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

TL;DR

This paper presents a fully data-free Physics-Informed Neural Network capable of solving compressible inviscid flows around a circular cylinder and demonstrates unprecedented stability and physical fidelity for data-free PINNs in extreme aerodynamics.

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.

Figures (8)

• Figure 1: Schematic of the proposed Hybrid Convolutional PINN architecture. The network processes input spatial coordinates via a Fourier feature embedding layer, followed by a stack of $L=6$ Hybrid Convolutional Blocks. Each block sequentially applies a large-kernel radial 1D convolution and an anisotropic azimuthal 2D convolution to embed directional physical inductive biases. Finally, the processed hidden states are mapped to the macroscopic flow variables ($\rho, u, v, p$) through a Multi-Layer Perceptron (MLP) decoder equipped with Softplus-based positivity constraints to ensure thermodynamic realizability.
• Figure 2: Schematic representation of the computational domain (left frame) and the spatial distribution of the composite loss function components (right frame). The physical boundary conditions are strictly enforced via localized penalty terms: freestream inflow ($\mathcal{L}_{in}$), solid slip wall ($\mathcal{L}_{wall}$), and geometric symmetry ($\mathcal{L}_{sym}$). To stabilize the purely data-free optimization, specific regional constraints are introduced: an upstream fixing mask ($\mathcal{L}_{mask}$) to prevent non-physical pre-shock oscillations, a localized nose region penalty ($\mathcal{L}_{nose}$) to enforce sharp shock resolution, and a pointwise stagnation anchor ($\mathcal{L}_{stag}$) to calibrate the macroscopic thermodynamic amplitude. Additionally, a Total Variation penalty ($\mathcal{L}_{tv}$) is applied azimuthally to suppress carbuncle instabilities, while the Mach-scaled Euler PDE residuals ($\mathcal{L}_{pde}$) and enthalpy conservation ($\mathcal{L}_{H}$) are evaluated globally across the continuous fluid domain.
• Figure 3: Snapshots of density contours across training epochs (ep$=10^3$, $8 \times 10^3$ and $1.1 \times 10^4$) obtained using baseline MLP.
• Figure 4: Contours of density, Mach number, temperature, and pressure for various Mach numbers ($\hbox{Ma}=3,5,9,11,13$ and $15$). Top halves ($0 \le Y$) show PINN type A predictions; bottom halves ($Y \le 0$) show CFD reference solutions.
• Figure 5: Profiles of physical quantities ($\rho$, $u$, $T$ and $p$) along the stagnation streamline ($Y=0$) comparing PINN type A predictions with CFD references.