and generalization robustness in karst-affected carbonate engineering applications.

This work advances the state of geomechanical strength prediction for heterogeneous carbonate formations by delivering the following key contributions:

1.1. Comprehensive Multi-Algorithm Comparison

We implement and rigorously compare four distinct feed-forward neural network architectures: radial basis function (RBF), Bayesian regularized (BR), scaled conjugate gradient (SCG), and Levenberg–Marquardt (LM), on the same Seybaplaya carbonate rock dataset. This direct side-by-side evaluation under uniform preprocessing and performance metrics has not been previously reported for these formations.

1.2. Statistically Validated Performance Ranking

By executing each model over 30 independent runs and applying the Friedman test with Benjamini–Hochberg correction, we objectively establish which algorithms differ significantly in predictive accuracy (median test-set MSE) and which perform comparably. This statistical layer provides practitioners with confidence bounds rather than anecdotal rankings.

1.3. Sensitivity-Driven Feature Importance Analysis

Leveraging Dimopoulos et al.’s partial derivatives method, we quantify the relative influence of water content, interconnected porosity, and real density on UCS predictions. Interconnected porosity emerges as the dominant driver (54.4%), followed by water content (30.9%) and density (14.7%), offering geotechnical insights into key weakening mechanisms.

1.4. Guidelines for Model Selection in Karst-Influenced Carbonates

Our results demonstrate that RBF networks deliver the highest overall accuracy (median R² = 0.975; RMSE = 1.313 MPa), while Bayesian regularization offers superior robustness to noise. Although faster to converge, SCG and LM methods exhibit slightly lower predictive power. These findings inform academic researchers and field engineers on optimal ANN choices for similar geological contexts.

1.5. Data Resource for Seybaplaya Formation

We present a meticulously curated database of 50 core specimens with water content, porosity, density, UCS measurements, and standardized testing protocols. This dataset supports reproducibility and future benchmarking in carbonate rock mechanics.

Through these contributions, the study identifies the most effective ANN strategies for UCS estimation in Seybaplaya carbonates and establishes a transparent, statistically grounded framework for future machine learning applications in geomechanics.

The paper is organized as follows. Section 2 reviews related works on UCS prediction models and ANN applications in rock mechanics. Section 3 details the materials and methods, covering study site location and sampling, rock material testing procedures, and dataset compilation. Section 4 introduces the four feed-forward neural network architectures implemented for UCS prediction. Section 5 outlines the performance evaluation metrics used to assess model accuracy and robustness. Section 6 presents the results and discussion: it begins with descriptive statistics of the experimental dataset, followed by model implementation and testing protocols for each ANN variant (RBF, Bayesian regularized, scaled conjugate gradient, and Levenberg–Marquardt), offers a comparative performance overview and sensitivity analysis, and concludes with a summary of key findings, practical implications, and suggestions for future research.

2. Related Works

2.1. Empirical and Regression-Based Models

Empirical correlations link UCS to porosity, density, or ultrasonic velocity, confirming negative porosity and positive density influences [6,7,8,9]. However, these models assume uniform textures and often fail in heterogeneous, water-saturated, or karst-affected carbonates like Seybaplaya.

2.2. Water Content, Porosity, and Density Effects

Water content reduces effective cohesion and increases pore pressure; porosity introduces voids that act as crack sites; and density reflects load-bearing capacity [10,11,12,13,14,15]. In karst-influenced formations, mineralogical and textural heterogeneity scatter simple UCS–property relationships.

2.3. Neural Network Applications in Geomechanics

ANNs capture nonlinear, multivariate interactions without predefined relationships [16,17]. Bayesian regularization [18] and Levenberg–Marquardt [19] ANNs outperform MLR with higher R² and lower RMSE in sedimentary and carbonate rocks [20].

2.4. Hybrid and Optimization Frameworks

ANFIS models achieve > 90% variance explanation for UCS and elastic modulus [21,22,23,24]. PSO-optimized ANNs yield R² > 0.95 on shale and sandstone [25,26,27,28]. Petrographic classification with sensitivity-driven feature selection produces parsimonious models (R² > 0.91) [29], and hybrid architectures with advanced input selection further improve accuracy [30,31,32,33].

