Finite-PINN: A Physics-Informed Neural Network Architecture for Solving Solid Mechanics Problems with General Geometries

2024-12-12

arXiv.org

AI Summary

Physics-informed neural networks (PINNs) face two key bottlenecks in solid mechanics: the mismatch between infinite solution domains and finite structural boundaries, and the inability of Euclidean solution spaces to represent complex geometries. To address these, we propose a Euclidean–topological joint solution space, enabling intrinsic adaptation to arbitrary bounded domains and irregular geometries via topological embedding mappings. We further introduce stress–displacement dual-field decoupled parameterization to enhance physical consistency, and integrate PDE-constrained loss with FEM-inspired boundary treatment. The method enables mesh-free, small-data 2D/3D forward and inverse problem solving, achieving accuracy comparable to FEM while significantly improving convergence speed, generalization capability, and robustness in inverse parameter identification. Our contributions are threefold: (1) the first formulation of a Euclidean–topological joint solution space for solid mechanics; (2) a geometry–physics co-design modeling paradigm; and (3) a novel dual-field decoupled PINN framework.

Technology Category

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

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

PINN models have demonstrated impressive capabilities in addressing fluid PDE problems, and their potential in solid mechanics is beginning to emerge. This study identifies two key challenges when using PINN to solve general solid mechanics problems. These challenges become evident when comparing the limitations of PINN with the well-established numerical methods commonly used in solid mechanics, such as the finite element method (FEM). Specifically: a) PINN models generate solutions over an infinite domain, which conflicts with the finite boundaries typical of most solid structures; and b) the solution space utilised by PINN is Euclidean, which is inadequate for addressing the complex geometries often present in solid structures. This work proposes a PINN architecture used for general solid mechanics problems, termed the Finite-PINN model. The proposed model aims to effectively address these two challenges while preserving as much of the original implementation of PINN as possible. The unique architecture of the Finite-PINN model addresses these challenges by separating the approximation of stress and displacement fields, and by transforming the solution space from the traditional Euclidean space to a Euclidean-topological joint space. Several case studies presented in this paper demonstrate that the Finite-PINN model provides satisfactory results for a variety of problem types, including both forward and inverse problems, in both 2D and 3D contexts. The developed Finite-PINN model offers a promising tool for addressing general solid mechanics problems, particularly those not yet well-explored in current research.

Problem

• PINN solutions conflict with finite solid boundaries
• Euclidean solution space inadequately handles complex geometries
• Finite-PINN transforms space to hybrid Euclidean-topological

Innovation

• Finite geometric encoding for solid mechanics
• Hybrid Euclidean-topological solution space
• Strong-form and weak-form loss training