Feature-engineered physics-conditioned DNN with validation-selected ridge residual correction for pore water pressure prediction in clays

1 Introduction

Pore water pressure is a central state variable in clayey soils because it controls effective stress, deformation, and stability (Chao et al., 2024a; Chao et al., 2024b). It is important in slopes, embankments, and other earth structures exposed to environmental cycling and loading (Fan et al., 2023; Dong et al., 2017). Related geotechnical studies have further shown that interface behavior and clay content can materially influence soil response under imposed loading and coupled hydraulic conditions (Wang Z. et al., 2025; Zhao et al., 2025). Traditional empirical and semi-empirical equations remain useful because they are transparent and easy to apply. However, most assume a fixed functional form and require separate calibration for a limited range of soils and loading paths. These assumptions become restrictive when soil type, axial strain, confining pressure, vertical overburden pressure, and drying-wetting history act together. Repeated wetting and drying can change suction, fabric, stiffness, permeability, and water retention. Loading can further alter the stress path and drainage response. The resulting pore-pressure curves may show soil-dependent amplitudes, nonlinear growth, local transitions, and hysteresis that are difficult to reproduce with a conventional equation.

Machine-learning methods provide an alternative way to represent these nonlinear relationships (Seide et al., 2011; RM et al., 2020; Wu et al., 2018). Deep neural networks can learn mappings without prescribing the full response form in advance. Nevertheless, a purely data-driven model creates concerns for geotechnical applications (Borkar and Karam, 2019; Hussein et al., 2020). Its internal reasoning may be difficult to interpret, and predictions can become unreliable outside the experimental domain. This problem is greater when the number of independent tests is small. Thousands of points sampled from a few curves do not represent thousands of independent experiments. Random point-level splitting can therefore place strongly correlated observations from the same curve in both training and testing subsets and produce an optimistic assessment. In addition, a network based only on raw variables must discover known hydro-mechanical interactions from limited data, which can lead to unstable learning and weak physical consistency.

This study addresses these issues through a staged physics-conditioned framework. Three related configurations are developed and compared. The Physics-Conditioned DNN (PC-DNN) uses the basic soil, deformation, environmental, and stress variables. The feature-engineered physics-conditioned DNN (FE-PC-DNN) adds nonlinear terms and interactions that represent coupled effects among strain, soil type, stress level, and drying-wetting cycles. The third configuration applies validation-selected residual correction to the FE-PC-DNN prediction. Because validation assigned the neural candidate zero weight and ridge regression full weight, the reported method is a feature-engineered physics-conditioned DNN with ridge residual correction, clipping, and validation-bias calibration to reduce systematic error (Muralidhar et al., 2020; Wang and Yu, 2025; Dong et al., 2022). Here, physics-guided and physics-conditioned mean that hydro-mechanical knowledge is used to select inputs, construct features, define a reference response, and restrict the correction. Governing equations and boundary-condition residuals are not imposed in the loss function. The framework is therefore different from a conventional physics-informed neural network. Its contribution lies in the coordinated workflow, rather than in any individual machine-learning component.

The database contains 42 complete curves for kaolin clay and Mercia Mudstone clay under different axial strains, drying-wetting cycles, vertical overburden pressures, and confining pressures (Ou et al., 2022; Chao et al., 2023a). Thirty curves are used for training, six for validation, and six for testing. The division is performed at the complete-curve level to prevent information leakage. Performance is evaluated using error metrics and prediction plots. SHAP analysis is used to relate the learned feature contributions to soil-mechanics principles (Abioye et al., 2025; Mahmood et al., 2025; Chao et al., 2025). A unified kernel-weighted B-spline empirical equation is also calibrated from the 36 development curves and evaluated once on the six locked test curves. This equation provides a reproducible estimator while retaining one formulation for both clay types. The study combines predictive modelling, leakage-resistant validation, interpretation, and an independently tested empirical expression. The conclusions are limited to the two-soil experimental domain, and external validation is required before field application (Chao et al., 2023b; Chao et al., 2024c).

2 Dataset and data analysis

2.1 Dataset description

The dataset used in this study was compiled from the original experimental data table associated with the laboratory programme reported in a previous experimental study (Chao et al., 2024a). In that programme, two clayey soils, namely, Kaolin Clay and Mercia Mudstone Clay, were tested to investigate their hydro-mechanical responses under drying–wetting cycles and loading conditions. The use of the original tabulated data ensures that the machine-learning models were trained on recorded experimental values rather than values digitized from published figures.

