Learning Parameterized Nonlinear Elasticity on Curved Surfaces

2026-04-13

AI Summary

This work addresses the limitations of traditional approaches to nonlinear elasticity on curved surfaces, which rely on symmetry assumptions and require repeated solves for each combination of geometric and material parameters, thereby struggling with continuously varying parameters or broken symmetries. The authors propose a unified solution framework based on physics-informed neural networks (PINNs), wherein the governing equations of nonlinear elasticity—formulated using differential geometry—and associated boundary conditions are hard-encoded into the loss function. This enables a single model to generalize across a continuous parameter space. Notably, the method achieves the first unified modeling of nonlinear elastic systems with fivefold defects on spherical surfaces across varying parameters, accurately reproducing known analytical and numerical solutions while successfully extrapolating to parameter combinations beyond the training range, thus overcoming key constraints of conventional solvers.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

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.

Problem

• nonlinear elasticity
• curved surfaces
• parameterized systems
• elastic manifolds
• scalability

Innovation

• physics-informed neural networks
• nonlinear elasticity
• curved manifolds
• parameterized modeling
• topological defects