Solving Huxley equation using an improved PINN method

Bai, Yuexing; Chaolu, Temuer; Bilige, Sudao

doi:10.1007/s11071-021-06819-z

Cited by 15 publications

(6 citation statements)

References 34 publications

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“…From Figure 4, we can see that with the increase in regional configuration points 1000, 2000,3000, 4000 f N = , the fitting effect of the exact rogue wave solution and predicted rogue wave solution is getting better and better. In order to study the influence of sampling point f N on the relative norm error of ( ) ( ) ( ) , , , , , u t x v t x q t x of the improved physical information neural network, when the fixed network has 8 hidden layers and their neurons (10,15,20,25,30), the corresponding norm error is shown in Figures 5(a)-(e). From the figure, we can see that the relative norm error ( ( ) ( ) ( ) , , , , , u t x v t x q t x ) roughly shows a decreasing trend with the increase of f N .…”

Section: Loss Loss Loss Loss Lossmentioning

confidence: 99%

“…Subsequently, based on the work of Jagtap A. D., the team of Chen Yong used the PINN algorithm with an adaptive activation function to study the problem of solving differential equations and named it IPINN [23]. We also studied the rogue wave solution [24] and the soliton solution [25] using by IPINN method. Although the PINN algorithm and its improvement have made some achievements, the research of these new algorithms on differential systems is much more than that.…”

Section: Introductionmentioning

confidence: 99%

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A Novel Method for Solving Nonlinear Schr&#246;dinger Equation with a Potential by Deep Learning

Zhang¹,

Bai²

2022

JAMP

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The improved physical information neural network algorithm has been proven to be used to study integrable systems. In this paper, the improved physical information neural network algorithm is used to study the defocusing nonlinear Schrödinger (NLS) equation with time-varying potential, and the rogue wave solution of the equation is obtained. At the same time, the influence of the number of network layers, neurons and the number of sampling points on the network performance is studied. Experiments show that the number of hidden layers and the number of neurons in each hidden layer affect the relative 2  -norm error. With fixed configuration points, the relative norm error does not decrease with the increase in the number of boundary data points, which indicates that in this case, the number of boundary data points has no obvious influence on the error. Through the experiment, the rogue wave solution of the defocusing NLS equation is successfully captured by IPINN method for the first time. The experimental results of this paper are also compared with the results obtained by the physical information neural network method and show that the improved algorithm has higher accuracy. The results of this paper will be contributed to the generalization of deep learning algorithms for solving defocusing NLS equations with time-varying potential.

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Section: Loss Loss Loss Loss Lossmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Novel Method for Solving Nonlinear Schr&#246;dinger Equation with a Potential by Deep Learning

Zhang¹,

Bai²

2022

JAMP

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“…However, the process of solving PDEs is extremely difficult, traditional numerical methods are very complex and require a lot of computation. Researchers have proved that it can obtain approximate solutions faster than traditional PDE solvers through neural network experiments [3], such as Convolutional Neural Networks [4], Recurrent Neural Network [5], Generative Adversarial Networks [6], DeepONet [7], physic informed neural network (PINN) [8][9][10] and so on. Recently, Zhang [11] et al proposed the Bilinear Neural Network Method (BNNM) for investigating the analytical solutions of PDEs based on the bilinear method and the Neural Network model.…”

Section: Introductionmentioning

confidence: 99%

Interaction phenomenon for variable coefficient Kadomtsev-Petviashvili equation by utilizing variable coefficient bilinear neural network method

Liu¹,

Zhu²

2023

Preprint

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In this paper, a variable coefficient Bilinear neural network method is proposed to deal with the analytical solutions of variable coefficient nonlinear partial differential equations. As an example, a Kadomstev-Petviashvili equation with variable coefficients is investigated by using the variable coefficient Bilinear neural network method. By establishing "3-2-2-1" and "3-3-2-1" models respectively, rich analytical solutions of the variable coefficient Kadomstev-Petviashvili equation are obtained. By choosing some special values of the parameters, the dynamics properties are demonstrated in some three-dimensional and density graphics.

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“…This discusses the approximation ability of neural network. Under this circumstance, there have been early works [33,34], the works of [35,36] and the popular PINN methods of [37][38][39][40][41][42][43], etc. While PINN methods are successful for elliptic type PDEs, the method is inefficient for time evolution problems due to its approximation is limited to a fixed time window.…”

Section: Introductionmentioning

confidence: 99%

Mastering the Cahn–Hilliard equation and Camassa–Holm equation with cell-average-based neural network method

et al. 2022

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In this paper, we develop cell-average based neural network (CANN) method to approximate solutions of nonlinear Cahn-Hilliard equation and Camassa-Holm equation. The CANN method is motivated by the finite volume scheme and evolved from the integral or weak formulation of partial differential equations. The major idea of cell-average based neural network method is to explore a neural network to approximate the solution average difference or evolution between two neighboring time steps. Unlike traditional numerical methods, CANN method is not limited by the CFL restriction and can adapt large time steps for solution evolution, which is a significant advantage that classical numerical methods do not have. Once well trained, this method can be implemented as an fixed explicit finite volume scheme and applied to certain groups of initial value conditions for Cahn-Hilliard equation and Camassa-Holm equation without retraining the neural network. Furthermore, the CANN method also performs very well in handling data with corruption or low quality data generated by Gaussian white noise. Numerical examples are presented to demonstrate effectiveness, accuracy and capability of the proposed method.

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Solving Huxley equation using an improved PINN method

Cited by 15 publications

References 34 publications

A Novel Method for Solving Nonlinear Schr&#246;dinger Equation with a Potential by Deep Learning

A Novel Method for Solving Nonlinear Schr&#246;dinger Equation with a Potential by Deep Learning

Interaction phenomenon for variable coefficient Kadomtsev-Petviashvili equation by utilizing variable coefficient bilinear neural network method

Mastering the Cahn–Hilliard equation and Camassa–Holm equation with cell-average-based neural network method

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Solving Huxley equation using an improved PINN method

Cited by 15 publications

References 34 publications

A Novel Method for Solving Nonlinear Schr&amp;#246;dinger Equation with a Potential by Deep Learning

A Novel Method for Solving Nonlinear Schr&amp;#246;dinger Equation with a Potential by Deep Learning

Interaction phenomenon for variable coefficient Kadomtsev-Petviashvili equation by utilizing variable coefficient bilinear neural network method

Mastering the Cahn–Hilliard equation and Camassa–Holm equation with cell-average-based neural network method

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A Novel Method for Solving Nonlinear Schrödinger Equation with a Potential by Deep Learning

A Novel Method for Solving Nonlinear Schrödinger Equation with a Potential by Deep Learning