PINN-based form-finding of compressive shells under vertical and horizontal loads: comparison with machine learning and optimization methods

The design of compression-only shells under combined vertical and horizontal loading requires the identification of geometries that satisfy membrane equilibrium while remaining fully compressive. Conventional form-finding methods achieve this through the constrained optimization of a concave Airy stress function, but this process can be computationally demanding and sensitive to initial conditions. To address these limitations, this paper presents a Physics-Informed Neural Network (PINN) framework for form-finding compressive shells subjected to both vertical and horizontal loads. The proposed PINN embeds the governing equations of membrane equilibrium directly into its loss function, enabling the network to infer the optimal Airy stress parameters that define the shell geometry while ensuring equilibrium and concavity. The approach allows the structural form to be learned in a physics-consistent manner, offering improved flexibility and interpretability compared to traditional optimization. To evaluate its performance, the PINN is benchmarked against both machine learning regression techniques, specifically XGBoost, Random Forest, and k-Nearest Neighbours, and two numerical optimizers. The comparative analysis highlights the PINN’s capability to accurately identify equilibrium-consistent forms while maintaining computational efficiency.