Shock-Centered Low-Rank Structure and Neural-Operator Representation of Rarefied Micro-Nozzle Flows

We examine the structure of Direct Simulation Monte Carlo (DSMC)-resolved internal compression layers in rarefied micro-nozzle flows and show that their apparent parametric complexity is largely a registration and finite-thickness scaling effect. A density-gradient diagnostic identifies the compression-layer station $x_s$, while a jump-based thickness ${\delta }_j=\mathrm{\Delta }\rho \mathrm{/}\max \mathrm{\mid }\mathrm{\partial }\rho \mathrm{/}\mathrm{\partial }x\mathrm{\mid }$ defines a shock-centered coordinate ${\xi }_j=(x-x_s)\mathrm{/}{\delta }_j$. In physical coordinates, the leading proper orthogonal decomposition (POD) mode of the centerline density profiles captures only $83.33\mathrm{\%}$ of the fluctuation energy, whereas the jump-scaled coordinate increases this value to $98.33\mathrm{\%}$. A two-dimensional shock-window POD further confirms that this compactness is not a centerline artifact: in the registered $({\xi }_j,\eta )$ frame, the first density mode captures $94.98\mathrm{\%}$ and the first two modes capture $99.05\mathrm{\%}$ of the fluctuation energy. The same region is identified by density-gradient and gradient-length Knudsen-number diagnostics, linking the reduced representation to localized short-gradient-length rarefaction rather than to shock motion alone. We then use this structure as an inductive bias in a shock-aligned Fusion–Deep Operator Network (DeepONet) surrogate for density, velocity components, temperature, Mach number, and pressure. For held-out back-pressure cases, density, temperature, and pressure errors remain below $6.8\mathrm{\%}$, $4.3\mathrm{\%}$, and $6.8\mathrm{\%}$, respectively, and the hardest case reduces the shock-window mean error from $9.75\mathrm{\%}$–$22.27\mathrm{\%}$ for standard baselines to $4.51\mathrm{\%}$. The results show that improved prediction follows from the reduced shock-centered structure of the DSMC fields rather than from network capacity alone.