We learn parameterized nonlinear elasticity on curved surfaces using a physics-informed neural network that enforces governing equations and boundary conditions directly through the loss function, enabling a single trained model to represent a continuous family of elastic equilibria across geometric and material parameters. Nonlinear elasticity on curved manifolds underlies the mechanics of crystalline shells, elastic membranes, and viral capsids, where curvature and topological defects determine equilibrium structure and stability. Traditional exact and finite element solvers rely on symmetry reduction and must be reinitialized for each parameter choice, limiting scalability when symmetry is broken or parameters vary. We validate the proposed learning-based solver on a benchmark problem from curved elasticity, namely the one-dimensional single disclination on a spheroidal surface with known exact and numerical solutions. The network accurately reproduces these solutions, including parameter combinations excluded from training, demonstrating generalization across geometry and material regimes. This study establishes a scalable framework for learning nonlinear elastic systems on curved manifolds and lays the groundwork for extensions to fully two-dimensional and multi-defect configurations relevant to protein shells and other curved elastic networks.

翻译：我们通过一种物理信息神经网络学习曲面上的参数化非线性弹性，该网络直接通过损失函数强制满足控制方程和边界条件，使得单个训练模型能够表示跨几何与材料参数的连续弹性平衡族。曲面流形上的非线性弹性构成了晶体壳、弹性膜和病毒衣壳的力学基础，在这些结构中，曲率和拓扑缺陷决定了平衡结构与稳定性。传统的精确解和有限元求解器依赖于对称性约化，且需为每个参数选择重新初始化，这在对称性破缺或参数变化时限制了可扩展性。我们基于曲面弹性中的一个基准问题——即椭球面上具有已知精确解与数值解的一维单根向错——验证了所提出的基于学习的求解器。该网络能准确复现这些解，包括训练中未涉及的参数组合，展现了跨几何与材料区间的泛化能力。本研究为在曲面流形上学习非线性弹性系统建立了可扩展框架，并为拓展至蛋白质壳及其他曲面弹性网络相关的全二维与多缺陷构型奠定了基础。