Solving nonlinear subsonic compressible flow in infinite domain via multi-stage neural networks

TL;DR

This study proposes a novel framework utilizing Physics-Informed Neural Networks (PINNs) to solve the full nonlinear compressible potential equation in an unbounded (infinite) domain and addresses the unbounded-domain and convergence challenges inherent in standard PINNs by incorporating a coordinate transformation and embedding physical asymptotic constraints directly into the network architecture.

Abstract

In aerodynamics, accurately modeling subsonic compressible flow over airfoils is critical for aircraft design. However, solving the governing nonlinear perturbation velocity potential equation presents computational challenges. Traditional approaches often rely on linearized equations or finite, truncated domains, which introduce non-negligible errors and limit applicability in real-world scenarios. In this study, we propose a novel framework utilizing Physics-Informed Neural Networks (PINNs) to solve the full nonlinear compressible potential equation in an unbounded (infinite) domain. We address the unbounded-domain and convergence challenges inherent in standard PINNs by incorporating a coordinate transformation and embedding physical asymptotic constraints directly into the network architecture. Furthermore, we employ a Multi-Stage PINN (MS-PINN) approach to iteratively minimize residuals, achieving solution accuracy approaching machine precision. We validate this framework by simulating flow over circular and elliptical geometries, comparing our results against traditional finite-domain and linearized solutions. Our findings quantify the noticeable discrepancies introduced by domain truncation and linearization, particularly at higher Mach numbers, and demonstrate that this new framework is a robust, high-fidelity tool for computational fluid dynamics.

Figures (9)

• Figure 1: Schematic of PINN framework for solving subsonic flow. The network consists of six hidden layers, each with 60 units, used to approximate the solution. The output $\phi$ is used to compute the loss functions through automatic differentiation, which are then employed to train the weights and biases of the activation function in the hidden layers. The right panel illustrates the imposed boundary conditions.
• Figure 2: Comparison of results between finite and infinite domains. ($a$) Analytical solution \ref{['eq:incom_ana']} in an infinite domain, visualized over the truncated region. ($b$) Comparison of solution profiles, accompanied by a log-log plot illustrating their asymptotic decay with distance. ($c$) Error distribution for the first-stage PINN solved on an infinite domain. ($d$) Error distribution for the first-stage PINN solved on a truncated finite domain.
• Figure 3: Solution profile under exponential mapping ($q=e^{-r}$). For incompressible flow, (a) shows $\phi(q,\beta)$ in the $(q,\beta)$ domain; (b) shows slices at $\beta=1$ and $\beta=0.5$, highlighting a sharp gradient near $q=0$.
• Figure 4: MS-PINN solution for incompressible flow. ($a$) Predicted potential $\phi$ for the Laplace equation \ref{['eq:laplace']}. ($b$) Loss curve over multi-stage training. ($c$) Equation residual and solution error after the first-stage training. ($d$) Corresponding results after the second-stage training.
• Figure 5: MS-PINN solutions for compressible flow. ($a$) Solution of the linear equation \ref{['eq:linear']} at $M_{\infty}=0.4$, shown via $u'=\partial\phi/\partial x$, together with the second-stage equation residual, $r_2(x,y)$, over the infinite domain and the corresponding loss curves. ($b$) Corresponding results for the nonlinear equation \ref{['eq:eqn1_pert']}.