mlsanityData to 14 Oct 2026

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

arXiv 2412.09453 ↗First posted 12 Dec 2024cs.CE cs.LG math.APcited by 1

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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.

Key facts. Cited by 1 paper 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.

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25 of 78 references.

References · 25

  1. 2020Implicit Neural Representations with Periodic Activation Functionscited by 2,238
  2. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  3. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
  4. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  5. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  6. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  7. 2021Physics-informed neural networks with hard constraints for inverse designcited by 280
  8. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 222
  9. 2021Exact imposition of boundary conditions with distance functions in physics-informed deep neural networkscited by 137
  10. 2019An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applicationscited by 118
  11. 2019Transfer learning enhanced physics informed neural network for phase-field modeling of fracturecited by 116
  12. 2021Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problemscited by 102
13 more
  1. 2022Physics-informed neural networks for inverse problems in supersonic flowscited by 76
  2. 2020Physics-informed neural network for ultrasound nondestructive quantification of surface breaking crackscited by 68
  3. 2021Physics informed neural networks for continuum micromechanicscited by 46
  4. 2022Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarioscited by 46
  5. 2020Physics-Informed Neural Networks for Nonhomogeneous Material Identification in Elasticity Imagingcited by 40
  6. 2022A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element methodcited by 33
  7. 2022Delta-PINNs: physics-informed neural networks on complex geometriescited by 28
  8. 2020Meshless physics-informed deep learning method for three-dimensional solid mechanicscited by 27
  9. 2021Solving Partial Differential Equations with Point Source Based on Physics-Informed Neural Networkscited by 14
  10. 2022Physics-Informed Neural Networks for Shell Structurescited by 13
  11. 2022Physics-informed neural networks for modeling rate- and temperature-dependent plasticitycited by 11
  12. 2023Identifying Constitutive Parameters for Complex Hyperelastic Materials using Physics-Informed Neural Networkscited by 3
  13. 2022Enriched physics-informed neural networks for in-plane crack problems: Theory and MATLAB codescited by 2

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