2.5. Comparative Machine Learning and Regression Studies in Geomechanics

Sabri et al. [34] compared MLR, SVR, Bi-LSTM, ANN-TLBO, and ANN-PSO on 54 petrographic samples; ANN-PSO achieved the highest training (R² = 0.9911) and testing (R² = 0.9868) metrics, highlighting quartz content’s dominance. Kochukrishnan et al. [35] showed that optimally tuned step-wise regression (R² ≈ 0.988/0.990) rivals linear regression (R² ≈ 0.980/0.986). Genetic and programming methods (GEP, LGP) outperform ANNs in model parsimony and MAE, while ELM excels over GEP and LSSVM (R ≈ 0.9642) with robust validation. Overall, hybrid ML, optimized regression, and genetic-programming approaches consistently yield high accuracy, but systematic, side-by-side evaluation of different ANN training algorithms for karst-influenced carbonates remains an open research need.

3. Materials and Methods

3.1. Study Site Location and Sampling

Under the new Mexican Mining Law, effective 24 September 1992, most mineral substances are now subject to analysis (Servicio Geológico Mexicano, 2021) [36]. Mining concessions are registered and administered through the Regional Delegation of the General Directorate of Mines in Puebla, Puebla, which oversees a mining office in Campeche [36].

The town of Seybaplaya (Campeche) lies within the “Yucatán Platform”, an extensive marine-sedimentary bedrock province on the Yucatán Peninsula. Formed by millions of years of marine sediment accumulation, this platform reaches an average thickness of approximately 200 m [36].

Locally, the Seybaplaya outcrop sequence (Qpt Cq–Cz) comprises bioclastic limestones, polymictic conglomerates, coquina, and calcareous breccia horizons. These units form an elongated, shore-parallel body belt roughly 15 km long and 1–2 km wide. The best exposures appear as subhorizontal to gently seaward-dipping strata, particularly near the town of Seybaplaya and along Federal Highway 180 between Haltunchén, Villa Madero, and Seybaplaya (Campeche Geological-Mining Map 1.00 × 10⁻³; 1:250,000).

This study focuses on the lithological and geomechanical characteristics of the Seybaplaya rock bench and its immediate surroundings, where an active quarry is located, as illustrated in Figure 1.

3.2. Rock Material Testing and Database

3.2.1. Quarry Exploration

The quarry known as “Seybaplaya bank rocks” shown in Figure 2 is strategically important due to its proximity to the deep-water port. This facilitates the transport of the crushed aggregates to markets in other Mexican states. These bank rocks yield various sizes of construction-grade aggregates suitable for building foundations and formulating asphalt mixtures for roadway applications.

3.2.2. Sample Size

In studies where the primary variable is qualitative and is reported as the proportion of the phenomenon in the reference population, the sample size is calculated using the following equation:

$$n={\displaystyle \frac{{z_{\propto }}^2\ast p\ast q}{e^2}}$$

(1)

where

n = sample size;

$z_{\propto }$ = statistical parameter corresponding to the chosen confidence level;

e = allowable estimation error;

p = probability of success (occurrence of the event under study);

q = 1 − p, the probability of failure (non-occurrence of the event under study).

For populations of unknown size or exceeding 10,000, the required sample size was calculated by assuming a maximum variance of p = 50% and q = 50% (no prior estimate of the event’s prevalence). A 90% confidence level was adopted, corresponding to ($z_{\propto }$ = 1.645).

$$n={\displaystyle \frac{{1.645}^2\ast 50\ast 50}{{12}^2}}=46.979$$

Therefore, a total of 50 samples were collected. Five sampling locations were chosen from the previously blasted bench, and their coordinates are recorded in

Table 1. Locations of the 50 samples.

Sample Range	UTM Coordinates (X, Y)
Samples 1–10	(741595, 2174717)
Samples 11–20	(741611, 2174838)
Samples 21–30	(741585, 2174851)
Samples 31–40	(741591, 2174750)
Samples 41–50	(741605, 2174803)

.

3.2.3. Laboratory Testing Methods

Rock cores were prepared and their dimensions and geometry verified in strict accordance with ASTM D4543-12 [37], which requires specimens to be straight, circular cylinders conforming to the following tolerances:

Length-to-Diameter Ratio: Between 2.0 and 2.5.

Minimum Diameter: [Specify análisi diameter in mm].

End Surfaces: Cylinder end faces shall be lapped or polished to produce flat surfaces within a flatness tolerance of 0.00254 cm.

These preparations guarantee consistency and precision in all subsequent tests, as demonstrated in Figure 3.

To characterize the properties of the rock bench, the following sequence of laboratory tests was implemented:

3.2.4. Uniaxial Compressive Strength (UCS)

The uniaxial compressive strength test (UCS) was performed according to ASTM D7012-23 [38]. The UCS was computed by dividing the peak axial load at failure by the specimen’s cross-sectional area.

$$UCS={\displaystyle \frac{P}{A}}$$

(2)

where

UCS = Uniaxial Compressive Strength.

P = Axial load.

A = Specimen’s cross-sectional area.

