Embedded Neural Boundary: Learning Near-Wall Impulse for Fast Flow Prediction

Learned correctors for coarse-grid flow solvers are typically volumetric: a large CNN corrects every cell and is trained by unrolling through a differentiable solver. Classical simulation concentrates its effort within a few cells of the wall; we embed the learned component there, as a neural boundary. For coarse embedded-boundary flow past a sphere, an impulse confined, before the pressure projection, to a three-cell near-wall band removes  60% of the near-wall velocity error, and most of what remains is static per operating condition. We shape the corrector accordingly: a small MLP predicts eight POD band coefficients per frame from the near-wall state and inflow condition, and a static bias field absorbs the rest. Its network adds 1% to the solver's wall-clock on a 24-thread CPU, against 190% for a U-Net. In closed loop it has 1.3–2.1× lower near-wall error than the best of five training variants of a volumetric solver-in-the-loop CNN with 177× its trainable parameters (21× counting fixed tables), and, with its impulse counted as body force, drag within 2–4% of the fine reference; the static field raises the margin to 2.6× and recovers 87–96% of a replayed full-field oracle's improvement. The same construction holds on 13 two-dimensional geometries and a spheroid at incidence for held-out Reynolds numbers.