The main variables include soil type, drying–wetting cycle number, overburden pressure, confining pressure, axial strain and pore-water pressure. The drying–wetting cycle numbers are 0, 1 and 3. The overburden pressures are 0, 25 and 50 kPa, while the confining pressures are 20, 35 and 50 kPa. Axial strain is used to describe the deformation process during loading, and pore-water pressure is taken as the prediction target.

In total, 42 complete experimental curves are included in the database. The curve, rather than the individual observation, is treated as the independent experimental unit. Thirty curves were used for training, six for validation, and six for testing. The same split was used for all three neural-network models. The six test curves were also locked during selection of the empirical-equation structure and its hyperparameters. After those settings had been fixed, the equation was recalibrated using the 36 development curves and evaluated once on the test subset. This curve-level protocol prevents strongly correlated points from one curve from appearing in different subsets. It evaluates internal generalization within the investigated two-soil laboratory domain, not external transfer to other soils or field conditions.

2.2 Data distribution analysis

Figure 1a shows the distribution of the main input variables, including soil type, confining pressure, overburden pressure, drying–wetting cycle number and axial strain. The dataset contains two clayey soils, Kaolin Clay and Mercia Mudstone Clay. The number of curves for Kaolin Clay is higher than that for Mercia Mudstone Clay, which means that the model needs to learn from a slightly unbalanced soil-type distribution. The confining pressure, overburden pressure and drying–wetting cycle number are distributed over several discrete experimental levels. This is consistent with the design of laboratory tests, where loading conditions are usually controlled at fixed values. Axial strain shows a relatively continuous distribution over the loading process, which provides the main deformation path for pore-water pressure prediction.

Figure 1b presents the distribution of the output variable, pore-water pressure. The values are not uniformly distributed, and several frequency peaks can be observed. This indicates that the pore-water pressure response is affected by different combinations of soil type, stress condition and drying–wetting history. Therefore, a nonlinear prediction model is needed to describe this behaviour (Chao et al., 2023c; Chao et al., 2024d).

Before modelling, records were screened for missing, non-numeric, or non-finite axial-strain and pore-pressure values. No valid pore-pressure observation was removed. Curve metadata were then cross-checked against the experimental file hierarchy. Four curve header records contained one condition field that was inconsistent with their confining-pressure folders and overburden-load filenames; these metadata were corrected before the curve-level split, while the measured strain and pore-pressure values were left unchanged. Soil type was encoded as a categorical variable, and numerical scaling parameters were estimated from the training subset and applied unchanged to validation and testing data. Complete curves, rather than individual records, were assigned to the three subsets to avoid information leakage (Li et al., 2023; Yu et al., 2024).

2.3 Correlation analysis

Figure 2 shows the Pearson correlation heatmap for the input and output features. The two soil-type indicators exhibit a correlation of −1.00 because they are encoded as mutually exclusive categorical variables; that is, a sample belongs to either Kaolin Clay or Mercia Mudstone Clay, but not both. The heatmap also indicates a clear relationship between soil type and pore-water pressure (Shi et al., 2023; Chao et al., 2024e). Kaolin Clay shows a positive correlation with pore-water pressure, whereas Mercia Mudstone Clay shows a negative correlation, suggesting that material type has a strong influence on the pore-water pressure response.

Axial strain is positively correlated with pore-water pressure. This is physically reasonable because pore-water pressure evolves with deformation during loading. As axial strain increases, the internal stress state and drainage response of the soil may change, leading to variation in pore-water pressure.

In Pearson heatmaps, the correlation between pore water pressure and confining pressure, vertical overburden pressure, and number of drying-wetting cycle is relatively small. This result does not imply that these variables are unimportant. Pearson correlation describes only linear relationships, whereas the effects of confining pressure, vertical overburden pressure, and drying-wetting cycles in clayey soils are often coupled with soil type and axial strain. For example, drying-wetting cycles may alter soil structure and stiffness, while different stress levels may affect the development of excess pore water pressure during deformation (Zhang et al., 2023; Yao et al., 2025).

The heatmap therefore suggests that pore-water pressure is not controlled by a single variable. Instead, it is governed by the combined effects of material type and loading path. This observation supports the use of DNN-based models to capture nonlinear relationships among the input features (Zhao G. et al., 2023; Zhao S. et al., 2023).