Testing was carried out on a Model 2000 Universal Testing Machine (Serial No. 011065) using standard uniaxial compression protocols.

Measure and record the specimen’s dimensions to calculate its cross-sectional area.

Verify that the Universal Testing Machine is properly zeroed and configured before testing.

Position the specimen centrally between the compression platens as shown in Figure 4.

Use the machine control software to define the test parameters and initiate the compression sequence.

Apply the axial load gradually at a constant rate until specimen failure.

Load the specimen to failure, as evidenced by crack initiation as illustrated in Figure 5.

Upon failure, remove the specimen, install a fresh sample, and repeat the procedure.

3.2.5. Water Content

This test determines the rock’s in situ moisture content—an indicator of retained water influenced by bench conditions, ambient humidity, and pore connectivity—according to ASTM D2216-10 [39], as shown in Figure 6.

$$\% Water Content={\displaystyle \frac{p_i-p_f}{p_f}}\times 100$$

(3)

where

p_i = initial (wet) mass of the sample, in grams;

p_f = final (oven dry) mass of the sample, in grams;

An average moisture content is calculated from multiple representative specimens. The step-by-step procedure for determining moisture content is as follows:

Sample preparation;

Initial weighing (wet weight);

Oven drying: maintain 105 ± 5 °C for 24 h;

Final weighing (dry weight);

Calculation of water content.

Figure 6. Procedure for determining the water content of rock samples.

Figure 6. Procedure for determining the water content of rock samples.

3.2.6. Real Density Test

The real density of the rock was measured by ASTM D854-23 [40], as depicted in Figure 7. This method defines true density as the ratio of the specimen’s actual mass to its true volume and calculates it using the following equation:

$$\mathsf{\gamma }={\displaystyle \frac{m}{V}}$$

(4)

where

γ = real density (g/cm³);

m = mass of the rock (g);

V = comprehensive analysis of the specimen (cm³).

Figure 7. Real density determination of rock specimens: (A) measurement of specimen mass; (B) determination of specimen volume.

Figure 7. Real density determination of rock specimens: (A) measurement of specimen mass; (B) determination of specimen volume.

3.2.7. The Procedures for Determining the Rock’s True Density

Weigh the sample and record its mass (m) in grams;

Measure and record the sample volume (V) in cubic centimeters (cm³).

3.2.8. Interconnected Porosity

ASTM D4404-18 [41,42] specifies the standard procedure for determining rock porosity, as visualized in Figure 8, expressed by the following equation:

$$Pore\%={\displaystyle \frac{Porevolume}{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}}\times 100$$

(5)

where

Pore volume (V_p) = volume of fluid absorbed by the rock sample;

Total volume (V_t) = the rock sample’s overall volume.

Figure 8. Determination of interconnected porosity: (A) measurement of pore volume via fluid uptake; (B) determination of total sample volume.

Figure 8. Determination of interconnected porosity: (A) measurement of pore volume via fluid uptake; (B) determination of total sample volume.

3.2.9. Database

Tests were performed on 50 cylindrical specimens (diameter = 5.08 cm; length-to-diameter ratio = 2–2.5), and for each sample, we recorded uniaxial compressive strength (UCS, MPa), moisture content (%), real density (g/cm³), and porosity (%) [43].

4. Neural Network Architectures for Uniaxial Compressive Strength Prediction

We implemented and compared four feed-forward neural network architectures to capture the complex, nonlinear dependence of uniaxial compressive strength (UCS) on water content, porosity, and density. Each model employs a distinct training algorithm and regularization scheme, which allows for a systematic evaluation of their convergence rates, generalization performance, and robustness to measurement noise:

4.1. Radial Basis Function (RBF) Neural Network

An RBF network is a single-hidden-layer feed-forward model that uses Gaussian basis functions centered on training patterns. Hidden neurons are added incrementally until a performance goal on the training set is achieved, and the output layer linearly combines these localized responses to predict UCS. RBFs train quickly and model highly nonlinear mappings, but they require careful tuning of the spread parameter to balance bias and variance [44].

4.2. Bayesian Regularized Feed-Forward Neural Network

This approach incorporates weight decay into a Bayesian framework, ensuring that the objective minimizes both squared error and a complexity term. By utilizing the Levenberg–Marquardt algorithm, hyperparameters that govern the balance between error and weight penalty are updated during training, resulting in smoother, more generalizable models without a held-out validation set. This method is particularly effective with noisy or limited data, albeit at the expense of increased per-epoch computation [45].

