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arXiv · 2111.09489

Data-driven discoveries of B\"acklund transforms and soliton evolution equations via deep neural network learning schemes

Zijian Zhou·Li Wang·Weifang Weng·Zhenya Yan
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Abstract

We introduce a deep neural network learning scheme to learn the B\"acklund transforms (BTs) of soliton evolution equations and an enhanced deep learning scheme for data-driven soliton equation discovery based on the known BTs, respectively. The first scheme takes advantage of some solution (or soliton equation) information to study the data-driven BT of sine-Gordon equation, and complex and real Miura transforms between the defocusing (focusing) mKdV equation and KdV equation, as well as the data-driven mKdV equation discovery via the Miura transforms. The second deep learning scheme uses the explicit/implicit BTs generating the higher-order solitons to train the data-driven discovery of mKdV and sine-Gordon equations, in which the high-order solution informations are more powerful for the enhanced leaning soliton equations with higher accurates.

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Zijian Zhou, Li Wang, Weifang Weng, Zhenya Yan. 2021-11-18. Data-driven discoveries of B\"acklund transforms and soliton evolution equations via deep neural network learning schemes. https://doi.org/10.1016/j.physleta.2022.128373

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