3 Methodology

3.1 Overall modelling framework

The overall framework is shown in Figure 3. Raw records and test-condition metadata are first checked. Soil type is encoded, and numerical variables are normalized with statistics estimated from the training subset. Complete curves are then assigned to the training, validation, and testing subsets (Chao et al., 2024f; Zhang et al., 2025; Armistead et al., 2026).

Three configurations are developed in sequence. PC-DNN uses the measured soil, strain, cycle, and stress variables. FE-PC-DNN adds nonlinear terms and hydro-mechanical interactions. The third configuration applies a validation-selected residual stage to out-of-fold FE-PC-DNN errors. Validation selected ridge regression, not the neural residual candidate, for the reported correction (Chao et al., 2024g; Wang X. et al., 2025). In this paper, physics-guided and physics-conditioned mean that physical knowledge shapes the inputs, feature construction, reference prediction, and correction limits. Governing equations are not enforced during optimization.

This definition differs from a conventional physics-informed neural network. A PINN places governing-equation residuals, boundary conditions, or initial conditions in the training objective. The present model does not use these penalties. Performance is assessed on held-out curves using error metrics, prediction plots, and generalization gaps (Ren et al., 2024; Zheng et al., 2024). SHAP and the calibrated empirical equation support interpretation and in-domain engineering screening (Yue et al., 2026; Zhao et al., 2026a; Zhao et al., 2026b).

3.2 FE-PC-DNN model

FE-PC-DNN uses 32 inputs. Table 1 gives their formulas and grouping. K and M are mutually exclusive indicators for kaolin clay and Mercia Mudstone clay. The other basic variables are axial strain εa, drying-wetting cycle number N, vertical overburden pressure pv, and confining pressure σ3. Engineered terms are formed before standardization. Their means and standard deviations are then estimated from the training subset and applied unchanged to validation and test data.

TABLE 1. Engineered inputs used by FE-PC-DNN.

The added terms have a direct modelling purpose. Powers and transformations of strain describe the curved loading path. Stress combinations describe the level and contrast of the applied pressures. Products with strain allow the stress and cycle effects to change during deformation. Soil-conditioned terms allow the two clays to follow different response paths. These features do not impose a constitutive law. They provide a structured input space in which the network can learn the remaining nonlinear relation.

3.3 PC-DNN model

The PC-DNN model was developed as the baseline network for pore-water-pressure prediction. PC-DNN denotes the Physics-Conditioned DNN. The word conditioned indicates that the model receives physically meaningful variables describing soil type, axial strain, drying-wetting history, vertical overburden pressure, and confining pressure. It does not indicate that a governing hydro-mechanical equation is imposed as a hard or soft constraint. The target variable is pore-water pressure (Chakma et al., 2025; Yu et al., 2025).

The input vector of the PC-DNN model is defined as the mathematical formulation of the PC-DNN model is summarized in Equations 1–4:where S denotes soil type, ε a is axial strain, N dw is the number of drying–wetting cycles, σ v is overburden pressure, and σ c is confining pressure. The model output is the predicted pore water pressure:where u ^ denotes the predicted pore water pressure, and f θ represents the DNN model with trainable parameters θ.

x = S , ε a , N d w , σ v , σ c (1)

u ^ = f θ x (2)

As shown in Figure 4, the PC-DNN consists of one input layer, four hidden layers, and one output layer. After feature encoding, the input layer contains six neurons. The four hidden layers contain 128, 128, 64, and 32 neurons, respectively. The output layer contains one neuron, which provides the predicted pore-water pressure. The calculation in each hidden layer can be expressed as:where W l and b l are the weight matrix and bias vector of layer l, and ϕ · is the activation function. The model is trained by minimizing the mean squared error between the measured and predicted pore water pressure:where u i and u ^ i are the measured and predicted pore water pressure values, respectively. The PC-DNN model provides a reference for evaluating the improvement introduced by the FE-PC-DNN and RC-FE-PC-DNN models.

h l = ϕ W l h l ‐ 1 + b l (3)

M S E = 1 n ∑ i = 1 n u i ‐ u ^ i 2 (4)

This baseline configuration uses only soil type, axial strain, drying-wetting cycle number, vertical overburden pressure, and confining pressure as input features. It therefore provides a transparent benchmark against which the benefits of feature engineering and residual correction can be assessed.