4.3. Scaled Conjugate Gradient (SCG) Neural Network

SCG is a second-order method that scales conjugate gradient directions using approximate curvature information, avoiding expensive line searches, thus converging faster than simple gradient descent. In practice, a one-hidden-layer net (15 neurons) is trained over 1000 epochs with a 70% train/15% validation/15% test. SCG balances speed and robustness for moderate-sized problems [46].

4.4. Levenberg–Marquardt (LM) Neural Network

LM blends Gauss–Newton and gradient-descent updates by adjusting a damping parameter μ: it reduces μ when the quadratic approximation holds and increases it when steps overshoot. When applied to a 15-neuron hidden layer, it typically converges in far fewer epochs than first-order methods. Regularization can be added for stability; however, its high memory demands limit the use of very large networks [47].

5. Performance Evaluation Metrics

To assess the accuracy of our machine learning models, we employ four standard measures: Mean Absolute Error (MAE), Mean Square Error (MSE), Mean Absolute Percentage Error (MAPE), and the Coefficient of Determination (COD).

5.1. Mean Absolute Error (MAE)

This metric captures the average absolute difference between predicted values and their authentic counterparts [48]. It is defined as follows:

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{n}}\sum _{i=1}^n |y_i-{\widehat{y}}_i|$$

(6)

Here, $y_i$ denotes the observed (true) value, ${\widehat{y}}_i$ represents the model’s predicted value, and $n$ is the total number of observations.

5.2. Root Mean Square Error (RMSE)

RMSE measures the square root of the average squared deviations between predictions and observations [49]:

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n (y_i-{\widehat{y}}_i{)}^2}$$

(7)

5.3. Mean Absolute Percentage Error (MAPE)

MAPE expresses the average absolute error as a percentage of the true values and its predicted values [50], and is calculated as follows:

$$MAPE={\displaystyle \frac{1}{n}}\sum _{i=1}^n {\displaystyle \frac{|y_i-{\widehat{y}}_i|}{y_i}}\times 100\%$$

(8)

5.4. Coefficient of Determination (COD)

Also known as R², COD indicates the fraction of the variance in the dependent variable that is predictable from the independent variables [51]. It is calculated by:

$$COD=1-{\displaystyle \frac{\sum _{i=1}^n (y_i-{\widehat{y}}_i{)}^2}{\sum _{i=1}^n (y_i-\overline{y}{)}^2}}$$

(9)

6. Results and Discussion

6.1. Descriptive Statistics of the Experimental Dataset Variables

Table 2. Descriptive statistics of input features (water content, porosity, density) and target UCS for neural network modeling.

	Water Content	Interconnected Porosity	Real Density	UCS
Mean	6.707	15.459	2.352	40.14
Std_dev	2.335	4.377	0.159	7.914
Min	3.73	9.73	1.91	11.4
1st_quartile	5.0	12.35	2.262	34.932
Median	6.235	14.565	2.355	40.485
3rd_quartile	7.552	17.168	2.478	44.845
Max	13.62	28.46	2.62	54.17

presents the descriptive statistics for the water content (%), interconnected porosity (%), real density (g/cm³), and uniaxial compressive strength (UCS, MPa). The mean of water content is 6.707% (SD = 2.335%), ranging from 3.73% to 13.62%, which indicates moderate variability in sample moisture levels. Interconnected porosity averages 15.459% (SD = 4.377%), with values spanning 9.73% to 28.46%, reflecting heterogeneity of specimens. Real density shows a tight distribution (mean = 2.352 g/cm³, SD = 0.159 g/cm³; min = 1.91 g/cm³, max = 2.62 g/cm³), suggesting relatively consistent lithological composition. The uniaxial compressive strength (UCS) displays significant variability (mean = 40.14 MPa, SD = 7.914 MPa; min = 11.4 MPa, max = 54.17 MPa), underscoring a wide range of mechanical strength. The interquartile ranges, water content (5.00–7.552%), porosity (12.35–17.168%), density (2.262–2.478 g/cm³), and UCS (34.932–44.845 MPa), further illustrate the data distribution and substantiate the inclusion of all four variables in the subsequent machine learning models.

Figure 9 illustrates the frequency distributions of the four principal variables—moisture content, interconnected porosity, real density, and uniaxial compressive strength (UCS)—employed in this study.

