Semi-analytic PINN methods for singularly perturbed boundary value
problems
- URL: http://arxiv.org/abs/2208.09145v1
- Date: Fri, 19 Aug 2022 04:26:40 GMT
- Title: Semi-analytic PINN methods for singularly perturbed boundary value
problems
- Authors: Gung-Min Gie, Youngjoon Hong, Chang-Yeol Jung
- Abstract summary: 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.
- Score: 0.8594140167290099
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: 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. The PINNs have shown
impressive performance in solving various differential equations including
time-dependent and multi-dimensional equations involved in a complex geometry
of the domain. However, when considering stiff differential equations, neural
networks in general fail to capture the sharp transition of solutions, due to
the spectral bias. To resolve this issue, here we develop the semi-analytic
PINN methods, enriched by using the so-called corrector functions obtained from
the boundary layer analysis. Our new enriched PINNs accurately predict
numerical solutions to the singular perturbation problems. Numerical
experiments include various types of singularly perturbed linear and nonlinear
differential equations.
Related papers
- Transformed Physics-Informed Neural Networks for The Convection-Diffusion Equation [0.0]
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.
arXiv Detail & Related papers (2024-09-12T00:24:21Z) - General-Kindred Physics-Informed Neural Network to the Solutions of Singularly Perturbed Differential Equations [11.121415128908566]
We propose the General-Kindred Physics-Informed Neural Network (GKPINN) for solving Singular Perturbation Differential Equations (SPDEs)
This approach utilizes prior knowledge of the boundary layer from the equation and establishes a novel network to assist PINN in approxing the boundary layer.
The research findings underscore the exceptional performance of our novel approach, GKPINN, which delivers a remarkable enhancement in reducing the $L$ error by two to four orders of magnitude compared to the established PINN methodology.
arXiv Detail & Related papers (2024-08-27T02:03:22Z) - 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) - Physics-Aware Neural Networks for Boundary Layer Linear Problems [0.0]
Physics-Informed Neural Networks (PINNs) approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost of a Neural Network.
This paper explores PINNs for linear PDEs whose solutions may present one or more boundary layers.
arXiv Detail & Related papers (2022-07-15T21:15:06Z) - 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) - 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) - Spectrally Adapted Physics-Informed Neural Networks for Solving
Unbounded Domain Problems [0.0]
In this work, we combine two classes of numerical methods: (i) physics-informed neural networks (PINNs) and (ii) adaptive spectral methods.
The numerical methods that we develop take advantage of the ability of physics-informed neural networks to easily implement high-order numerical schemes to efficiently solve PDEs.
We then show how recently introduced adaptive techniques for spectral methods can be integrated into PINN-based PDE solvers to obtain numerical solutions of unbounded domain problems that cannot be efficiently approximated by standard PINNs.
arXiv Detail & Related papers (2022-02-06T05:25:22Z) - Characterizing possible failure modes in physics-informed neural
networks [55.83255669840384]
Recent work in scientific machine learning has developed so-called physics-informed neural network (PINN) models.
We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena even for simple PDEs.
We show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN's setup makes the loss landscape very hard to optimize.
arXiv Detail & Related papers (2021-09-02T16:06:45Z) - Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable
domain decomposition approach for solving differential equations [20.277873724720987]
We propose a new, scalable approach for solving large problems relating to differential equations called Finite Basis PINNs (FBPINNs)
FBPINNs are inspired by classical finite element methods, where the solution of the differential equation is expressed as the sum of a finite set of basis functions with compact support.
In FBPINNs neural networks are used to learn these basis functions, which are defined over small, overlapping subdomain problems.
arXiv Detail & Related papers (2021-07-16T13:03:47Z) - 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.