3.4 Ridge-corrected FE-PC-DNN (RC-FE-PC-DNN)

The reported RC-FE-PC-DNN configuration combines a reference prediction, the nonlinear FE-PC-DNN estimate, and a validation-selected bounded ridge residual correction (Chen et al., 2026a; Chen et al., 2026b; Peng et al., 2026). The first two terms describe the main pore-pressure response. The correction term r in Equation 5 is a low-amplitude regularized linear adjustment to repeatable error that remains after the main prediction.

u ^ R C = u ^ p h y s + u ^ F E + r ^ (5)

Residual targets were obtained from five-fold out-of-fold FE-PC-DNN predictions on the training curves. Two candidate correctors were examined: a compact neural network and ridge regression. Their weights were selected on the validation curves. The selected weights were αnn = 0 and αridge = 1. Consequently, the reported model is a feature-engineered physics-conditioned DNN with ridge residual correction. The candidate neural corrector did not contribute to the reported RC-FE-PC-DNN predictions.

The ridge term estimates a smooth condition-dependent bias rather than a new physical process. Its correction was clipped to ±0.419394 kPa. This limit was the 97.fifth percentile of the absolute out-of-fold correction in the selected run. The remaining mean validation residual was −0.198128 kPa. A bias weight of 0.5 gave the applied offset of −0.099064 kPa. These settings were fixed before the test curves were evaluated. They are numerical safeguards, not universal soil parameters.

From a soil-mechanics perspective, the remaining residual pattern is consistent with small departures between the measured response and the variables used to describe the coupled hydro-mechanical state. Drying and rewetting can rearrange clay aggregates, modify pore-size connectivity and water-retention behaviour, and alter the balance between contractive tendency and drainage during subsequent loading; these changes are only represented indirectly by soil type and cycle number in the present inputs. Likewise, the influence of confining and vertical pressures may vary along the strain path as stiffness degrades and the effective-stress state evolves, leaving systematic under- or overprediction in particular condition ranges. The ridge correction therefore acts as a low-amplitude adjustment for these unresolved but smoothly varying effects. Its bounded magnitude, compared with the overall pore-pressure range, indicates that the principal response is still carried by the physics-conditioned and feature-engineered components. The sign of an individual correction should not, however, be interpreted as direct evidence of a specific fabric or drainage mechanism, because the residual also contains experimental scatter and omitted variables.

3.5 Training strategy and hyperparameter setting

PC-DNN and FE-PC-DNN were trained with early stopping and random restarts. PC-DNN used four hidden layers with 128, 128, 64, and 32 neurons. FE-PC-DNN used 160, 128, 64, and 32 neurons. The larger feature set was controlled with a lower learning rate, dropout, and weight decay. The selected settings are listed in Table 2.

TABLE 2. Hyperparameters of the three DNN-based models.

The residual stage was selected separately. A compact neural candidate was trained as a possible nonlinear corrector, but validation blending assigned it zero weight. Ridge regression received full weight. Residual clipping and the half-bias correction were then applied. This selection is reported directly so that the final model is not described as a residual neural network when its accepted correction is linear.

3.6 Verification of overfitting prevention

Overfitting was checked with the complete-curve split and with the differences among training, validation, and test R ² values. PC-DNN and FE-PC-DNN used dropout, random restarts, and validation-based early stopping. FE-PC-DNN also used weight decay. RC-FE-PC-DNN used out-of-fold residual targets, ridge selection, clipping, and bias calibration. Table 3 reports both the safeguards and the observed generalization gaps.

TABLE 3. Regularization measures and quantitative generalization gaps.

RC-FE-PC-DNN has the smallest training-test and validation-test R ² gaps. The neural residual candidate reached its lowest validation loss at epoch 1, but its final weight was zero. That epoch is therefore not the convergence point of the accepted residual correction. The previous wording was misleading and has been removed. The selected ridge corrector has no epoch-based convergence curve.

3.7 Model evaluation metrics

The prediction performance of the proposed models was evaluated using several statistical metrics, including the coefficient of determination (R ²), mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and symmetric mean absolute percentage error (sMAPE). These metrics evaluate model performance from different perspectives, including goodness of fit, absolute error, and relative error.

