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.

翻译：单边膜结构的形态寻优通常通过有限元方法求解平衡方程来实现。本文研究物理信息神经网络作为一种替代方法，通过在神经网络训练过程中最小化配点处残差来强制满足平衡方程，而非求解基于网格的离散化系统。该方法特别适用于基于膜平衡分析的形态寻优问题，其中未知膜面由带有狄利克雷边界条件的二阶椭圆偏微分方程控制。本文提出并比较了两种物理信息神经网络公式：软边界条件方法，通过惩罚项施加边界条件；以及硬边界条件方法，通过距离函数和升函数在构造中精确满足边界条件。通过三个具有不同几何复杂度的案例评估了该方法，包括纯压和纯拉应力状态，以及组合自重、集中竖向载荷和水平作用。两种公式生成的膜面与基于有限元方法的偏微分方程求解器所得解高度吻合。硬边界条件公式产生的误差更小且残差分布更平滑（尤其在边界附近），表明精确强制狄利克雷条件可提升整体精度。软边界条件公式仍能产生具有结构意义的解，并在优先考虑实现简单性且允许边界数据适度松弛时具有吸引力。总体而言，结果表明物理信息神经网络是膜平衡分析形态寻优的一种可行替代方案。