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

arXiv: https://arxiv.org/abs/2412.09453 · HTML: https://mlsanity.com/paper/2412.09453/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 78 references that are in the map.
- First posted on arXiv 12 Dec 2024.

## Builds on (most cited first, 25 of 25)

- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks](https://mlsanity.com/paper/2005.03596/) (2020)
- [Physics informed neural networks for continuum micromechanics](https://mlsanity.com/paper/2110.07374/) (2021)
- [Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios](https://mlsanity.com/paper/2205.07731/) (2022)
- [Physics-Informed Neural Networks for Nonhomogeneous Material Identification in Elasticity Imaging](https://mlsanity.com/paper/2009.04525/) (2020)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [Delta-PINNs: physics-informed neural networks on complex geometries](https://mlsanity.com/paper/2209.03984/) (2022)
- [Meshless physics-informed deep learning method for three-dimensional solid mechanics](https://mlsanity.com/paper/2012.01547/) (2020)
- [Solving Partial Differential Equations with Point Source Based on Physics-Informed Neural Networks](https://mlsanity.com/paper/2111.01394/) (2021)
- [Physics-Informed Neural Networks for Shell Structures](https://mlsanity.com/paper/2207.14291/) (2022)
- [Physics-informed neural networks for modeling rate- and temperature-dependent plasticity](https://mlsanity.com/paper/2201.08363/) (2022)
- [Identifying Constitutive Parameters for Complex Hyperelastic Materials using Physics-Informed Neural Networks](https://mlsanity.com/paper/2308.15640/) (2023)
- [Enriched physics-informed neural networks for in-plane crack problems: Theory and MATLAB codes](https://mlsanity.com/paper/2206.08750/) (2022)

## Built on by (most cited first, 1 of 1)

- [Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learning](https://mlsanity.com/paper/2511.06802/) (2025)

## Abstract (arXiv)

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.