Deep Learning for Hyperbolic Conservation Laws with Non‐convex Flux. (10th March 2021)

• Record Type: Journal Article
• Authors: Minbashian, Hadi; Giesselmann, Jan
• Other Names: Gräßle C. guestEditor.; Mlinarić P. guestEditor.
• Is Part Of: Proceedings in applied mathematics and mechanics. Volume 20(2020)Supplement 1
• Page Start: n/a
• Page End: n/a
• Publication Date: 2021-03-10
• Subjects: Applied mathematics -- Periodicals; Engineering mathematics -- Periodicals; Mathematical physics -- Periodicals; 519
• DOI: 10.1002/pamm.202000347
• Languages: English
• ISSNs: 1617-7061

Abstract

Abstract: In this work, we investigate the capabilities of deep neural networks for solving hyperbolic conservation laws with non‐convex flux functions. The behaviour of solutions to these problems depends on the underlying small‐scale regularization. In many applications concerning phase transition phenomena, the regularization terms consist of diffusion and dispersion which are kept in balance in the limit. This may lead to the development of both classical and non‐classical (or undercompressive) shock waves at the same time which makes the development of approximation schemes that converge towards the appropriate weak solution of these problems challenging. Here, we consider a scalar conservation law with cubic flux function as a toy model and present preliminary results of an ongoing work to study the capabilities of a deep learning algorithm called PINNs proposed in [1] for solving this problem. It consists of a feed‐forward network with a hyperbolic tangent activation function along with an additional layer to enforce the differential equation.