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

Abstract

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 \(\delta_j=\Delta\rho/\max|\partial\rho/\partial x|\) defines a shock-centered coordinate \(\xi_j=(x-x_s)/\delta_j\). In physical coordinates, the leading proper orthogonal decomposition (POD) mode of the centerline density profiles captures only \(83.33\%\) of the fluctuation energy, whereas the jump-scaled coordinate increases this value to \(98.33\%\). 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 \(94.98\%\) and the first two modes capture \(99.05\%\) 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 \(6.8\%\), \(4.3\%\), and \(6.8\%\), respectively, and the hardest case reduces the shock-window mean error from \(9.75\%\)--\(22.27\%\) for standard baselines to \(4.51\%\). The results show that improved prediction follows from the reduced shock-centered structure of the DSMC fields rather than from network capacity alone.