A Physics Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticity

• URL: http://arxiv.org/abs/2108.07243v1
• Date: Mon, 16 Aug 2021 17:19:50 GMT
• Title: A Physics Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticity
• Authors: Mohammad Vahab and Ehsan Haghighat and Maryam Khaleghi and Nasser Khalili
• Abstract summary: We explore an application of the Physics Informed Neural Networks (PINNs) in conjunction with Airy stress functions and Fourier series. We find that enriching feature space using Airy stress functions can significantly improve the accuracy of PINN solutions for biharmonic PDEs.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We explore an application of the Physics Informed Neural Networks (PINNs) in conjunction with Airy stress functions and Fourier series to find optimal solutions to a few reference biharmonic problems of elasticity and elastic plate theory. Biharmonic relations are fourth-order partial differential equations (PDEs) that are challenging to solve using classical numerical methods, and have not been addressed using PINNs. Our work highlights a novel application of classical analytical methods to guide the construction of efficient neural networks with the minimal number of parameters that are very accurate and fast to evaluate. In particular, we find that enriching feature space using Airy stress functions can significantly improve the accuracy of PINN solutions for biharmonic PDEs.

Related papers

• Solving Poisson Equations using Neural Walk-on-Spheres [80.1675792181381]
  We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. We demonstrate the superiority of NWoS in accuracy, speed, and computational costs.
  arXiv  Detail & Related papers  (2024-06-05T17:59:22Z)
• Enriched Physics-informed Neural Networks for Dynamic Poisson-Nernst-Planck Systems [0.8192907805418583]
  This paper proposes a meshless deep learning algorithm, enriched physics-informed neural networks (EPINNs) to solve dynamic Poisson-Nernst-Planck (PNP) equations. The EPINNs takes the traditional physics-informed neural networks as the foundation framework, and adds the adaptive loss weight to balance the loss functions. Numerical results indicate that the new method has better applicability than traditional numerical methods in solving such coupled nonlinear systems.
  arXiv  Detail & Related papers  (2024-02-01T02:57:07Z)
• Separable Physics-Informed Neural Networks for the solution of elasticity problems [0.0]
  A method for solving elasticity problems based on separable physics-informed neural networks (SPINN) in conjunction with the deep energy method (DEM) is presented. Numerical experiments have been carried out for a number of problems showing that this method has a significantly higher convergence rate and accuracy than the vanilla physics-informed neural networks (PINN) and even SPINN.
  arXiv  Detail & Related papers  (2024-01-24T14:34:59Z)
• Implicit Stochastic Gradient Descent for Training Physics-informed Neural Networks [51.92362217307946]
  Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems. PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
  arXiv  Detail & Related papers  (2023-03-03T08:17:47Z)
• Physics-aware deep learning framework for linear elasticity [0.0]
  The paper presents an efficient and robust data-driven deep learning (DL) computational framework for linear continuum elasticity problems. For an accurate representation of the field variables, a multi-objective loss function is proposed. Several benchmark problems including the Airimaty solution to elasticity and the Kirchhoff-Love plate problem are solved.
  arXiv  Detail & Related papers  (2023-02-19T20:33:32Z)
• RBF-MGN:Solving spatiotemporal PDEs with Physics-informed Graph Neural Network [4.425915683879297]
  We propose a novel framework based on graph neural networks (GNNs) and radial basis function finite difference (RBF-FD) RBF-FD is used to construct a high-precision difference format of the differential equations to guide model training. We illustrate the generalizability, accuracy, and efficiency of the proposed algorithms on different PDE parameters.
  arXiv  Detail & Related papers  (2022-12-06T10:08:02Z)
• Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations [64.78260098263489]
  In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs) We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
  arXiv  Detail & Related papers  (2022-10-14T15:01:32Z)
• Momentum Diminishes the Effect of Spectral Bias in Physics-Informed Neural Networks [72.09574528342732]
  Physics-informed neural network (PINN) algorithms have shown promising results in solving a wide range of problems involving partial differential equations (PDEs) They often fail to converge to desirable solutions when the target function contains high-frequency features, due to a phenomenon known as spectral bias. In the present work, we exploit neural tangent kernels (NTKs) to investigate the training dynamics of PINNs evolving under gradient descent with momentum (SGDM)
  arXiv  Detail & Related papers  (2022-06-29T19:03:10Z)
• A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method [0.0]
  Physics-informed neural networks (PINNs) are capable of finding the solution for a given boundary value problem. We employ several ideas from the finite element method (FEM) to enhance the performance of existing PINNs in engineering problems.
  arXiv  Detail & Related papers  (2022-06-27T08:18:08Z)
• Auto-PINN: Understanding and Optimizing Physics-Informed Neural Architecture [77.59766598165551]
  Physics-informed neural networks (PINNs) are revolutionizing science and engineering practice by bringing together the power of deep learning to bear on scientific computation. Here, we propose Auto-PINN, which employs Neural Architecture Search (NAS) techniques to PINN design. A comprehensive set of pre-experiments using standard PDE benchmarks allows us to probe the structure-performance relationship in PINNs.
  arXiv  Detail & Related papers  (2022-05-27T03:24:31Z)
• Improved Training of Physics-Informed Neural Networks with Model Ensembles [81.38804205212425]
  We propose to expand the solution interval gradually to make the PINN converge to the correct solution. All ensemble members converge to the same solution in the vicinity of observed data. We show experimentally that the proposed method can improve the accuracy of the found solution.
  arXiv  Detail & Related papers  (2022-04-11T14:05:34Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.