Artificial neural network methods for the solution of second order boundary value problems

Cosmin Anitescu, Elena Atroshchenko, Naif Alajlan, Timon Rabczuk

Research output: Contribution to journalArticlepeer-review

552 Citations (Scopus)


We present a method for solving partial differential equations using artificial neural networks and an adaptive collocation strategy. In this procedure, a coarse grid of training points is used at the initial training stages, while more points are added at later stages based on the value of the residual at a larger set of evaluation points. This method increases the robustness of the neural network approximation and can result in significant computational savings, particularly when the solution is non-smooth. Numerical results are presented for benchmark problems for scalar-valued PDEs, namely Poisson and Helmholtz equations, as well as for an inverse acoustics problem.

Original languageEnglish
Pages (from-to)345-359
Number of pages15
JournalComputers, Materials and Continua
Issue number1
Publication statusPublished - 2019

Bibliographical note

Funding Information:
Acknowledgements: N. Alajlan and T. Rabczuk acknowledge the Distinguished Scientist Fellowship Program (DSFP) at King Saud University for supporting this work.

Publisher Copyright:
Copyright © 2019 Tech Science Press.


  • Adaptive collocation
  • Artificial neural networks
  • Deep learning
  • Inverse problems

ASJC Scopus subject areas

  • Biomaterials
  • Modelling and Simulation
  • Mechanics of Materials
  • Computer Science Applications
  • Electrical and Electronic Engineering


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