Finite-PINN: A Physics-Informed Neural Network with Finite Geometric Encoding for Solid Mechanics

TL;DR

This paper introduces Finite-PINN, a novel physics-informed neural network architecture that incorporates finite geometric encoding to better handle complex geometries and finite domains in solid mechanics problems.

Contribution

The study proposes a Finite-PINN model that integrates finite geometric encoding into PINNs, addressing domain finiteness and complex geometries in solid mechanics.

Findings

Efficiently approximates solutions for forward solid mechanics problems.

Reconstructs full-field solutions from sparse observations in inverse problems.

Retains core PINN framework while enhancing geometric handling.

Abstract

PINN models have demonstrated 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 presents a PINN architecture for general solid mechanics problems, referred to as the Finite-PINN model. The model is designed to effectively tackle two key challenges, while retaining as much of the original PINN framework as possible. To this end, the Finite-PINN incorporates finite geometric encoding into the neural network inputs, thereby transforming the solution space from a conventional Euclidean space into a hybrid Euclidean-topological space. The model is comprehensively trained using both strong-form and weak-form loss formulations, enabling its application to a wide range of forward and inverse problems in solid mechanics. For forward problems, the Finite-PINN model efficiently approximates solutions to solid mechanics problems when the geometric information of a given structure has been preprocessed. For inverse problems, it effectively reconstructs full-field solutions from very sparse observations by embedding both physical laws and geometric information within its architecture.