Revision as of 13:27, 22 November 2022

Summary

In this work, we present an application of modern deep learning methodologies to the numerical solution of two dimensional hyperbolic partial differential equations in transport models. More specifically, we employ a supervised deep neural network that takes into account of initial-boundary value problems for a scalar, 2D inviscid Burgers model including the case with Riemann data, whose solutions develop discontinuity, containing both shock wave and rarefaction wave. We also apply the proposed PINN approach to the linear advection equation with periodic sinusoidal initial condition. Our results suggest that a relatively simple deep learning model was capable of achieving promising results in the linear advection and inviscid Burgers equation with rarefaction, providing numerical evidence of good approximation of weakentropy solutions to the case of nonlinear 2D inviscid Burgers model. For the Riemann problems, the neural network performed better when rarefaction wave is predominant. The premises underlying these preliminary results as an integrated physics-informed deep learning approach are promising. However, there are hints of evidence suggesting specific fine tuning on the PINN methodology for solving hyperbolic-transport problems in the presence of shock formation in the solutions.

Abstract

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