Deep neural network for solving differential equations motivated by Legendre-Galerkin approximation

• URL: http://arxiv.org/abs/2010.12975v1
• Date: Sat, 24 Oct 2020 20:25:09 GMT
• Title: Deep neural network for solving differential equations motivated by Legendre-Galerkin approximation
• Authors: Bryce Chudomelka and Youngjoon Hong and Hyunwoo Kim and Jinyoung Park
• Abstract summary: We explore the performance and accuracy of various neural architectures on both linear and nonlinear differential equations. We implement a novel Legendre-Galerkin Deep Neural Network (LGNet) algorithm to predict solutions to differential equations.
• Score: 16.64525769134209
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is associated with solving these systems. We explore the performance and accuracy of various neural architectures on both linear and nonlinear differential equations by creating accurate training sets with the spectral element method. Next, we implement a novel Legendre-Galerkin Deep Neural Network (LGNet) algorithm to predict solutions to differential equations. By constructing a set of a linear combination of the Legendre basis, we predict the corresponding coefficients, $\alpha_i$ which successfully approximate the solution as a sum of smooth basis functions $u \simeq \sum_{i=0}^{N} \alpha_i \varphi_i$. As a computational example, linear and nonlinear models with Dirichlet or Neumann boundary conditions are considered.

Related papers

• Divergence-free algorithms for solving nonlinear differential equations on quantum computers [0.27624021966289597]
  We propose algorithms of divergence-free simulation for nonlinear differential equations in quantum computers. The solution of nonlinear differential equations free from evolution time constraints opens the door to practical applications of quantum computers.
  arXiv  Detail & Related papers  (2024-11-25T09:47:24Z)
• A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations [0.6040014326756179]
  We propose a novel backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward differential equations. The deep neural network (DNN) models are trained not only on the inputs and labels but also the differentials of the corresponding labels.
  arXiv  Detail & Related papers  (2024-04-12T13:05:35Z)
• NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition [67.46012350241969]
  This paper proposes a general acceleration methodology called NeuralStagger. It decomposing the original learning tasks into several coarser-resolution subtasks. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations.
  arXiv  Detail & Related papers  (2023-02-20T19:36:52Z)
• A Recursively Recurrent Neural Network (R2N2) Architecture for Learning Iterative Algorithms [64.3064050603721]
  We generalize Runge-Kutta neural network to a recurrent neural network (R2N2) superstructure for the design of customized iterative algorithms. We demonstrate that regular training of the weight parameters inside the proposed superstructure on input/output data of various computational problem classes yields similar iterations to Krylov solvers for linear equation systems, Newton-Krylov solvers for nonlinear equation systems, and Runge-Kutta solvers for ordinary differential equations.
  arXiv  Detail & Related papers  (2022-11-22T16:30:33Z)
• DEQGAN: Learning the Loss Function for PINNs with Generative Adversarial Networks [1.0499611180329804]
  This work presents Differential Equation GAN (DEQGAN), a novel method for solving differential equations using generative adversarial networks. We show that DEQGAN achieves multiple orders of magnitude lower mean squared errors than PINNs. We also show that DEQGAN achieves solution accuracies that are competitive with popular numerical methods.
  arXiv  Detail & Related papers  (2022-09-15T06:39:47Z)
• 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)
• Meta-Solver for Neural Ordinary Differential Equations [77.8918415523446]
  We investigate how the variability in solvers' space can improve neural ODEs performance. We show that the right choice of solver parameterization can significantly affect neural ODEs models in terms of robustness to adversarial attacks.
  arXiv  Detail & Related papers  (2021-03-15T17:26:34Z)
• Large-scale Neural Solvers for Partial Differential Equations [48.7576911714538]
  Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. Recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. We examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) We discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.
  arXiv  Detail & Related papers  (2020-09-08T13:26:51Z)
• Unsupervised Learning of Solutions to Differential Equations with Generative Adversarial Networks [1.1470070927586016]
  We develop a novel method for solving differential equations with unsupervised neural networks. We show that our method, which we call Differential Equation GAN (DEQGAN), can obtain multiple orders of magnitude lower mean squared errors.
  arXiv  Detail & Related papers  (2020-07-21T23:36:36Z)
• 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.