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Physics-informed neural networks method in high-dimensional integrable systems

School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China

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and

Yong Chen

https://orcid.org/0000-0002-6008-6542

School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

E-mail Address: ychen@sei.ecnu.edu.cn

Corresponding author.

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https://doi.org/10.1142/S021798492150531XCited by:39 (Source: Crossref)

Abstract

In this paper, the physics-informed neural networks (PINNs) are applied to high-dimensional system to solve the $(N + 1)$ -dimensional initial-boundary value problem with $2 N + 1$ hyperplane boundaries. This method is used to solve the most classic (2+1)-dimensional integrable Kadomtsev–Petviashvili (KP) equation and (3+1)-dimensional reduced KP equation. The dynamics of (2+1)-dimensional local waves such as solitons, breathers, lump and resonance rogue are reproduced. Numerical results display that the magnitude of the error is much smaller than the wave height itself, so it is considered that the classical solutions in these integrable systems are well obtained based on the data-driven mechanism.

Keywords:

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