An artificial neural network based deep collocation method for the solution of transient linear and nonlinear partial differential equations

Abstract

A 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.

Graphical abstract

Keywords

collocation method / artificial neural networks / deep machine learning / Sine–Gordon equation / transient wave equation / dynamic scalar and elasto-dynamic equation / Runge–Kutta method

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Electronic Supplementary Material

Supplementary material is available in the online version of this article at https://doi.org/10.1007/s11709-024-1011-4 and is accessible for authorized users.

Acknowledgements

Budarapu thankfully acknowledges the funds from the Department of Science and Technology (DST), Science and Engineering Research Board (SERB), India (No. SRG/2019/001581).

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Competing interests

The authors declare that they have no competing interests.

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