Data-Free PINNs for Compressible Flows: Mitigating Spectral Bias and Gradient Pathologies via Mach-Guided Scaling and Hybrid Convolutions
This addresses the problem of simulating extreme aerodynamics without data for researchers in computational fluid dynamics, but it is incremental as it builds on existing PINN methods with specific architectural and optimization improvements.
This paper tackled solving compressible inviscid flows up to Mach 15 around a circular cylinder without using data, by developing a data-free Physics-Informed Neural Network (PINN) that captures detached bow shocks with stability and physical fidelity, though with slightly thicker shock waves compared to computational fluid dynamics.
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.