Adopting Computational Fluid Dynamics concepts for Physics-Informed Neural Networks

Kurzfassung

Aerodynamic flows can be described by the compressible Navier-Stokes equations which can be simplified to the compressible Euler equations when neglecting the viscous terms. In engineering applications, solutions to the corresponding boundary value problems are important, for example, to draw conclusions about the aerodynamic forces. Classical methods, often based on finite-volume discretization strategies, are a valuable tool for this task. However, transferring these classical approaches to potentially advantageous hardware like graphic processing units and quantum computers, promising a significant speed-up, seems to be challenging. Recently, neural networks have been adapted as an alternative approach for the approximation of solutions to partial differential equations. We investigate the physics-informed neural network approach as a method for solving the compressible Euler equations, with the intention of determining whether this approach can also be implemented better on future hardware. Unlike classical neural networks, physics-informed neural networks directly incorporate a partial differential equation into the loss function during the network's training process. This enables the neural network to approximate the solution to the partial differential equation. However, obtaining accurate solutions to the compressible Euler equations employing the physics-informed neural network methodology has shown to be challenging. In this article, we demonstrate how computational concepts, well-known from classical methods, such as artificial viscosity and mesh transformation, can be adapted for physics-informed neural networks. Based on the inviscid Burgers' equation, we derive shock capturing methods which can be transferred to successfully solve the compressible Euler equations. We apply these approaches to a sub- and a transonic test case and compare the method with finite-volume results.