Transformed Physics-Informed Neural Networks for The Convection-Diffusion Equation

• URL: http://arxiv.org/abs/2409.07671v1
• Date: Thu, 12 Sep 2024 00:24:21 GMT
• Title: Transformed Physics-Informed Neural Networks for The Convection-Diffusion Equation
• Authors: Jiajing Guan, Howard Elman,
• Abstract summary: Singularly perturbed problems have solutions with steep boundary layers that are hard to resolve numerically. Traditional numerical methods, such as Finite Difference Methods, require a refined mesh to obtain stable and accurate solutions. We consider the use of Physics-Informed Neural Networks (PINNs) to produce numerical solutions of singularly perturbed problems.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Singularly perturbed problems are known to have solutions with steep boundary layers that are hard to resolve numerically. Traditional numerical methods, such as Finite Difference Methods (FDMs), require a refined mesh to obtain stable and accurate solutions. As Physics-Informed Neural Networks (PINNs) have been shown to successfully approximate solutions to differential equations from various fields, it is natural to examine their performance on singularly perturbed problems. The convection-diffusion equation is a representative example of such a class of problems, and we consider the use of PINNs to produce numerical solutions of this equation. We study two ways to use PINNS: as a method for correcting oscillatory discrete solutions obtained using FDMs, and as a method for modifying reduced solutions of unperturbed problems. For both methods, we also examine the use of input transformation to enhance accuracy, and we explain the behavior of input transformations analytically, with the help of neural tangent kernels.

Related papers

• Discovery of Quasi-Integrable Equations from traveling-wave data using the Physics-Informed Neural Networks [0.0]
  PINNs are used to study vortex solutions in 2+1 dimensional nonlinear partial differential equations. We consider PINNs with conservation laws (referred to as cPINNs), deformations of the initial profiles, and a friction approach to improve the identification's resolution.
  arXiv  Detail & Related papers  (2024-10-23T08:29:13Z)
• Augmented neural forms with parametric boundary-matching operators for solving ordinary differential equations [0.0]
  This paper introduces a formalism for systematically crafting proper neural forms with boundary matches that are amenable to optimization. It describes a novel technique for converting problems with Neumann or Robin conditions into equivalent problems with parametric Dirichlet conditions. The proposed augmented neural forms approach was tested on a set of diverse problems, encompassing first- and second-order ordinary differential equations, as well as first-order systems.
  arXiv  Detail & Related papers  (2024-04-30T11:10:34Z)
• Enhanced physics-informed neural networks with domain scaling and residual correction methods for multi-frequency elliptic problems [11.707981310045742]
  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.
  arXiv  Detail & Related papers  (2023-11-07T06:08:47Z)
• 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)
• HomPINNs: homotopy physics-informed neural networks for solving the inverse problems of nonlinear differential equations with multiple solutions [6.89453634946458]
  We propose homotopy physics-informed neural networks (HomPINNs) to solve inverse problems of nonlinear differential equations (DEs) The proposed framework begins with the use of NNs to simultaneously approximate unlabeled observations across diverse solutions while adhering to DE constraints. Our findings demonstrate that the proposed method is scalable and adaptable, providing an effective solution for solving DEs with multiple solutions and unknown parameters.
  arXiv  Detail & Related papers  (2023-04-06T01:20:23Z)
• 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)
• Semi-analytic PINN methods for singularly perturbed boundary value problems [0.8594140167290099]
  We propose a new semi-analytic physics informed neural network (PINN) to solve singularly perturbed boundary value problems. The PINN is a scientific machine learning framework that offers a promising perspective for finding numerical solutions to partial differential equations.
  arXiv  Detail & Related papers  (2022-08-19T04:26:40Z)
• Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditions [68.8204255655161]
  We develop a physics-informed neural network (PINN) model for parabolic problems with a sharply perturbed initial condition. Localized large gradients in the ADE solution make the (common in PINN) Latin hypercube sampling of the equation's residual highly inefficient. We propose criteria for weights in the loss function that produce a more accurate PINN solution than those obtained with the weights selected via other methods.
  arXiv  Detail & Related papers  (2022-08-18T05:00:24Z)
• 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)
• Conditional physics informed neural networks [85.48030573849712]
  We introduce conditional PINNs (physics informed neural networks) for estimating the solution of classes of eigenvalue problems. We show that a single deep neural network can learn the solution of partial differential equations for an entire class of problems.
  arXiv  Detail & Related papers  (2021-04-06T18:29:14Z)
• 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.