The coefficient of determination, R 2, was used to measure the agreement between the measured and predicted pore water pressure values. The model evaluation metrics are defined in Equations 6–10:where y i is the measured value, y ^ i is the predicted value, y ¯ is the mean value of the measured data, and n is the number of samples. A higher R 2 indicates better agreement between prediction and measurement.

R 2 = 1 ‐ ∑ i = 1 n y i ‐ y ^ i 2 ∑ i = 1 n y i ‐ y ¯ 2 (6)

The MSE was used to measure the average squared prediction error:

M S E = 1 n ∑ i = 1 n y i ‐ y ^ i 2 (7)

MSE gives greater weight to large errors because the residuals are squared, while RMSE has the same unit as the target variable and is easier to interpret physically.

The MAE was used to measure the average absolute deviation between measured and predicted values:

M A E = 1 n ∑ i = 1 n y i ‐ y ^ i (8)

Compared with RMSE, MAE is less sensitive to large individual errors and provides a direct measure of average prediction deviation.

The MAPE was used to evaluate the relative prediction error as a percentage:

M A P E = 100 % n ∑ i = 1 n y i ‐ y ^ i y i (9)

The sMAPE was also used to describe the relative error in a more balanced form:

s M A P E = 100 % n ∑ i = 1 n y i ‐ y ^ i y i + y ^ i / 2 (10)

MAPE expresses the error as a percentage of the measured value, while sMAPE reduces the asymmetry caused by using only the measured value as the denominator.

4 Results

4.1 Predictive performance

Figures 5–7 compare measured and predicted pore-water pressure for the training, validation, and test subsets. All three models follow the main response trend. The comparison is made at the same complete-curve split for each model.

Figure 5 shows the training results. PC-DNN provides the baseline. FE-PC-DNN gives a slightly closer fit after the engineered terms are added. RC-FE-PC-DNN has a marginally lower training R ² than FE-PC-DNN. This is not evidence of poorer generalization because the residual stage was selected from validation behaviour rather than training fit alone.

Figure 6 shows the validation results. FE-PC-DNN improves the representation of the loading conditions. RC-FE-PC-DNN further reduces the remaining condition-dependent bias. Its validation R ² is close to its training and test values.

Figure 7 gives the test results. RC-FE-PC-DNN has the closest overall agreement among the three models. The result supports prediction on held-out curves within the two-soil laboratory domain. It does not establish transfer to new soils or field conditions.

Figure 8 summarizes the error metrics. On the test subset, MSE decreased from 2.13 kPa² for PC-DNN and 1.89 kPa² for FE-PC-DNN to 1.58 kPa² for RC-FE-PC-DNN. RMSE decreased from 1.46 to 1.38 kPa–1.26 kPa. MAE decreased from 0.90 to 0.91 kPa–0.82 kPa sMAPE decreased from 9.33% to 8.87%–8.75%.

RC-FE-PC-DNN gives the lowest test errors and the smallest R ² gaps. These results support the combined use of structured features and controlled residual correction. The conclusion is limited to the conditions represented by the 42 curves.

4.2 Training behaviour and generalization

Figures 9a,b show the training and validation loss curves for PC-DNN and FE-PC-DNN. PC-DNN reached its best validation state at epoch 231. FE-PC-DNN reached its best state at epoch 44. Figure 9c compares the absolute R ² gaps across the three models.

No epoch curve is reported for the accepted residual corrector because validation selected ridge regression. The compact neural residual candidate reached its lowest validation loss at epoch 1, but αnn = 0. It did not enter the final prediction. This distinction removes the earlier impression that the complete RC-FE-PC-DNN was trained in one epoch.

The quantitative gaps support the regularization result. For RC-FE-PC-DNN, the absolute training-test and validation-test R ² gaps are 0.0031 and 0.0028. The corresponding gaps are about 0.010 for PC-DNN and FE-PC-DNN. Thus, the residual correction improves consistency without relying on an excessive training fit.

The loss curves and the gap comparison should be read together. Early stopping controls the two neural backbones. The final residual stage is a separately selected ridge correction. Its clipping and bias terms limit the size and mean of the adjustment.

4.3 Model explanation

SHAP was used to describe how each input contributes to the model output. A positive SHAP value raises the predicted pore-water pressure relative to the reference value. A negative value lowers it. SHAP measures model attribution. It does not prove a causal soil mechanism.

