Solving differential equations using deep neural networks

NEUROCOMPUTING

卷:399

Article

Michoski, Craig¹  Milosavljevic, Milos²  Oliver, Todd¹  Hatch, David R.³ 

[1] Univ Texas Austin, Oden Inst Computat Engn & Sci, Austin, TX 78712 USA

[2] Univ Texas Austin, Dept Astron, Austin, TX 78712 USA

[3] Univ Texas Austin, Inst Fus Studies, Austin, TX 78712 USA

关键词: Deep neural networks;    Differential equations;    Partial differential equations;    Nonlinear;    Shocks;    Data analytics;    Optimization;   

DOI  :  10.1016/j.neucom.2020.02.015

来源: Elsevier

【 摘 要 】

Recent work on solving partial differential equations (PDEs) with deep neural networks (DNNs) is presented. The paper reviews and extends some of these methods while carefully analyzing a fundamental feature in numerical PDEs and nonlinear analysis: irregular solutions. First, the Sod shock tube solution to the compressible Euler equations is discussed and analyzed. This analysis includes a comparison of a DNN-based approach with conventional finite element and finite volume methods, and demonstrates that the DNN is competitive in terms of degrees of freedom required for a given accuracy. Further, the DNNbased approach is extended to consider performance improvements and simultaneous parameter space exploration. Next, a shock solution to compressible magnetohydrodynamics (MHD) is solved for, and used in a scenario where experimental data is utilized to enhance a PDE system that is a priori insufficient to validate against the observed/experimental data. This is accomplished by enriching the model PDE system with source terms that are then inferred via supervised training with synthetic experimental data. The resulting DNN framework for PDEs enables straightforward system prototyping and natural integration of large data sets (be they synthetic or experimental), all while simultaneously enabling single-pass exploration of an entire parameter space. Published by Elsevier B.V.

【 授权许可】

Free