Deep neural networks for solving forward and inverse problems of (2+1)-dimensional nonlinear wave equations with rational solitons

• URL: http://arxiv.org/abs/2112.14040v1
• Date: Tue, 28 Dec 2021 08:41:40 GMT
• Title: Deep neural networks for solving forward and inverse problems of (2+1)-dimensional nonlinear wave equations with rational solitons
• Authors: Zijian Zhou, Li Wang, and Zhenya Yan
• Abstract summary: Inverse problems of the (2+1)-dimensional KP-I equation and spin-NLS equation are studied via deep learning. The main idea of the data-driven forward and inverse problems is to use the deep neural networks with the activation function to approximate the solutions of the considered (2+1)-dimensional nonlinear wave equations.
• Score: 6.529583356640745
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we investigate the forward problems on the data-driven rational solitons for the (2+1)-dimensional KP-I equation and spin-nonlinear Schr\"odinger (spin-NLS) equation via the deep neural networks leaning. Moreover, the inverse problems of the (2+1)-dimensional KP-I equation and spin-NLS equation are studied via deep learning. The main idea of the data-driven forward and inverse problems is to use the deep neural networks with the activation function to approximate the solutions of the considered (2+1)-dimensional nonlinear wave equations by optimizing the chosen loss functions related to the considered nonlinear wave equations.

Related papers

• Solving Poisson Equations using Neural Walk-on-Spheres [80.1675792181381]
  We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. We demonstrate the superiority of NWoS in accuracy, speed, and computational costs.
  arXiv  Detail & Related papers  (2024-06-05T17:59:22Z)
• Physics-constrained convolutional neural networks for inverse problems in spatiotemporal partial differential equations [5.0401589279256065]
  We propose a physics-constrained convolutional neural network (PCCNN) to solve two types of inverse problems in partial differential equations (PDEs) We analyse both linear nonlinear convection-diffusion equations and the Navier-Stokes equations, which govern the chaotic dynamics of turbulent flows. We find that the PC-CNN correctly recovers the true solution for a variety of biases, which are parameterised as non-temporal functions.
  arXiv  Detail & Related papers  (2024-01-18T13:51:48Z)
• Error Analysis of Physics-Informed Neural Networks for Approximating Dynamic PDEs of Second Order in Time [1.123111111659464]
  We consider the approximation of a class of dynamic partial differential equations (PDE) of second order in time by the physics-informed neural network (PINN) approach. Our analyses show that, with feed-forward neural networks having two hidden layers and the $tanh$ activation function, the PINN approximation errors for the solution field can be effectively bounded by the training loss and the number of training data points. We present ample numerical experiments with the new PINN algorithm for the wave equation, the Sine-Gordon equation and the linear elastodynamic equation, which show that the method can capture
  arXiv  Detail & Related papers  (2023-03-22T00:51:11Z)
• Accelerating the training of single-layer binary neural networks using the HHL quantum algorithm [58.720142291102135]
  We show that useful information can be extracted from the quantum-mechanical implementation of Harrow-Hassidim-Lloyd (HHL) This paper shows, however, that useful information can be extracted from the quantum-mechanical implementation of HHL, and used to reduce the complexity of finding the solution on the classical side.
  arXiv  Detail & Related papers  (2022-10-23T11:58:05Z)
• Neural Basis Functions for Accelerating Solutions to High Mach Euler Equations [63.8376359764052]
  We propose an approach to solving partial differential equations (PDEs) using a set of neural networks. We regress a set of neural networks onto a reduced order Proper Orthogonal Decomposition (POD) basis. These networks are then used in combination with a branch network that ingests the parameters of the prescribed PDE to compute a reduced order approximation to the PDE.
  arXiv  Detail & Related papers  (2022-08-02T18:27:13Z)
• Deep Random Vortex Method for Simulation and Inference of Navier-Stokes Equations [69.5454078868963]
  Navier-Stokes equations are significant partial differential equations that describe the motion of fluids such as liquids and air. With the development of AI techniques, several approaches have been designed to integrate deep neural networks in simulating and inferring the fluid dynamics governed by incompressible Navier-Stokes equations. We propose the emphDeep Random Vortex Method (DRVM), which combines the neural network with a random vortex dynamics system equivalent to the Navier-Stokes equation.
  arXiv  Detail & Related papers  (2022-06-20T04:58:09Z)
• Learning Physics-Informed Neural Networks without Stacked Back-propagation [82.26566759276105]
  We develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is two orders of magnitude faster.
  arXiv  Detail & Related papers  (2022-02-18T18:07:54Z)
• Inverse Problem of Nonlinear Schr\"odinger Equation as Learning of Convolutional Neural Network [5.676923179244324]
  It is shown that one can obtain a relatively accurate estimate of the considered parameters using the proposed method. It provides a natural framework in inverse problems of partial differential equations with deep learning.
  arXiv  Detail & Related papers  (2021-07-19T02:54:37Z)
• Data-driven peakon and periodic peakon travelling wave solutions of some nonlinear dispersive equations via deep learning [7.400475825464313]
  We apply the multi-layer physics-informed neural networks (PINNs) deep learning to study the data-driven peakon and periodic peakon solutions of some well-known nonlinear dispersion equations. Results will be useful to further study the peakon solutions and corresponding experimental design of nonlinear dispersive equations.
  arXiv  Detail & Related papers  (2021-01-12T09:50:28Z)
• Data-driven rogue waves and parameter discovery in the defocusing NLS equation with a potential using the PINN deep learning [7.400475825464313]
  We use the multi-layer PINN deep learning method to study the data-driven rogue wave solutions of the defocusing nonlinear Schr"odinger (NLS) equation with the time-dependent potential. Results will be useful to further discuss the rogue wave solutions of the defocusing NLS equation with a potential in the study of deep learning neural networks.
  arXiv  Detail & Related papers  (2020-12-18T00:09:21Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)

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.