Enhanced physics-informed neural networks with domain scaling and residual correction methods for multi-frequency elliptic problems

• URL: http://arxiv.org/abs/2311.03746v1
• Date: Tue, 7 Nov 2023 06:08:47 GMT
• Title: Enhanced physics-informed neural networks with domain scaling and residual correction methods for multi-frequency elliptic problems
• Authors: Deok-Kyu Jang, Hyea Hyun Kim, Kyungsoo Kim
• Abstract summary: Neural network approximation methods are developed for elliptic partial differential equations with multi-frequency solutions. The efficiency and accuracy of the proposed methods are demonstrated for multi-frequency model problems.
• Score: 11.707981310045742
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: In this paper, neural network approximation methods are developed for elliptic partial differential equations with multi-frequency solutions. Neural network work approximation methods have advantages over classical approaches in that they can be applied without much concerns on the form of the differential equations or the shape or dimension of the problem domain. When applied to problems with multi-frequency solutions, the performance and accuracy of neural network approximation methods are strongly affected by the contrast of the high- and low-frequency parts in the solutions. To address this issue, domain scaling and residual correction methods are proposed. The efficiency and accuracy of the proposed methods are demonstrated for multi-frequency model problems.

Related papers

• Chebyshev Spectral Neural Networks for Solving Partial Differential Equations [0.0]
  The study uses a feedforward neural network model and error backpropagation principles, utilizing automatic differentiation (AD) to compute the loss function. The numerical efficiency and accuracy of the CSNN model are investigated through testing on elliptic partial differential equations, and it is compared with the well-known Physics-Informed Neural Network(PINN) method.
  arXiv  Detail & Related papers  (2024-06-06T05:31:45Z)
• Physics-Informed Generator-Encoder Adversarial Networks with Latent Space Matching for Stochastic Differential Equations [14.999611448900822]
  We propose a new class of physics-informed neural networks to address the challenges posed by forward, inverse, and mixed problems in differential equations. Our model consists of two key components: the generator and the encoder, both updated alternately by gradient descent. In contrast to previous approaches, we employ an indirect matching that operates within the lower-dimensional latent feature space.
  arXiv  Detail & Related papers  (2023-11-03T04:29:49Z)
• Solutions to Elliptic and Parabolic Problems via Finite Difference Based Unsupervised Small Linear Convolutional Neural Networks [1.124958340749622]
  We propose a fully unsupervised approach, requiring no training data, to estimate finite difference solutions for PDEs directly via small linear convolutional neural networks. Our proposed approach uses substantially fewer parameters than similar finite difference-based approaches while also demonstrating comparable accuracy to the true solution for several selected elliptic and parabolic problems.
  arXiv  Detail & Related papers  (2023-11-01T03:15:10Z)
• An Optimization-based Deep Equilibrium Model for Hyperspectral Image Deconvolution with Convergence Guarantees [71.57324258813675]
  We propose a novel methodology for addressing the hyperspectral image deconvolution problem. A new optimization problem is formulated, leveraging a learnable regularizer in the form of a neural network. The derived iterative solver is then expressed as a fixed-point calculation problem within the Deep Equilibrium framework.
  arXiv  Detail & Related papers  (2023-06-10T08:25:16Z)
• A Block-Coordinate Approach of Multi-level Optimization with an Application to Physics-Informed Neural Networks [0.0]
  We propose a multi-level algorithm for the solution of nonlinear optimization problems and analyze its evaluation complexity. We apply it to the solution of partial differential equations using physics-informed neural networks (PINNs) and show on a few test problems that the approach results in better solutions and significant computational savings.
  arXiv  Detail & Related papers  (2023-05-23T19:12:02Z)
• A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks [52.5899851000193]
  We develop an ODE based IVP solver which prevents the network from getting ill-conditioned and runs in time linear in the number of parameters. We show that current methods based on this approach suffer from two key issues. First, following the ODE produces an uncontrolled growth in the conditioning of the problem, ultimately leading to unacceptably large numerical errors.
  arXiv  Detail & Related papers  (2023-04-28T17:28:18Z)
• 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)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Hierarchical Learning to Solve Partial Differential Equations Using Physics-Informed Neural Networks [2.0305676256390934]
  We propose a hierarchical approach to improve the convergence rate and accuracy of the neural network solution to partial differential equations. We validate the efficiency and robustness of the proposed hierarchical approach through a suite of linear and nonlinear partial differential equations.
  arXiv  Detail & Related papers  (2021-12-02T13:53:42Z)
• Physics informed neural networks for continuum micromechanics [68.8204255655161]
  Recently, physics informed neural networks have successfully been applied to a broad variety of problems in applied mathematics and engineering. Due to the global approximation, physics informed neural networks have difficulties in displaying localized effects and strong non-linear solutions by optimization. It is shown, that the domain decomposition approach is able to accurately resolve nonlinear stress, displacement and energy fields in heterogeneous microstructures obtained from real-world $mu$CT-scans.
  arXiv  Detail & Related papers  (2021-10-14T14:05:19Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

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.