Physics-Aware Neural Networks for Boundary Layer Linear Problems

• URL: http://arxiv.org/abs/2208.12559v1
• Date: Fri, 15 Jul 2022 21:15:06 GMT
• Title: Physics-Aware Neural Networks for Boundary Layer Linear Problems
• Authors: Antonio Tadeu Azevedo Gomes and Larissa Miguez da Silva and Frederic Valentin
• Abstract summary: 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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most pieces of work in the area of PINNs tackle non-linear PDEs. Nevertheless, many interesting problems involving linear PDEs may benefit from PINNs; these include parametric studies, multi-query problems, and parabolic (transient) PDEs. The purpose of this paper is to explore PINNs for linear PDEs whose solutions may present one or more boundary layers. More specifically, we analyze the steady-state reaction-advection-diffusion equation in regimes in which the diffusive coefficient is small in comparison with the reactive or advective coefficients. We show that adding information about these coefficients as predictor variables in a PINN results in better prediction models than in a PINN that only uses spatial information as predictor variables. This finding may be instrumental in multiscale problems where the coefficients of the PDEs present high variability in small spatiotemporal regions of the domain, and therefore PINNs may be employed together with domain decomposition techniques to efficiently approximate the PDEs locally at each partition of the spatiotemporal domain, without resorting to different learned PINN models at each of these partitions.

Related papers

• Functional Tensor Decompositions for Physics-Informed Neural Networks [8.66932181641177]
  Physics-Informed Neural Networks (PINNs) have shown continuous and increasing promise in approximating partial differential equations (PDEs) We propose a generalized PINN version of the classical variable separable method. Our methodology significantly enhances the performance of PINNs, as evidenced by improved results on complex high-dimensional PDEs.
  arXiv  Detail & Related papers  (2024-08-23T14:24:43Z)
• Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks [17.69666422395703]
  Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) In this paper, we establish rigorous analysis of the physics-informed convolutional neural network (PICNN) for solving PDEs on the sphere.
  arXiv  Detail & Related papers  (2023-08-18T14:58:23Z)
• 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)
• 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)
• Revisiting PINNs: Generative Adversarial Physics-informed Neural Networks and Point-weighting Method [70.19159220248805]
  Physics-informed neural networks (PINNs) provide a deep learning framework for numerically solving partial differential equations (PDEs) We propose the generative adversarial neural network (GA-PINN), which integrates the generative adversarial (GA) mechanism with the structure of PINNs. Inspired from the weighting strategy of the Adaboost method, we then introduce a point-weighting (PW) method to improve the training efficiency of PINNs.
  arXiv  Detail & Related papers  (2022-05-18T06:50:44Z)
• Lie Point Symmetry Data Augmentation for Neural PDE Solvers [69.72427135610106]
  We present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity. In the context of PDEs, it turns out that we are able to quantitatively derive an exhaustive list of data transformations. We show how it can easily be deployed to improve neural PDE solver sample complexity by an order of magnitude.
  arXiv  Detail & Related papers  (2022-02-15T18:43:17Z)
• 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)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs [0.0]
  We derive rigorous bounds on the error incurred by PINNs in approximating the solutions of a large class of parabolic PDEs. We construct neural networks, whose PINN residual (generalization error) can be made as small as desired. These results enable us to provide a comprehensive error analysis for PINNs in approximating Kolmogorov PDEs.
  arXiv  Detail & Related papers  (2021-06-28T08:37:56Z)
• dNNsolve: an efficient NN-based PDE solver [62.997667081978825]
  We introduce dNNsolve, that makes use of dual Neural Networks to solve ODEs/PDEs. We show that dNNsolve is capable of solving a broad range of ODEs/PDEs in 1, 2 and 3 spacetime dimensions.
  arXiv  Detail & Related papers  (2021-03-15T19:14:41Z)
• 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.