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A unified deep artificial neural network approach to partial differential equations in complex geometries
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2018 (English)In: Neurocomputing, ISSN 0925-2312, E-ISSN 1872-8286, Vol. 317, p. 28-41Article in journal (Refereed) Published
Abstract [en]

In this paper, we use deep feedforward artificial neural networks to approximate solutions to partial differential equations in complex geometries. We show how to modify the backpropagation algorithm to compute the partial derivatives of the network output with respect to the space variables which is needed to approximate the differential operator. The method is based on an ansatz for the solution which requires nothing but feedforward neural networks and an unconstrained gradient based optimization method such as gradient descent or a quasi-Newton method. We show an example where classical mesh based methods cannot be used and neural networks can be seen as an attractive alternative. Finally, we highlight the benefits of deep compared to shallow neural networks and device some other convergence enhancing techniques.

Place, publisher, year, edition, pages
2018. Vol. 317, p. 28-41
National Category
Computational Mathematics
Research subject
Mathematics with specialization in Applied Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-362369DOI: 10.1016/j.neucom.2018.06.056ISI: 000444237900003OAI: oai:DiVA.org:uu-362369DiVA, id: diva2:1253206
Funder
Göran Gustafsson Foundation for Research in Natural Sciences and MedicineAvailable from: 2018-08-23 Created: 2018-10-04 Last updated: 2018-11-14Bibliographically approved

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Publisher's full texthttps://arxiv.org/abs/1711.06464

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Berg, JensNyström, Kaj

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