• Water Content (%): The moisture content exhibits an approximately bell-shaped distribution centered at 6–7%, with the majority (≈70%) falling between 4% and 9%. A slight rightward skew is apparent, as a small number of specimens reach up to 13–14%. This skew suggests occasional high-moisture outliers, which may influence rock weakening.
• Interconnected Porosity (%): Porosity values are concentrated between 10% and 20%, peaking around 12–15%. The distribution is mildly right-skewed, with a tail extending toward 28%, indicating that most rocks share similar characteristics, while a few exhibit significantly higher void space.
• Real Density (g/cm³): Density measurements cluster tightly between 2.2 and 2.5 g/cm³, with a modal of 2.3–2.4 g/cm³ and very few values below 2.1 g/cm³ or above 2.6 g/cm³. This narrow spread reflects a relatively homogeneous lithology across the rock bank.
• Uniaxial Compressive Strength (UCS, MPa): The strength distribution spans 11 MPa to 60 MPa, with about ≈75% between 30 MPa and 50 MPa. The histogram shows a mild right skew, driven by a few robust specimens. The central tendency around 35–45 MPa underscores the variability in mechanical resistance, which is essential for robust predictive modeling.

Overall, these histograms reveal that real density is relatively uniform. In contrast, water content, porosity, and UCS exhibit greater variability and mild skewness—distributional characteristics that underscore the need for machine learning algorithms capable of modeling nonlinear relationships and robustly handling outliers in our predictive framework.

Figure 10 depicts the pairwise relationships among water content (%), interconnected porosity (%), real density (g/cm³), and uniaxial compressive strength (UCS, MPa) in a matrix of scatterplots with histograms on the diagonal. The following observations emerge:

6.1.1. Diagonal Histograms

• Water content concentrates between 4% and 9%, confirming the moderate moisture range noted previously.
• Porosity predominantly ranges from 12% to 17%, with only a few samples exhibiting values above 25%.
• Density is tightly clustered near 2.3–2.5 g/cm³.
• UCS shows a dominant band from 30 MPa to 50 MPa, matching the earlier histogram.

6.1.2. Water Content vs. Porosity

• A strong positive linear (r ≈ 0.85) indicates that wetter specimens tend to exhibit higher porosity, likely reflecting pore filling by moisture (upper-left scatter).

6.1.3. Water Content vs. Density

• A moderate negative correlation (r ≈ –0.65) indicates that samples with higher moisture content generally exhibit lower dry density (lower-left cluster on the scatter plot), as expected when porosity increases

6.1.4. Water Content vs. UCS

• A weak to moderate negative correlation indicates that higher moisture content is generally associated with reduced mechanical strength. However, the considerable scatter suggests that additional factors also affect UCS.

6.1.5. Porosity vs. Density

• Porosity and density are inversely related (r ≈ –0.72), confirming that greater void space corresponds to lower bulk density (second-row, first-column scatter).

6.1.6. Porosity vs. UCS

• A negative correlation (r ≈ –0.78) indicates that samples with higher porosity exhibit lower UCS, underscoring porosity’s critical role in strength reduction.

6.1.7. Density vs. UCS

• A strong positive correlation (r ≈ 0.80) indicates that denser rocks resist compressive loading more effectively, making density one of the most predictive features for UCS (bottom-right scatter).

These pairwise patterns confirm the expected geomechanical interdependencies—moisture and porosity weaken the rock, whereas density enhances strength—and validate the selection of these four variables as primary inputs for subsequent machine learning models.

Figure 11 presents the pairwise Pearson correlation coefficients between moisture content (%), interconnected porosity (%), real density (g/cm³), and uniaxial compressive strength (UCS, MPa). The color-coded matrix, with overlaid coefficient values, highlights the following key relationships:

• Water Content vs. Porosity (r = 0.992): A near-perfect positive correlation indicates that specimens with higher moisture content exhibit increased interconnected porosity, reflecting the pore network’s enhanced fluid-retention capacity.
• Moisture Content vs. Real Density (r = –0.861): A strong inverse relationship indicates that specimens retaining more moisture exhibit lower density, consistent with increased pore volume reducing the mass per unit volume.
• Moisture Content vs. UCS (r = –0.649): A moderate inverse correlation indicates that higher moisture content typically reduces compressive strength, although additional factors also influence UCS variability.
• Porosity vs. Density (r = –0.805): Porosity and density are strongly inversely related, confirming that an increased void fraction corresponds to reduced material compactness.
• Porosity vs. UCS (r = –0.568): A moderate negative correlation indicates that increased porosity compromises compressive strength, underscoring pore structure as a primary weakening mechanism.
• Density vs. UCS (r = 0.929): A robust positive correlation indicates denser rocks resist compressive loading more effectively, making real density the most predictive univariate feature for UCS.

These findings validate the anticipated geomechanical interdependencies and support the inclusion of all four variables in our multivariate predictive models.

On the other hand, Figure 12 shows a parallel coordinate plot with four vertical axes—water content (%), interconnected porosity (%), real density (g/cm³), and uniaxial compressive strength (UCS, MPa) simultaneously mapping each sample’s values to highlight multidimensional relationships and sample heterogeneity.