Figure 10 presents the global SHAP results for RC-FE-PC-DNN. The beeswarm plot shows the direction and spread of the contributions. The bar plot shows the mean absolute SHAP value for each main variable.

Soil type has the widest SHAP range. It represents differences in mineralogy, fabric, pore-size distribution, compressibility, permeability, and water retention. Axial strain is the second main feature because pore pressure changes as deformation develops. Confining pressure also contributes by changing mean stress and lateral restraint.

Drying-wetting cycle number and vertical pressure have smaller global contributions in this database. This does not make them physically unimportant. Their effects can depend on soil type, strain level, and the other stress variable. The global plot alone cannot show these conditional patterns.

Figure 11 provides a complementary nonlinear check. Model-agnostic KernelSHAP was applied to the unified empirical equation in Section 4.4. This equation reproduced the six locked test curves with R ² = 0.982. The new plots do not replace the RC-FE-PC-DNN global SHAP analysis. They are used to check whether the dominant directions remain visible in a separate, reproducible in-domain predictor.

The strain effect is strongly nonlinear. Its SHAP value rises rapidly over the early loading range and then approaches a plateau. The trend also differs between the two soils. The confining-pressure contribution is condition-dependent rather than monotonic across 20, 35, and 50 kPa, and its spread changes with strain. Cycle effects are smaller and differ between the soils. Vertical pressure has little direct contribution in this equation. These patterns describe the tested grid and should not be treated as universal thresholds.

4.4 Empirical formula

To provide a reproducible engineering expression, the former low-order polynomial was replaced by one unified kernel-weighted B-spline equation. The equation structure and its hyperparameters were selected using only the 30 training and 6 validation curves. After the settings had been fixed, the equation was recalibrated on all 36 development curves. The six test curves remained locked until the final evaluation. One formulation is used for both clay types; separate soil-specific equations were not fitted.

Compared with a fixed low-order polynomial, the cubic B-spline basis does not force one global curvature over the complete strain range. This is useful for curves under coupled drying-wetting cycles and loading, where an initially rapid pore-pressure change may be followed by a transition region and a gradual plateau, and where the location and sharpness of these features vary with soil and stress condition. Because each B-spline basis function has local support, the fitted curve can adjust one strain interval without introducing unnecessary distortion in the remaining intervals, while the kernel neighbour weights allow the coefficient set to change smoothly across nearby combinations of soil type, confining pressure, vertical pressure, and cycle number. The resulting equation therefore retains a single reproducible form but represents nonlinear strain-dependent evolution and condition-to-condition variation more flexibly than a conventional polynomial within the calibrated domain.

Let ε denote axial strain and let z = (S, σ₃, pᵥ, N) describe soil type, confining pressure, vertical overburden pressure, and drying-wetting cycle number. Soil type is coded as S = 1 for kaolin clay and S = 0 for Mercia Mudstone clay. Each development curve i is represented by ten cubic B-spline coefficients aᵢ_m. The base knots are 0, 2.142857, 4.285714, 6.428571, 8.571429, 10.714286, 12.857143, and 15%. The unified predictor is:

u ^ emp ε , z = ∑ m = 1 10 B m 3 ε ∑ i ∈ N 3 z w i z a i m (11)

For a requested condition z, N₃(z) contains the three nearest development curves in the scaled condition space. Their normalized kernel weights are:

w i z = exp ‐ D i 2 z 2 ∑ j ∈ N 3 z ⁡ exp ‐ D j 2 z 2 (12)

The scaled distance used to select the neighbours and calculate the weights is:

D i 2 z = S ‐ S i 0.05 2 + σ 3 ‐ σ 3 , i 5 2 + p v ‐ p v , i 100 2 + N ‐ N i 1 2 (13)

Here, σ3 and pv are expressed in kPa. The length scales, spline settings, and neighbour count are listed in Table 4. The reference identifier has the form K/M-conf-over-cycle. For example, K-20–25–3 denotes kaolin clay at 20 kPa confining pressure, 25 kPa vertical pressure, and three drying-wetting cycles. Tables 5, 6 give the ten coefficients for all 36 development curves. Equations 11–13 and Tables 4–6 allow the equation to be reproduced without the neural-network code.

TABLE 4. Parameters of the unified kernel-weighted B-spline empirical equation.

TABLE 5. Reference-curve B-spline coefficients a₁-a₅.

TABLE 6. Reference-curve B-spline coefficients a₆-a₁₀.

