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  

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

作　　者：郑淑雯 高振 袁春鑫 Zheng Shuwen;Gao Zhen;YuanChunxin(School of Mathematical Sciences,Ocean University of China,Qingdao 266100,China)

机构地区：[1]中国海洋大学数学科学学院,青岛266100

出　　处：《数值计算与计算机应用》2022年第4期400-414,共15页Journal on Numerical Methods and Computer Applications

基　　金：国家自然科学基金(11871443);山东省高等学校“青创科技计划”(2019KJI002);中央高校基本科研业务费(202042004)资助.

摘　　要：在基于反向传播(BackPropagation,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 basismethod,RBM)引入了长短期记忆神经网络(LongShort-TermMemory,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(RecurrentNeuralNetwork,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 Propa-gation network(BP-RBM).The non-intrusive reduction basis method is introduced to LSTM network which inherits Recurrent Neural Network(RNN)'s memorability,parameter shar-ing,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.More-over,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 showthat,compared with BP-RBM,the error between the reduced order solution recovered bythis method and the high-fidelity snapshot solution can be reduced ten times.

关 键 词：非侵入式约化基方法 长短期记忆神经网络 非线性波方程 海洋内孤立波 

分 类 号：O175.29[理学—数学] TP183[理学—基础数学]