A unified deep artificial neural network approach to partial differential equations in complex geometries

NEUROCOMPUTING

卷:317

Article

Berg, Jens¹  Nystrom, Kaj¹ 

[1] Uppsala Univ, Dept Math, SE-75105 Uppsala, Sweden

关键词: Deep neural networks;    Partial differential equations;    Advection;    Diffusion;    Complex geometries;   

DOI  :  10.1016/j.neucom.2018.06.056

来源: Elsevier

【 摘 要 】

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. (C) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

Free