Shock-Aware Physics-Guided Fusion-DeepONet Operator for Rarefied Micro-Nozzle Flows

TL;DR

This work tackles fast, accurate surrogate modeling of rarefied micro-nozzle flows with shocks by devising a shock-aware Fusion-DeepONet neural operator. The method couples a physics-guided trunk featuring shock-aligned coordinates (signed distance, soft indicator, and multi-scale Gaussian envelopes) with a branch conditioned on nozzle pressure ratio, fused multiplicatively to predict velocity fields across geometric and operating-condition variations. A two-phase curriculum with distance- and gradient-based weighting concentrates learning on high-gradient regions near shocks, achieving high fidelity against DSMC data and showing strong interpolation and extrapolation performance, including Burgers’ equation as a canonical test. The surrogate delivers substantial runtime reductions (training+inference on GPUs < 30 minutes for the same flow configurations) and demonstrates robustness to unseen back-pressure and throat-geometry changes, making it a promising design-tool for rapid uncertainty quantification and optimization in rarefied propulsion systems.

Abstract

We present a comprehensive, physics aware deep learning framework for constructing fast and accurate surrogate models of rarefied, shock containing micro nozzle flows. The framework integrates three key components, a Fusion DeepONet operator learning architecture for capturing parameter dependencies, a physics-guided feature space that embeds a shock-aligned coordinate system, and a two-phase curriculum strategy emphasizing high-gradient regions. To demonstrate the generality and inductive bias of the proposed framework, we first validate it on the canonical viscous Burgers equation, which exhibits advective steepening and shock like gradients.

Figures (17)

• Figure 1: Burgers' equation --- Interpolation test. The proposed shock-aware Fusion-DeepONet closely matches the numerical reference at several time instances, including the steep-front region.
• Figure 2: Burgers' equation --- Extrapolation test (test viscosity outside the training range). The model retains accuracy and phase alignment near high gradients, indicating robust out-of-distribution generalization induced by the shock-aware features and curriculum.
• Figure 3: Schematic of the pressure-driven micro-nozzle with downstream plume region. A symmetry condition is imposed along the top centerline, so only half of the geometry is modeled by our DSMC solver. At the inlet, the total conditions $(P_{\mathrm{in}},\,T_{\mathrm{in}},\,U_{\mathrm{in}})$ are prescribed; at the outlet, the back pressure $P_{\mathrm{out}}$ is imposed. The overall axial length is $L$.
• Figure 4: DSMC reference fields in the converging–diverging nozzle. Color contours show (a,c) streamwise velocity $U$ and (b,d) transverse velocity $V$ at two back pressures. At $P_{\mathrm{back}}=15kPa$, the shock/steep-gradient layer is closer to the throat with milder downstream compression; at $P_{\mathrm{back}}=15kPa$, the normal-shock structure is stronger and farther downstream with a larger post-shock deceleration in $U$ and enhanced $V$ gradients near the diffuser wall. These maps serve as the ground-truth targets for evaluating the neural predictor.
• Figure 5: Model loss (Huber) versus epoch on a logarithmic scale for the nozzle problem. The blue curve denotes Train and the orange curve denotes Validation. Both losses decrease steadily and remain close throughout training, indicating good generalization and no evident overfitting.