Physics-informed neural networks for predicting ultimate bearing capacity of layered soils under complex loading conditions

Towards physics-consistent machine learning models: A geomechanics-based artificial neural network for high-cyclic soil response

Estimating long-term soil deformation is crucial for the safe design of offshore foundations, railway substructures, pavement layers, and other cyclically loaded geotechnical systems. Traditional constitutive formulations, such as the High-Cycle Accumulation (HCA) model, provide reliable simulations but rely on empirically calibrated parameters that require extensive experimental campaigns for their determination. To overcome this shortcoming, it is proposed to design a Geomechanics-based Artificial Neural Network (GANN) that bypasses the need for calibration parameters and instead uses common soil descriptors. Two steps are required for this purpose: (i) develop a foundational GANN capable of accurately predicting accumulated strain evolution for a given soil; and (ii) generalize this GANN so that it can operate for a wide variety of soils using simple input variables such as the ones related to grain size distribution and state boundary surface. Accordingly, this manuscript aims to address this first step by considering two different soils as examples, namely Karlsruhe fine sand and Australian superfine silica sand. For each soil, a synthetic database is generated from the HCA model to train a GANN architecture, enabling it to learn the fundamental relationships governing strain accumulation. The methodology integrates data-driven learning with a physics-informed loss function, ensuring that the predicted strain evolution remains consistent with soil mechanics principles. This GANN comprises multiple fully connected layers using several activation functions, with outputs structured to capture strain accumulation over a high number of cycles. Validation against synthetic and experimental data confirms excellent performance, supporting this first step toward a generalized GANN framework.

Bayesian-Optimized Ensemble Machine Learning for Predicting Settlement of Cohesionless Soil Under Loads

Geotechnical Evaluation of Gradient-Based Neural Networks for Factor of Safety Prediction in Homogeneous Soil Slopes Under Hydraulic Variability

Slope stability assessment remains a fundamental challenge in geotechnical engineering because of the complex nonlinear interactions among soil properties, slope geometry, and hydraulic conditions, particularly variations in pore-water pressure. This study investigates the reliability of Artificial Neural Network–Multi-Layer Perceptron (ANN–MLP) models for predicting the Factor of Safety (FoS) of homogeneous soil slopes through a systematic comparison of three gradient-based optimization algorithms: Adam, Mini-Batch Gradient Descent (MBGD), and Nesterov Accelerated Gradient (NAG). A database comprising 2014 slope cases, compiled from published studies and numerically generated using Limit Equilibrium Method (LEM) and Finite Element Method (FEM) analyses, was used for model development and k-fold cross-validation. Beyond statistical evaluation, the developed models were validated using two classical dry-slope benchmark frameworks based on the Taylor stability charts and Bishop–Morgenstern stability coefficients, followed by two documented engineering case studies from Hulu Kelang and Pahang, Malaysia, to assess predictive performance under both dry and variable hydraulic conditions. Adam achieved the highest cross-validated predictive accuracy (R2 = 0.988; RMSE = 0.212), whereas MBGD demonstrated the closest overall agreement with the reference LEM solutions across the validation cases and under increasing pore-water pressure ratios. NAG generally produced more conservative predictions while exhibiting greater sensitivity to hyperparameter selection. All models successfully reproduced the expected nonlinear reduction in FoS with increasing pore-water pressure, consistent with established geotechnical behaviour. The results demonstrate that optimizer selection significantly influences ANN–MLP prediction behaviour and that properly validated gradient-based ANN models can serve as efficient decision-support tools for rapid slope stability assessment under hydraulic variability.

A Physics-Data Hybrid Model for Predicting Earth Pressure Evolution in Buried Horizontal Cylindrical Tanks

Buried horizontal cylindrical tanks are susceptible to stress instabilities, such as shell buckling and weld fatigue, under nonuniform ground settlement. Classical Terzaghi-based earth pressure theories simplify key parameters into static constants, rendering them inadequate for capturing the dynamic soil-tank interaction and parameter evolution induced by settlement. To address this limitation, a physics-data hybrid model (PDHM) is developed by embedding a genetic programming (GP) module into a three-dimensional (3D) analytical earth pressure framework for medium-dense sand conditions. This approach leverages the symbolic regression capability of GP to derive explicit nonlinear expressions for the dynamic load-bearing width B and lateral pressure coefficient K , thereby unifying data-driven adaptability with physical constraints. To ensure transparency and reproducibility, a standalone GP model is constructed as a purely data-driven baseline, utilizing identical feature inputs, preprocessing procedures, and training-testing protocols. Results demonstrate that the PDHM reduces prediction errors by 82.1% to 96.1% compared to the baseline, with decreases in RMSE and MAPE of 51.89% and 92.34%, respectively. Consequently, the PDHM offers an interpretable, generalizable, and computationally efficient tool for analyzing stress evolution and supporting the safety assessment of buried horizontal cylindrical tanks in medium-dense sand conditions.

Integrating deep neural network with elastoplastic analysis: a hybrid approach for slope stability analysis

Conventional data-driven methods for slope stability analysis often exhibit an over-reliance on data while neglecting underlying physical principles. To address this limitation, this study proposes a physics-informed neural network (PINN) framework that integrates a neural network surrogate with the elastoplastic deformation mechanism of soil slopes. The governing equations incorporating the elastoplastic constitutive model based on the Mohr–Coulomb yield criterion, along with the boundary conditions, are embedded into the training framework of the PINN. The PINN functions as a surrogate model that requires no pre-constructed training dataset and automatically satisfies both the governing equations and the boundary conditions. Consequently, the developed PINN can directly predict the displacement field of a slope and automatically derive the associated stress–strain fields that comply with the deformation mechanism. These outputs are coupled with the multi-initial point sequential quadratic programming (MSQP) algorithm and the slip surface stress analysis (SSSA) method, enabling the efficient identification of the critical slip surface and the calculation of the corresponding factor of safety (FOS). The proposed method is validated through two illustrative examples. Comparisons of the results with those from commercial software confirm the high accuracy of the proposed method in predicting the stress–strain response and the FOS. This study provides a data-driven and physics-informed paradigm for slope stability analysis grounded in clear physical mechanisms.

A Physics-informed Neural Network Approach for Robust Buckling Load Prediction and Reliability-Based Design of Thin Truncated Conical Shells

Thin-walled truncated conical shells are widely used in aerospace, marine, offshore, and lightweight infrastructure systems due to their high strength-to-weight ratio and geometric efficiency. Their buckling resistance under axial compression, however, is highly sensitive to geometric imperfections, manufacturing tolerances, material variability, and nonlinear instability effects. Conventional design procedures rely on conservative knockdown factors (KDFs), such as those recommended in NASA SP-8019, which do not explicitly account for shell geometry, fabrication quality, data uncertainty, or target reliability. This study develops a physics-informed neural network (PiNN) framework for predicting critical buckling loads of thin truncated conical shells and integrates the trained surrogate within a reliability-based design (RBD) formulation. The model combines geometric and material descriptors with mechanics-informed features derived from shell stability theory and the localized reduced stiffness method (LRSM). A physics-informed loss function penalizes mechanically inadmissible predictions exceeding the theoretical elastic buckling load. The framework is trained and evaluated using 133 experimental Mylar conical shell tests under axial compression. Compared with a conventional deep neural network (DNN), the PiNN improves predictive accuracy, reduces mean absolute error, and enhances physical consistency. The trained PiNN is then used to evaluate reliability indices and calibrate safety-consistent KDFs for prescribed target reliability levels. Results demonstrate that the PiNN-RBD framework provides an efficient approach for uncertainty-aware design of imperfection-sensitive shell structures.