Data-Free PINNs for Compressible Flows: Mitigating Spectral Bias and Gradient Pathologies via Mach-Guided Scaling and Hybrid Convolutions

TL;DR

This paper introduces a data-free PINN framework for compressible flows that employs Mach-guided residual scaling, hybrid convolutions, and analytical shock solutions to improve stability and accuracy in high-speed aerodynamics.

Contribution

It proposes a novel hybrid architecture and dynamic residual scaling strategy for stable, data-free simulation of high Mach number flows with explicit shock capturing.

Findings

Successfully captures detached bow shocks without data

Achieves stable training across Mach regimes up to 15

Demonstrates improved physical fidelity over traditional PINNs

Abstract

This paper presents a fully data-free Physics-Informed Neural Network (PINN) capable of solving compressible inviscid flows (ranging from supersonic to hypersonic, up to Ma=15, where Ma is the Mach number) around a circular cylinder. To overcome the spatial blindness of standard Multi-Layer Perceptrons, a structured hybrid architecture combining radial 1D convolutions with anisotropic azimuthal 2D convolutions is proposed to embed directional inductive biases. For stable optimization across disparate flow regimes, a regime-dependent, Mach-number-guided dynamic residual scaling strategy is introduced. Crucially, this approach scales down residuals to mitigate extreme gradient stiffness in high-Mach regimes, while applying penalty multipliers to overcome the inherent spectral bias and explicitly enforce weak shock discontinuities in low-supersonic flows. Furthermore, to establish a global thermodynamic anchor essential for stable shock wave capturing, exact analytical solutions at the stagnation point are embedded into the loss formulation. This is coupled with a novel "Upstream Fixing" boundary loss and a Total Variation (TV) loss to explicitly suppress upstream noise and the non-physical carbuncle phenomenon. The proposed framework successfully captures the detached bow shock without referential data. While the requisite artificial viscosity yields a slightly thicker shock wave compared to computational fluid dynamics, the proposed method demonstrates unprecedented stability and physical fidelity for data-free PINNs in extreme aerodynamics.