• High-strength subset (rightmost lines): Samples with UCS > 45 MPa (the upper bundle on the right axis) consistently correspond to high real density (2.5–2.7 g/cm³) and low moisture content (3–6%) and porosity (9–15%), reaffirming that dense, low-porosity rocks exhibit superior compressive resistance.
• Low-strength subset: Samples with UCS < 25 MPa align with lower real density (1.9–2.2 g/cm³) and elevated moisture content (8–14%) and porosity (18–28%), highlighting how increased pore volume and moisture compromise compressive strength.
• Intermediate cluster: The majority of samples fall within mid-range values—moisture content 5–8%, porosity 12–18%, real density 2.3–2.5 g/cm³, and UCS 30–45 MPa—reflecting the dataset’s central tendency and confirming these intervals as representative for model training.
• Outliers: A few lines diverge sharply, such as one sample with exceptionally high porosity (>25%) yet moderate UCS (~30 MPa), suggesting localized lithological variations or measurement anomalies worth further geological investigation.

Overall, the parallel coordinate plot visually confirms the inverse relationship between moisture content/porosity and density/UCS, clearly delineating distinct strength–density–porosity regimes that inform feature selection and stratified modeling strategies in our subsequent machine learning workflows.

6.2. Model Implementation and Training Protocols

To ensure a fair comparison of four neural network models for predicting uniaxial compressive strength (UCS) from moisture content, interconnected porosity, and real density, each algorithm was implemented in MATLAB R2024a using standardized preprocessing, consistent performance metrics, and carefully selected hyperparameters.

Table 3. Summary of Model Configurations, Training Functions, and Evaluation Criteria.

Model	MATLAB Function	Topology	Data Split	Preprocessing	Key Hyperparameters	Performance Metrics
RBF network	newrb	Up to 25 Gaussian units	80% train/20% test	Inputs z-score; outputs mapminmax [0,1]	Spread = 0.8; goal MSE = 1 × 10−3	RMSE, R2
Bayesian regularized NN	trainbr	1 hidden layer, 15 neurons	80%/20%	z-score; mapminmax	Max epochs = 1 000; α, β auto-tuned	MAE, COD
SCG network	trainscg	1 hidden layer, 15 neurons	70%/15%/15%	z-score; mapminmax	λ0 = 1.0 × 10−3;σ = 1.0 × 10−6; reg = 0.01	RMSE, MAPE
LM network	trainlm	1 hidden layer, 15 neurons	70%/15%/15%	z-score; mapminmax	μ0 = 1.0 × 10−3; μ↓ = 0.1; μ↑ = 10; reg = 0.1	COD, RMSE

summarizes the data splits, network architectures, training functions, and evaluation metrics; detailed descriptions follow.

6.2.1. Radial Basis Function (RBF) Neural Network

Implemented with MATLAB’s newrb, the network grew hidden Gaussian units one at a time (up to 25) until the training MSE fell below 1 × 10⁻³ (spread = 0.8). Inputs were standardized (z-score) and outputs normalized to [0,1] via mapminmax. The model achieved R² = 0.972 and RMSE = 1.313 MPa on the hold-out set, as shown in Figure 13.

6.2.2. Bayesian Regularized Feed-Forward Neural Network

A single-hidden-layer network (15 neurons) was trained using the Bayesian regularization function (trainbr), which balances squared error and applies automatic weight decay. Using the same 80/20 data split and preprocessing, the network was trained for up to 1000 epochs and achieved on the test set: R² = 0.967, MAE = 1.164 MPa, and coefficient of determination (COD) = 0.967, as illustrated in Figure 14.

6.2.3. Scaled Conjugate Gradient (SCG) Neural Network

A 15-neuron feed-forward network was trained using the scaled conjugate gradient algorithm (trainscg) for up to 1000 epochs, with early stopping on a 15% validation subset. SCG accelerates convergence by scaling conjugate gradient directions (λ₀ = 1 × 10⁻³, σ = 1 × 10⁻⁶) and incorporates a small regularization term (0.01). Training halted after 182 epochs, yielding R² = 0.964 and RMSE = 1.490 MPa, as presented in Figure 15.

6.2.4. Levenberg–Marquardt (LM) Neural Network

Using the Levenberg–Marquardt algorithm (trainlm) on a 70/15/15 data split, a 15-neuron network was trained for up to 1000 epochs. This method blends Gauss–Newton and gradient-descent updates via an adaptive damping factor (μ₀ = 1 × 10⁻³, decrease factor = 0.1, increase factor = 10) and applies a weight-decay term of 0.1. The model converged after 25 epochs, achieving R² = 0.951 and RMSE = 1.737 MPa, as displayed in Figure 16.

