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

A Framework Integrating the Dynamic Stiffness Matrix with Physics-Informed Neural Networks for Solving Eigenvalue Problems and Analysing Dynamic Response

This paper introduces a framework that integrates the dynamic stiffness matrix (DSM) with physics-informed neural networks (PINN). The DSM-PINN embeds physical constraints within the model and demonstrates robustness, particularly when addressing limited datasets across diverse investigations. In this approach, deep neural network outputs approximate the displacement fields of element nodes. Unlike the finite element method (FEM), the element shape functions are homogeneous solutions to the governing partial differential equation, forming the basis of the exact dynamic stiffness matrix, thereby avoiding high-order derivative terms. This matrix also serves as a frequency-domain spectral element, resulting in a strong-form PINN. The loss function is produced by connecting neural networks with dynamic stiffness matrices. We focus on utilising PINNs to resolve eigenvalue problems by employing the Wittrick-Williams algorithm, which overcomes the challenge of neural networks failing to converge to higher-order eigenvalues. Additionally, the frequency domain-PINN method is used to analyse structural dynamic responses under moving and impulsive loads, addressing the limitation of neural networks in handling complex numbers. Theoretical convergence stability of the suggested approach is also analysed even DSM is an indefinite matrix after implementing the boundary condition. The numerical results validate the practicality and efficacy of the recommended approach.

A hybrid ANN–FEM framework for high accuracy slope stability prediction

Purpose. To develop a hybrid methodology that integrates Artificial Neural Networks (ANN) with Finite Element Method (FEM) simulations for the rapid and accurate prediction of slope stability. Methodology. A dataset of 1,000 FEM simulations was generated by systematically varying seven key input parameters: slope geometry (height and angle) and soil properties (cohesion, friction angle, unit weight, pore water pressure ratio, and reinforcement type). An ANN model with a (7-10-1) feedforward architecture was trained on this data. Findings. The model demonstrated exceptional predictive performance, achieving a near-perfect correlation coefficient (R  0.999997) and an extremely low mean squared error (MSE = 3.6828  10-6), showing close agreement with the FEM-computed factors of safety (FOS). A comprehensive sensitivity analysis based on analysis of variance identified the pore water pressure ratio as the dominant controlling parameter, contributing approximately 77 % to the variability of FOS, followed by cohesion with a contribution of about 13 %. Complementary correlation analysis revealed that cohesion exhibits the strongest linear correlation with FOS (r = 0.83), whereas the pore water pressure ratio shows a relatively weak linear correlation, highlighting its pronounced nonlinear and interaction-driven influence on slope stability. These results demonstrate that the proposed ANN–FEM hybrid framework provides a powerful, efficient, and reliable tool for slope stability assessment and parametric analysis. The methodology is particularly well suited for engineering applications requiring rapid decision-making, large-scale evaluations, and uncertainty analysis. Originality. The core originality of this research is its development of a robust ANN–FEM hybrid framework applied to a large, systematically generated dataset of 1,000 slope simulations. Unlike many studies, it comprehensively incorporates seven critical input variables, including the often underrepresented pore water pressure. Furthermore, its scientific rigor is enhanced by a dual interpretability strategy that combines analysis of variance for quantifying parameter contribution and correlation heatmaps for distinguishing linear effects from nonlinear ones, providing deeper insight into slope stability mechanisms. Practical value. This study provides engineers with a fast and reliable tool to predict slope safety in seconds instead of running time-consuming FEM simulations, making it highly valuable for real-time decision-making and large parametric studies. Practically, it also shows that controlling pore water pressure (through drainage) is the most effective risk-reduction strategy, while cohesion offers a predictable way to improve slope stability in design.

Calibration of neural viscoelastic models via full-field data

We propose an unsupervised learning framework for calibrating a physics-augmented neural network (PANN) for small-strain viscoelasticity via full-field data. It only requires quantities that are directly accessible in real experiments for training, namely global reaction forces and surface displacements. The underlying PANN is embedded in the generalized standard materials theory, in which two scalar-valued potentials render the constitutive model thermodynamically consistent by construction, while invariant-based representations of the free energy and the dual dissipation potential additionally ensure material symmetry. Considering a thin specimen under the plane stress assumption, we formulate a constrained optimization problem based on the equilibrium gap method in combination with quasi-Newton optimizers and automatic differentiation. Thereby, the unknown out-of-plane strain follows from the plane stress condition and the evolution of the internal variables is captured by an implicit time integration scheme. The resulting system of nonlinear equations is solved via a local Newton iteration at quadrature point and time step. To drastically reduce the computational cost of training, the backward adjoint method is employed to compute the gradient of the target loss, instead of backpropagating through all Newton iteration steps. The proposed framework is demonstrated for synthetic data, including noisy displacements and forces, showing excellent agreement across a wide range of deformation rates and load paths.

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

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, ensuring that the predicted strain evolution remains consistent with soil mechanics principles.

Gradient-Based Equation Adaptive Weighting in Physics-Informed Neural Networks for Water Hammer Analysis

To address the issues of optimization instability and imbalance in the contributions of multiple governing equations in conventional Physics-Informed Neural Networks (PINNs) for hydraulic transient problems, a gradient-based equation adaptive weighting strategy is proposed in this study. This strategy is incorporated into the PINN framework, referred to as GEAW-PINNs (gradient-based equation adaptive weighting in Physics-Informed Neural Networks), for predicting pressure and flow velocity during water hammer events. In GEAW-PINNs, the loss terms associated with different governing equations in the partial differential equation (PDE) constraints are dynamically weighted, thereby enhancing training stability. In the model construction, the classical governing equations of water hammer are employed to establish the PDE constraints, in which the Brunone model is incorporated. Meanwhile, the corresponding model coefficient is treated as a trainable parameter, enabling simultaneous parameter inversion and prediction of pressure and flow velocity. High-accuracy numerical solutions are generated as reference data to validate the proposed framework. The results demonstrate that GEAW-PINN effectively improves the stability of PINNs for the prediction of pressure in water hammer phenomena, thereby enhancing overall optimization performance and prediction accuracy. For the reservoir–pipeline–valve system, the proposed method achieved relative errors of only 0.00742 for pressure and 0.0183 for velocity. And the proposed method can also provide accurate predictions in complex pipe network systems. For the pipeline network system, the absolute prediction errors were approximately 15 for pressure and 0.05 for velocity. Finally, the robustness of the proposed method was evaluated under different random seeds and 25 dB noise. The prediction error exhibited little variation across different random seeds, with a variance of only 1.1429×10−7 and 6.87×10−7. Under 25 dB noise, the prediction error increased only slightly to 9.88×10−3 and 2.5×10−2. This study provides a practical example for achieving stable PINN training in multi-physics coupled problems.

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