Solving Nonlinear Equation of Subsonic Compressible Flow via PINNs

Understanding subsonic compressible flow over airfoils is fundamental to aerodynamics. However, accurately solving the governing nonlinear perturbation-velocity potential equations in infinite domains remains a major challenge. A common simplification is to solve linearized equations in truncated finite domains, but this approach inevitably introduces errors and limits practical applicability.

In this study, we employ an advanced physics-informed neural network (PINN) framework to solve the nonlinear subsonic compressible flow equations over various airfoil geometries in infinite domains. Standard PINNs are insufficient for this task, so we introduce several key innovations. First, by analyzing the asymptotic behavior of the solution at infinity, we design a novel coordinate transformation that maps the infinite domain to a normalized finite one. Second, we embed the solution’s characteristic features into the network architecture to enhance convergence. Finally, we implement a multi-stage training scheme that significantly reduces training error and achieves unprecedented accuracy.

We quantitatively compare our results with conventional finite-domain, linearized solutions, explicitly identifying and analyzing the errors introduced by domain truncation and equation linearization.