Abstract
In this paper, with the aid of symbolic computation system Python and based on the deep neural network (DNN), automatic differentiation (AD), and limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization algorithms, we discussed the modified Korteweg-de Vries (mkdv) equation to obtain numerical solutions. From the predicted solution and the expected solution, the resulting prediction error reaches10−6. The method that we used in this paper had demonstrated the powerful mathematical and physical ability of deep learning to flexibly simulate the physical dynamic state represented by differential equations and also opens the way for us to understand more physical phenomena later.
Highlights
In recent years, nonlinear phenomena have been widely used in fields such as mathematics, physics, chemistry, biology, finance, and engineering technology
Because a large number of mathematical models of scientific and engineering problems are reduced to the problem for determining solutions of ordinary differential equations (ODEs) and partial differential equations (PDEs) and the problems are complex and the amount of calculation is huge, except for a few special types of differential equations that can be solved by analytical methods, the analytical expressions to be obtained are extremely difficult in most cases
From the results obtained in the experiment, some novel and important developments for searching for analytical solitary wave solutions for PDE were investigated
Summary
Nonlinear phenomena have been widely used in fields such as mathematics, physics, chemistry, biology, finance, and engineering technology. With the development of deep learning in recent years, Professor Karniadakis from the Department of Applied Mathematics at Brown University and his collaborators reexamined the technology and developed a set of deep learning algorithm frameworks based on the original It was named “physics-informed neural networks (PINN)” and was first used to solve forward and inverse problems of partial differential equations. From the results obtained in the experiment, some novel and important developments for searching for analytical solitary wave solutions for PDE were investigated The results of this manuscript may well complement the existing literature as the following: extended and modified direct algebraic method, extended mapping method, and Seadawy techniques to find solutions for some nonlinear partial differential equations such as dispersive solitary wave solutions of Kadomtsev-PetviashviliBurgers dynamical equations [13]; the elliptic function, bright and dark solitons, and solitary wave solutions of higher-order NLSE [14]; abundant lump solution and interaction phenomenon of (3 + 1)-dimensional generalized Kadomtsev-Petviashvili equation [15]; describing the bidirectional propagation of small amplitude long capillary. This study is of significance for the later study of soliton solutions
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