A physics-constrained deep residual network for solving the sine-Gordon equation |
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Authors: | Jun Li Yong Chen |
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Institution: | (Shanghai Key Laboratory of Trustworthy Computing,East China Normal University,Shanghai,200062,China;School of Mathematical Sciences,Shanghai Key Laboratory of PMMP,Shanghai Key Laboratory of Trustworthy Computing,East China Normal University,Shanghai,200062,China;College of Mathematics and Systems Science,Shandong University of Science and Technology,Qingdao,266590,China;Department of Physics,Zhejiang Normal University,Jinhua,321004,China) |
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Abstract: | Despite some empirical successes for solving nonlinear evolution equations using deep learning,there are several unresolved issues. First, it could not uncover the dynamical behaviors of some equations where highly nonlinear source terms are included very well. Second, the gradient exploding and vanishing problems often occur for the traditional feedforward neural networks.In this paper, we propose a new architecture that combines the deep residual neural network with some underlying physical laws. Using the sine-Gordon equation as an example, we show that the numerical result is in good agreement with the exact soliton solution. In addition, a lot of numerical experiments show that the model is robust under small perturbations to a certain extent. |
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Keywords: | sine-Gordon equation deep residual network soliton integrable system |
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