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基于长短期记忆神经网络的非侵入式约化基方法在非线性波问题中的应用

郑淑雯, 高振, 袁春鑫   

  1. 中国海洋大学数学科学学院, 青岛 266100
  • 收稿日期:2021-06-12 发布日期:2022-12-08
  • 通讯作者: 高振, Email: zhengao@ouc.edu.cn.
  • 基金资助:
    国家自然科学基金(11871443), 山东省高等学校“青创科技计划”(2019KJI002)和中央高校基本科研业务费(202042004)资助.

郑淑雯, 高振, 袁春鑫. 基于长短期记忆神经网络的非侵入式约化基方法在非线性波问题中的应用[J]. 数值计算与计算机应用, 2022, 43(4): 400-414.

Zheng Shuwen, Gao Zhen, Yuan Chunxin. LSTM NETWORK BASED NON-INTRUSIVE REDUCED BASIS METHOD FOR NONLINEAR WAVE EQUATIONS[J]. Journal on Numerica Methods and Computer Applications, 2022, 43(4): 400-414.

LSTM NETWORK BASED NON-INTRUSIVE REDUCED BASIS METHOD FOR NONLINEAR WAVE EQUATIONS

Zheng Shuwen, Gao Zhen, Yuan Chunxin   

  1. School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China
  • Received:2021-06-12 Published:2022-12-08
在基于反向传播(Back Propagation BP)网络的非侵入式约化基方法(BP-RBM)的基础上非侵入式约化基方法(Reduced basis method RBM)引入了长短期记忆神经网络(Long Short-Term Memory LSTM)提出了基于LSTM网络的非侵入式约化基方法(LSTM-RBM).该网络在继承循环神经网络(Recurrent Neural Network RNN)的可记忆性参数共享性图灵完备性等特性的基础上同时解决了RNN在长时间序列训练过程中存在的梯度消失和梯度爆炸问题.LSTM-RBM解决了BP-RBM无法准确求解的具有复杂非线性特性的非线性波问题例如二维Navier-Stokes方程和海洋内孤立波问题.此外在求解一般的非线性波问题中该方法相比BP-RBM在处理由非线性性质产生的大梯度结构上更有优势.数值测试结果表明相比于BP-RBM该方法恢复的降阶解与高保真快照解的误差可以缩小10倍左右.
The non-intrusive reduction basis method based on Long Short-Term Memory network (LSTM-RBM) is proposed on the non-intrusive reduction basis method based on Back Propagation network (BP-RBM). The non-intrusive reduction basis method is introduced to LSTM network which inherits Recurrent Neural Network (RNN)'s memorability, parameter sharing, turing completeness and other characteristics, and solves the gradient vanishing and gradient explosion problems existing in the long sequence training of RNN. The problem that BP-RBM cannot solve the complex nonlinear wave equation accurately is solved, such as the two-dimensional Navier-Stokes equation and the ocean internal wave problem. Moreover, in solving general nonlinear wave problems, this method is more advantageous than BP-RBM in dealing with large gradient areas. The results of numerical experiments show that, compared with BP-RBM, the error between the reduced order solution recovered by this method and the high-fidelity snapshot solution can be reduced ten times.

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