Abstract
<p style="margin-right:19px; margin-left:10px; text-align:justify"><span style="font-size:12pt"><span style="text-justify:inter-ideograph"><span style="font-family:"Times New Roman",serif"><span style="font-size:14.0pt">Due to the growth of the artificial intelligence, complicated fluid flow problem such as advection diffusion equation or Navier-Stokes equation can now be solved adequately by various machine learning techniques. Physics Informed Neural Network (PINN) method [1] is one of these methods and has been proven to be successful in applying to various nonlinear partial differential equations. However, it was found that PINN often encounters numerical instability and inaccuracy due to the presence of strong advection effects in fluid flow problems, such as advection diffusion equation with high Peclet number of Navier Stokes equations with high Reynolds number. To address this issue, this study employs a revised PINN method inspiring from the well-known weak boundary condition method [2] and streamline-upwind Petrov-Galerkin (SUPG) stabilization method [3] in the conventional Galerkin formulation. The revision of the PINN converts these weak forms into strong form version that can be applied along with PINN collocation method. The effectiveness of the proposed method is benchmarked by various advection dominated advection diffusion equations, and can be further extended to the Navier-Stokes equations.</span></span></span></span></p><p style="margin-right:19px; margin-left:10px; text-align:justify"> </p><p style="margin-right:19px; margin-left:10px; text-align:justify"><span style="font-size:12pt"><span style="text-justify:inter-ideograph"><span style="font-family:"Times New Roman",serif"><b><span style="font-size:14.0pt">References</span></b></span></span></span></p><p style="margin-right:19px; margin-left:10px; text-align:justify"><span style="font-size:12pt"><span style="text-justify:inter-ideograph"><span style="font-family:"Times New Roman",serif"><span style="font-size:14.0pt">[1] Raissi, Maziar, Paris Perdikaris, and George E. Karniadakis. "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations." Journal of Computational Physics 378 (2019): 686-707.</span></span></span></span></p><p style="margin-right:19px; margin-left:10px; text-align:justify"><span style="font-size:12pt"><span style="text-justify:inter-ideograph"><span style="font-family:"Times New Roman",serif"><span style="font-size:14.0pt">[2] Bazilevs, Yuri, and Thomas JR Hughes. "Weak imposition of Dirichlet boundary conditions in fluid mechanics." Computers & Fluids 36.1 (2007): 12-26.</span></span></span></span></p><p style="margin-right:19px; margin-left:10px; text-align:justify"><span style="font-size:12pt"><span style="text-justify:inter-ideograph"><span style="font-family:"Times New Roman",serif"><span style="font-size:14.0pt">[3] Brooks, Alexander N., and Thomas JR Hughes. "Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations." Computer Methods in Applied Mechanics and Engineering 32.1-3 (1982): 199-259.</span></span></span></span></p>