Physics-informed neural networks for shock capturing in inviscid flows around an airfoil
Abstract
Physics-informed neural networks (PINNs) have shown promise in solving partial differential equations (PDEs). However, shock capturing remains challenging for PINNs, particularly in steady flows around an airfoil, where shocks may not even be captured. In this study, we point out that the steady Euler equations used to construct the loss function impose weak constraints, which make it difficult to correct the continuous function approximation preference of neural networks, causing gradient descent to converge to a smooth local optimum. Based on this insight, we propose to strengthen the physical constraints by reconstructing steady shock capturing as a temporal evolution process governed by the unsteady Euler equations. The temporal evolution law of the flow field guides gradient descent, thereby alleviating its tendency to converge to smooth local optima and ensuring shock formation. We introduce a PDE loss function incorporating a pseudo-time stepping mechanism to obtain the steady state solution without solving the unsteady Euler equations over a long time interval. In addition, we develop a simplified formulation of the Euler equations to reduce the complexity of the PDE loss function, thereby improving shock capturing accuracy. By solving four forward problems involving different flow conditions and geometries, we verify that the proposed method achieves high accuracy in shock capturing while reducing the wall time to one-seventh of that required by existing advanced methods. Moreover, the proposed method demonstrates its capability to solve a parametric problem involving the angle of attack and Mach number without relying on any labeled data.
Related
- Accelerating High‐Fidelity Airfoil Design via Physics‐Informed Video Diffusion Model
- Accelerating noisy variational quantum algorithms with physics-informed denoising networks
- Physics-informed neural networks for soliton solutions and parameter identification of the two-mode Korteweg–de Vries equation
- Lazy training of quantum physics informed neural networks
- Lecture notes on Physics Informed Neural Networks, Neural Operators, and their applications
BibTeX
@misc{song2026physicsinformed,
title = {{Physics-informed neural networks for shock capturing in inviscid flows around an airfoil}},
author = {Jiahao Song and Wenbo Cao and Weiwei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.08130},
eprint = {2607.08130},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08130},
}
RIS
TY - UNPB
TI - Physics-informed neural networks for shock capturing in inviscid flows around an airfoil
AU - Jiahao Song
AU - Wenbo Cao
AU - Weiwei Zhang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2607.08130
UR - https://arxiv.org/abs/2607.08130
AB - Physics-informed neural networks (PINNs) have shown promise in solving partial differential equations (PDEs). However, shock capturing remains challenging for PINNs, particularly in steady flows around an airfoil, where shocks may not even be captured. In this study, we point out that the steady Euler equations used to construct the loss function impose weak constraints, which make it difficult to correct the continuous function approximation preference of neural networks, causing gradient descent to converge to a smooth local optimum. Based on this insight, we propose to strengthen the physical constraints by reconstructing steady shock capturing as a temporal evolution process governed by the unsteady Euler equations. The temporal evolution law of the flow field guides gradient descent, thereby alleviating its tendency to converge to smooth local optima and ensuring shock formation. We introduce a PDE loss function incorporating a pseudo-time stepping mechanism to obtain the steady state solution without solving the unsteady Euler equations over a long time interval. In addition, we develop a simplified formulation of the Euler equations to reduce the complexity of the PDE loss function, thereby improving shock capturing accuracy. By solving four forward problems involving different flow conditions and geometries, we verify that the proposed method achieves high accuracy in shock capturing while reducing the wall time to one-seventh of that required by existing advanced methods. Moreover, the proposed method demonstrates its capability to solve a parametric problem involving the angle of attack and Mach number without relying on any labeled data.
ER -