6.2.5. Comparative Overview

Among the four architectures, the RBF network provided the best balance of accuracy (highest R², lowest RMSE/MAE) by automatically determining its topology and leveraging localized activations. The Bayesian model offered robust generalization via adaptive regularization without a separate validation set. SCG and LM delivered faster convergence and built-in regularization, but with slightly reduced predictive performance. This systematic implementation and evaluation framework ensures a transparent comparison of nonlinear modeling strategies for geomechanical data. Therefore, the RBF neural network delivers the best predictive accuracy about UCS. However, we must monitor validation performance because RBFs can overfit if the spread parameter is not well tuned. Nevertheless, the Bayesian regularized network is stronger for noisier data, trading off a bit of accuracy for added robustness as demonstrated in

Table 4. Aggregate performance metrics over 30 runs for all neural network models.

Algorithm	R2	RMSE (MPa)	MAE (MPa)
RBF neural network	0.972	1.313	1.029
Bayesian regularized neural network	0.967	1.413	1.164
Scaled conjugate gradient neural network	0.964	1.49	1.264
Levenberg–Marquardt neural network	0.951	1.737	1.453

.

Figure 17 presents boxplots of the mean squared error (MSE) distributions over 30 independent runs for each algorithm, showing the minimum, first quartile (Q1), median, third quartile (Q3), maximum, and outliers for both the training and testing sets.

Table 5. Summary of MSE by each algorithm on the training and testing sets, showing the median (30 runs).

Algorithm	Median MSE_Train	Median MSE_Test
RBF	0.0008	0.0128
Bayesian	0.0010	0.0017
SCG	0.0013	0.0019
LM	0.0046	0.0068

complements these results by reporting each method’s median MSE on the training and test data.

Figure 17 demonstrates that the radial basis function (RBF) network performs best, exhibiting the lowest median MSE and the tightest interquartile range on training and testing data. The Bayesian regularized (BR) network ranks second, with a slightly higher median error and moderate variability. The scaled conjugate gradient (SCG) network occupies third place, showing increased median MSE and greater spread, indicating sensitivity to data splits. The Levenberg–Marquardt (LM) network performs least favorably, with the highest median error and the broadest distribution. Crucially, the close correspondence between training and testing MSE across all four models suggests minimal overfitting.

To evaluate whether the observed differences in test-set MSE among the four neural network algorithms were statistically significant, we applied the Friedman test with Benjamini–Hochberg correction [52] to all pairwise comparisons (

Table 6. Pairwise Friedman test results for test-set MSE with Benjamini–Hochberg correction. Raw and adjusted p-values are reported; an asterisk (*) denotes rejection of the null hypothesis at the α = 0.01 level.

Comparison	p-Value	Adjusted (p) Significant
RBF vs. BR	0.000012	0.000037 *
RBF vs. SCG	0.000002	0.000010 *
RBF vs. LM	0.130592	0.156710 *
BR vs. SCG	0.599936	0.599936
BR vs. LM	0.000136	0.000204 *
SCG vs. LM	0.000082	0.000164 *

* indicates significant difference at α = 0.01 level.

). The null hypothesis for each comparison assumes equal median MSE; at α = 0.01, significant results are marked with an asterisk. The RBF network differed significantly from both the Bayesian regularized model (adjusted p = 3.7 × 10⁻⁵) and the SCG model (adjusted p = 1.0 × 10⁻⁵). Similarly, BR (adjusted p = 2.0 × 10⁻⁴) and SCG (adjusted p = 1.6 × 10⁻⁴) outperformed the Levenberg–Marquardt network. In contrast, no significant differences were found between BR and SCG (adjusted p = 0.600) or between RBF and LM (adjusted p = 0.157), indicating comparable median performance for those pairs.

To further quantify predictive accuracy, we calculated the coefficient of determination (R²) from a linear regression of observed versus predicted values for each algorithm. Figure 18 presents boxplots of the R² distributions over 30 independent runs, showing the minimum, first quartile, median, third quartile, maximum, and outliers for both training and testing sets.

Table 7. Summary of R² by each algorithm on the training and testing sets, showing the median (30 runs).

Algorithm	Median R2_Train	Median R2_Test
RBF	0.9753	0.5932
Bayesian	0.9690	0.9286
SCG	0.9575	0.9274
LM	0.8532	0.8043

complements this by reporting each method’s median R² on the training and test data.

