Author:
Mishra Abhishek,Anitescu Cosmin,Budarapu Pattabhi Ramaiah,Natarajan Sundararajan,Vundavilli Pandu Ranga,Rabczuk Timon
Abstract
AbstractA combined deep machine learning (DML) and collocation based approach to solve the partial differential equations using artificial neural networks is proposed. The developed method is applied to solve problems governed by the Sine–Gordon equation (SGE), the scalar wave equation and elasto-dynamics. Two methods are studied: one is a space-time formulation and the other is a semi-discrete method based on an implicit Runge–Kutta (RK) time integration. The methodology is implemented using the Tensorflow framework and it is tested on several numerical examples. Based on the results, the relative normalized error was observed to be less than 5% in all cases.
Publisher
Springer Science and Business Media LLC
Reference35 articles.
1. Xu L, Hui W, Zeng Z. The algorithm of neural networks on the initial value problems in ordinary differential equations. In: Proceedings of the 2nd IEEE Conference on Industrial Electronics and Applications. New York: Institute of Electrical and Electronics Engineers, 2007, 813–816
2. Mall S, Chakraverty S. Comparison of artificial neural network architecture in solving ordinary differential equations. Advances in Artificial Neural Systems, 2013, 2013: 12–12
3. Yadav N, Yadav A, Kumar M. An Introduction to Neural Network Methods for Differential Equations. Berlin: Springer, 2015
4. Lagaris I E, Likas A, Fotiadis D I. Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks, 1998, 9(5): 987–1000
5. Dissertation for the Doctoral Degree;K Rudd,2013