2019/05/10 by Craig Michoski, Michoski, Craig, Miloš Milosavljević +6
Engineering · Physics and Astronomy · #Computational Fluid Dynamics and Aerodynamics #Computational Physics (physics.comp-ph) #Data Analysis #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics and Turbulent Flows #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Statistics and Probability (physics.data-an)
paper · pdf · doi:10.48550/arxiv.1905.04351
openalex publication_date 2019/05/10 · openalex created_date 2022/07/29 · openalex updated_date 2026/08/01
Recent work has introduced a simple numerical method for solving partial\ndifferential equations (PDEs) with deep neural networks (DNNs). This paper\nreviews and extends the method while applying it to analyze one of the most\nfundamental features in numerical PDEs and nonlinear analysis: irregular\nsolutions. First, the Sod shock tube solution to compressible Euler equations\nis discussed, analyzed, and then compared to conventional finite element and\nfinite volume methods. These methods are extended to consider performance\nimprovements and simultaneous parameter space exploration. Next, a shock\nsolution to compressible magnetohydrodynamics (MHD) is solved for, and used in\na scenario where experimental data is utilized to enhance a PDE system that is\n\a priori insufficient to validate against the observed/experimental\ndata. This is accomplished by enriching the model PDE system with source terms\nand using supervised training on synthetic experimental data. The resulting DNN\nframework for PDEs seems to demonstrate almost fantastical ease of system\nprototyping, natural integration of large data sets (be they synthetic or\nexperimental), all while simultaneously enabling single-pass exploration of the\nentire parameter space.\n