Physics-informed neural networks for form-finding of unilateral membrane structures

The form-finding of unilateral membrane structures is often addressed by solving equilibrium equations using Finite Element Methods (FEMs). In this paper, Physics-Informed Neural Networks (PINNs) are investigated as an alternative, in which the equilibrium equation is enforced by minimizing its residual at collocation points through neural-network training, rather than by solving a mesh-based discretized system. This alternative is particularly suitable for form-finding problems based on Membrane Equilibrium Analysis (MEA), where the unknown is a membrane governed by a second-order elliptic Partial Differential Equation (PDE) with Dirichlet boundary conditions. This paper proposes PINNs for the form-finding of unilateral membrane structures governed by prescribed Airy-stress fields and compares two formulations: a soft-Boundary Condition (soft-BC) approach, in which the boundary conditions are enforced through a penalty term, and a hard-BC approach, in which they are satisfied by construction through distance and lift functions. The two approaches are assessed on three case studies of different geometrical complexity, involving both compression-only and tension-only stress states, as well as combined self-weight, concentrated vertical loads, and horizontal actions. The results show that both formulations provide membrane surfaces in close agreement with solutions obtained with a PDE solver based on the FEM. The hard-BC formulation yields smaller errors and a smoother spatial distribution of the residual, especially near the boundary, indicating that the exact treatment of the Dirichlet conditions has a direct influence on the overall solution accuracy. The soft-BC formulation, however, still provides structurally meaningful solutions and remains attractive when a simpler implementation is preferred and a limited relaxation of the boundary data is acceptable. Overall, the study demonstrates that PINNs are a viable numerical alternative for MEA-based form-finding and that the choice between soft and hard boundary enforcement depends on the prescribed boundary condition in the structural problem.