Guided Observation-Space Projections for Constrained Flow Matching

Generative models of physical fields are often required at inference time to satisfy observations or physical constraints that were not enforced during training. Projection-based samplers meet them by correcting the predicted final sample onto the constraint set at every step, but a gradient-guidance step taken beforehand is largely lost in that correction and in the interpolation that follows. We introduce guided observation-space projection (GOSP), which keeps a guided displacement across repeated projection by updating the source sample with the current state, and characterise the guidance that survives a linear projection. Because the projection runs at every step, we solve it in observation space without forming the constraint Jacobian, with a matrix-free and a structured kernel-Gram solver. On Navier–Stokes benchmarks GOSP improves reconstruction from sparse and moving observations while keeping the constraint residuals of a projection sampler.