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

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.