The equation was evaluated on 6,006 observations from the six locked test curves. It achieved R ² = 0.982, RMSE = 0.894 kPa, MAE = 0.716 kPa, and sMAPE = 6.03%. Figure 12 shows the measured-predicted agreement and the six curve comparisons. No test observation was used to select the spline knots, distance scales, neighbour count, or coefficient-fitting procedure. This is an internal test within the investigated grid, not external validation for new soils or field conditions.

5 Discussion

The staged comparison separates the effect of each modelling step. FE-PC-DNN adds explicit strain, stress, cycle, and soil interactions. The final correction is ridge-based because validation gave the neural candidate zero weight. The method is therefore accurately framed as a feature-engineered physics-conditioned DNN with validation-selected ridge residual correction, rather than as a deep residual-correction architecture. The RC-FE-PC-DNN abbreviation is retained only as a compact label for the evaluated configuration.

The framework is physics-guided, not physics-informed in the PINN sense. Physical knowledge is used to choose variables, build features, define a reference response, and limit the correction. No governing hydro-mechanical equation is placed in the loss function. The model is intended to organize data-driven learning. It does not replace a constitutive model.

The regularization evidence is quantitative. RC-FE-PC-DNN has smaller training-test and validation-test R ² gaps than the other two models. Out-of-fold residual targets reduce direct fitting to the training errors. The 97.fifth-percentile clipping limit bounds isolated corrections. Half of the remaining mean validation residual is removed by bias calibration. The discarded neural candidate reached its best state at epoch 1, but that epoch has no role in the final ridge correction.

The interpretation results agree with soil-mechanics reasoning. Soil type represents major differences in fabric and hydraulic behaviour. Strain reflects fabric change and volumetric tendency during loading. Confining pressure changes mean effective stress and limits lateral deformation. The dependence plots also show that these effects are nonlinear and conditional. In particular, the strain response levels off at larger deformation, and the confining-pressure effect changes with strain. These observations apply only to the tested grid and do not establish causality.

The unified empirical equation offers a second route for rapid in-domain estimation. Cubic B-splines preserve the strain-path shape, and kernel weights use the three closest development conditions. Its high test accuracy is therefore best interpreted as interpolation within a dense experimental grid. It should not be presented as a general equation for other soils or loading paths.

In practice, the predicted pore pressure can support preliminary effective-stress calculations, comparison of slope-stability scenarios, and sensitivity checks in soil-water interaction studies. The framework can also provide initial values or screening cases before a coupled numerical analysis. These uses require inputs within the tested ranges. The database contains only two clays and discrete stress and cycle levels. It does not include independent laboratories, field monitoring, drainage rate, suction, or time-dependent variables. Site investigation and coupled hydro-mechanical analysis therefore remain necessary for design decisions.

6 Conclusion

This study developed a staged physics-conditioned DNN framework for pore-water-pressure prediction under drying-wetting cycles and loading. PC-DNN used the basic variables, and FE-PC-DNN added 32 structured inputs. The final configuration augmented FE-PC-DNN with validation-selected ridge residual correction.

The ridge-corrected FE-PC-DNN gave the lowest test MSE, RMSE, MAE, and sMAPE, together with the smallest training-test and validation-test R ² gaps. Validation selected αnn = 0 and αridge = 1; the accepted correction was therefore linear, clipped to ±0.419394 kPa, and combined with an applied validation-bias offset of −0.099064 kPa. The final method is a feature-engineered physics-conditioned DNN with a regularized linear residual correction.

Global SHAP identified soil type, axial strain, and confining pressure as the main variables. Complementary dependence plots showed a nonlinear strain response and condition-dependent stress effects. These patterns are consistent with the roles of soil fabric, deformation, drainage response, and effective stress, but they do not prove causality.

The unified empirical equation achieved R ² = 0.982, RMSE = 0.894 kPa, MAE = 0.716 kPa, and sMAPE = 6.03% on six locked test curves. Its parameters are fully reported. Both the neural framework and the equation remain in-domain tools for the two-soil laboratory grid. Broader use requires independent soils, wider stress and drainage paths, and field-scale validation.

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Footnotes

Edited by: Ahmed Kenawy, Mansoura University, Egypt

Reviewed by: Mawuko Luke Yaw Ankah, University of Mines and Technology, Ghana

Zecheng Wang, East China University of Technology, China