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

Abstract

Form-finding of unilateral membrane structures is commonly addressed by solving equilibrium equations with Finite Element Methods (FEMs). This paper investigates Physics-Informed Neural Networks (PINNs) as an alternative, where the equilibrium equation is enforced by minimizing its residual at collocation points during neural-network training rather than by solving a mesh-based discretized system. This approach is well suited to form-finding problems based on Membrane Equilibrium Analysis (MEA), in which the unknown membrane surface is governed by a second-order elliptic Partial Differential Equation (PDE) with Dirichlet boundary conditions. Two PINN formulations are proposed and compared: a soft-Boundary Condition (soft-BC) approach, where the boundary conditions are imposed through a penalty term, and a hard-BC approach, where they are satisfied exactly by construction through distance and lift functions. The methods are assessed on three case studies with different geometrical complexity, including compression-only and tension-only stress states, and combined self-weight, concentrated vertical loads, and horizontal actions. Both formulations produce membrane surfaces in close agreement with solutions obtained using an FEM-based PDE solver. The hard-BC formulation gives smaller errors and a smoother residual distribution, especially near the boundary, showing that exact enforcement of the Dirichlet conditions improves overall accuracy. The soft-BC formulation still provides structurally meaningful solutions and remains attractive when simpler implementation is preferred and limited relaxation of the boundary data is acceptable. Overall, the results show that PINNs are a viable alternative for MEA-based form-finding.