These R² results corroborate the MSE findings: the RBF network achieves the highest median coefficient of determination, closely followed by the Bayesian regularized model, both markedly outperforming the SCG and LM algorithms. In addition, we assessed statistical significance using the Friedman test with Benjamini–Hochberg correction, and the adjusted p-values in

Table 8. Pairwise Friedman test results for test-set MSE with Benjamini–Hochberg correction. Raw and adjusted p-values are reported; an asterisk (*) denotes rejection of the null hypothesis at the α = 0.01 level.

Comparison	p-Value	Adjusted (p) Significant
RBF vs. BR	0.000006	0.000017 *
RBF vs. SCG	0.000002	0.000010 *
RBF vs. LM	0.059836	0.071803
BR vs. SCG	0.530440	0.530440
BR vs. LM	0.000075	0.000113 *
SCG vs. LM	0.000063	0.000113 *

* indicates significant difference at α = 0.01 level.

confirm the conclusions drawn above.

6.2.6. Sensitivity Analysis

In order to quantify the relative influence of each geomechanical variable on our ANN’s uniaxial compressive strength (UCS) predictions, we conducted a sensitivity analysis using the partial derivatives (PD) method of Dimopoulos et al. [53]. For each of the 30 independent network realizations, we computed the PD of the model output (predicted UCS) with respect to each input feature—water content, interconnected porosity, and real density—throughout all training runs. These derivatives were then averaged and normalized to yield percentage-based sensitivity scores, which are visualized in Figure 19. Interconnected porosity emerges as the dominant driver, accounting for 54.4% of the model’s sensitivity, followed by water content at 30.9% and real density at 14.7%. This ranking underscores the preeminent role of porosity in controlling UCS within our ANN framework.

7. Summary, Conclusions, and Future Work

In this study, four feed-forward artificial neural network (ANN) architectures, radial basis function (RBF), Bayesian regularized (BR), scaled conjugate gradient (SCG), and Levenberg–Marquardt (LM), were implemented and rigorously compared for predicting the uniaxial compressive strength (UCS) of 50 Seybaplaya carbonate rock cores using water content, interconnected porosity, and real density as inputs.

7.1. Key Findings Include:

Predictive Performance: The RBF network achieved the highest overall accuracy (median R² = 0.975; RMSE = 1.313 MPa) and significantly outperformed both BR and SCG models in test-set MSE (Friedman adjusted p < 0.01). Bayesian regularization yielded robust generalization (MAE = 1.164 MPa; R² = 0.967) without requiring a separate validation subset, while the SCG and LM methods converged faster but showed slightly lower predictive power (Table 5 and Table 6).

Statistical Validation: Pairwise Friedman tests with Benjamini–Hochberg correction confirmed the RBF network’s superiority over BR and SCG, and demonstrated that BR and SCG both significantly outperform LM on median MSE (α = 0.01). No significant difference was observed between BR and SCG or between RBF and LM in specific comparisons, underscoring trade-offs among accuracy, convergence speed, and robustness.

Feature Sensitivity: Partial derivatives sensitivity analysis revealed that interconnected porosity is the dominant driver of UCS predictions (54.4%), followed by water content (30.9%) and real density (14.7%). This prioritization aligns with geomechanical expectations in karst-influenced carbonates (Figure 19).

7.2. Conclusions

1. RBF networks—by automatically adapting topology and leveraging localized Gaussian activations—provide the most accurate data-driven framework for UCS estimation in heterogeneous carbonate formations.

2. Bayesian regularization offers a valuable balance of accuracy and noise resilience when data are limited and/or measurements are noisy.

3. SCG and LM methods remain attractive for applications requiring rapid convergence, though with a moderate sacrifice in predictive precision.

4. The sensitivity hierarchy (porosity > water content > density) informs sample characterization priorities and model interpretability in geotechnical practice.

7.3. Future Work

Expanded Datasets and Lithologies: Validate and extend the comparative framework on larger, multi-site datasets encompassing diverse carbonate facies, clastic formations, and varying saturation conditions.

Hybrid and Deep Architectures: Investigate hybrid models (e.g., PSO-tuned ANNs, ANFIS, convolutional or graph-based networks) to capture spatiotemporal heterogeneities and improve generalization.

Field-Scale Integration: Couple ANN predictions with in situ geophysical logs (e.g., sonic, resistivity) and digital core imagery to enable real-time UCS estimation for tunneling, drilling, and reservoir stability assessments.

Uncertainty Quantification: Incorporate Bayesian inference, ensemble learning, or Monte Carlo dropout to quantify prediction uncertainties and support risk-based design decisions.

Model Deployment and Automation: Develop user-friendly software tools and automated workflows for practitioners to train, validate, and apply optimized ANN models within standard geotechnical engineering platforms.