Exploration of Physics-Informed Neural Networks for Compressible Flows in Aerodynamics

Abstract

The solution of partial differential equations is relevant for many disciplines in science and engineering. Airflow, for example over wings, can be modeled using nonlinear transport equations, such as the compressible Navier-Stokes equations or, in the simplified inviscid case, the compressible Euler equations. Numerical methods are used to predict the aerodynamic flow patterns including shock waves, which are particularly challenging to capture, and the resulting aerodynamic forces and moments. Hence, these numerical techniques have become a crucial tool for aerodynamic vehicle design. To this end, well-established methods, such as finite volume methods, predict the solution at discrete points in the domain. These methods require fine grids resulting in millions of degrees of freedom, which is challenging even today's computational hardware. Especially in multi-query scenarios, such as design exploration and optimization, the cumulative computational cost of performing numerous individual simulations can be prohibitive. Alternative numerical methods are recently gaining popularity as partial differential equation solvers, employing artificial neural networks as global continuous ansatz functions for the solution approximation. These so-called physics-informed neural networks are deep neural networks trained with a loss function that directly incorporates partial differential equations. Especially for engineering applications, this approach holds promise because a single deep neural network could substitute many classical simulations in multi-query scenarios. However, the capturing of shock waves is particularly challenging for the neural network-based approach. This work explores the application of physics-informed neural networks as an alternative approach for solving partial differential equations with a focus on compressible flow applications, governed by the Euler equations. Inspired by classical computational fluid dynamics methods, stabilization techniques for shock waves, based on artificial viscosity and sensor functions are developed and combined with mesh transformations. These techniques are validated on compressible flow problems in subsonic, transonic and supersonic conditions, yielding reasonable prediction accuracies. Furthermore, parametrizations of the boundary conditions and geometry are integrated into the input space of the network, enabling the approximation of solutions across the entire parameter space. The parametric models obtain similar prediction accuracies as their non-parametric counterparts and once trained, can be evaluated at negligible computational cost. The developed techniques significantly advance the state of the art of physics-informed neural network models in aerodynamics, especially for applications in transonic flows. This work shows that artificial viscosity can be a valuable tool that reliably stabilizes solutions including shocks and improves the accuracy of the method when dealing with transonic and supersonic flows. In addition, it is illustrated that parametric physics-informed neural network models are promising candidates for multi-query scenarios. While physics-informed neural network can, at this point, not outperform established numerical techniques in terms of computational efficiency and accuracy, they do excel in approximating parametric problems. With future developments and extensions to accommodate more complex, industrially relevant geometries and the Reynolds-averaged Navier Stokes equations, the approach has the potential to become a valuable tool for applications in aerodynamic design and other engineering applications.