Accurate and efficient computation of structural behavior is crucial, particularly with the rise of Digital Twin (DT) systems requiring real-time simulations. Finite Element Method (FEM), while widely used, has limitations: it's time-consuming, challenging to integrate into DT systems, struggles with infinite domain problems, and cannot easily solve inverse problems. Artificial intelligence (AI), especially deep learning, offers potential solutions. Data-driven models, however, lack physical interpretability. Physics-informed neural networks (PINNs) overcome this by incorporating PDEs, enabling solutions even with unknown boundary conditions. However, applying PINNs to structural mechanics, involving high-order PDEs, remains challenging due to accuracy and convergence issues. This research proposes ml-PINN to address these challenges in bending structure calculations.

The application of AI, particularly deep learning, in structural engineering is growing, with uses in damage identification, model optimization, and performance prediction. While data-driven models are prevalent, their lack of interpretability is a drawback. SHAP values can help explain predictions, but directly incorporating physics knowledge into the learning process is more desirable. PINNs have emerged as a method to include PDE knowledge, improving accuracy and efficiency compared to FEM, particularly when boundary conditions are unknown. PINNs have seen applications in various areas but face difficulties when applied to high-order PDEs in structural mechanics, motivating this study.

The proposed ml-PINN framework tackles the complexity of high-order PDEs by decomposing them into lower-order equations. This multi-level approach involves multiple neural networks, each handling a simplified aspect of the physics (geometric, constitutive, and equilibrium relations). For beam structures (1D), the fourth-order PDE is converted into five first-order equations; for shell structures (2D), the decomposition involves second-order PDEs. Each neural network is trained independently and then combined using a loss function that includes both PDE residuals and boundary condition constraints. The automatic differentiation technique is employed to calculate the derivatives. The entire system is optimized using the ADAM algorithm to minimize the total loss function. The effectiveness of the method is evaluated through comparative analyses with FEM results for different loading conditions on beam and shell structures. A sensitivity analysis is performed to determine the optimal number of collocation points (training data points) for both beam and shell structures.

The ml-PINN framework demonstrated high accuracy in predicting the deformation of both beam and shell structures under various loading conditions. For beam structures, the relative error between ml-PINN and FEM results was less than 0.5%, indicating excellent agreement. For shell structures, the relative error remained below 2%, even with slightly larger deviations observed at corner regions where more constraints were present. A sensitivity analysis showed that the accuracy improved with an increasing number of collocation points. For beam structures, 25 collocation points provided a good balance between accuracy and efficiency, while for shell structures, 1000 points were optimal. The ml-PINN achieved a four-fold increase in computational speed compared to the classical PINN. A further key finding highlighted the capability of the ml-PINN framework to enable generalization learning. By incorporating loading conditions (displacement and uniform load) as input parameters, the ml-PINN demonstrated efficient and accurate prediction under various loading cases without retraining, showcasing its potential as a robust and efficient surrogate model.

The ml-PINN approach successfully addresses the limitations of classical PINNs in handling high-order PDEs in structural mechanics. The decomposition into lower-order equations, coupled with the independent training of multiple neural networks, significantly improved accuracy and computational efficiency. The superior performance compared to FEM is noteworthy, especially considering the challenges FEM faces in real-time simulation and inverse problem solving. The generalization learning capability demonstrated in the study emphasizes the practical applicability of ml-PINN as a versatile tool for structural analysis and prediction, suitable for embedding within digital twin systems.

This study presents a novel ml-PINN framework that significantly enhances the accuracy and computational efficiency of solving high-order PDEs in structural mechanics. The multi-level approach successfully addresses convergence and accuracy issues associated with classical PINNs. The results demonstrate the potential of ml-PINN as a promising paradigm for structural mechanics computation and its suitability for integration into digital twin systems for real-time simulations. Future research could focus on extending the ml-PINN framework to more complex structural elements and nonlinear material behaviors, as well as exploring its application in dynamic analysis and inverse problems.

The current study primarily focuses on linear elastic materials and relatively simple structural configurations. The time-consuming training process of ml-PINN, although improved compared to classical PINN, remains a limitation. Further research is needed to explore strategies for optimizing the training process and extending the framework to more complex material models and loading scenarios. The accuracy at the corners of shell structures showed slightly larger errors compared to the internal regions. This warrants further investigation into boundary condition implementation and training strategies.