Making Pretrained CFD Surrogates Steerable: Sparse Representations for Test-Time Phase Synchronization

Graph-based surrogate models provide fast alternatives to high-fidelity CFD solvers, but their opaque latent spaces and limited controllability restrict use in safety-critical settings. In oscillatory flows their characteristic failure is phase drift, where the surrogate keeps producing the right vortex structures at the wrong times. We ask whether a frozen surrogate can be corrected at test time by acting on its pretrained internal representation, without retraining, and which properties of that representation make this possible. We propose a post-hoc phase-steering framework: sparse-autoencoder (SAE) features of the frozen embeddings that carry a coherent oscillation are rotated against their own analytic-signal (Hilbert) quadrature partners in a low-rank coefficient space, with a smooth time-varying phase offset and an amplitude gain as the only learned parameters. The same pipeline is applied in SAE, PCA, DMD-based, and raw embedding spaces, isolating representation quality as the variable. On held-out cylinder-wake trajectories with unseen meshes, geometries, and inflow conditions, SAE steering closes about 90% of an imposed phase gap and of the surrogate's naturally accumulated drift ($7.6 \rightarrow 0.7$ frames), leading every dense alternative on every metric and closing more of the phase gap on every single trajectory, because its wake-localized features keep the intervention contained. Standard static interventions (scaling, additive offsets, clamping) fail entirely: acting on a dynamical representation requires an intervention that respects its dynamics.