arXiv:2610.11459v1 Announce Type: new Abstract: Outside the linear-Gaussian regime, conditional sampling from Gaussian processes (GPs) is challenging. Recent methods such as FlowGP (Moss et al., (2026)) can condition on arbitrary non-linear and non-Gaussian statements, but at considerable cost: an expensive iterative and high-dimensional diffusion that requires hand-specified kernel hyperparameters. In this paper, we alleviate two significant drawbacks of FlowGP by (1) introducing kernel approximations that enable scaling to high-resolution domains and (2) proposing a way to obtain the marginal likelihood by measuring the work needed to steer the diffusion towards conditioning statements. We enable, for the first time, hyperparameter optimisation within FlowGP and demonstrate our approach on probabilistic downscaling from areal summary statistics, PDE solution inference on irregular domains, and recovery of sea level anomaly fields from non-Gaussian satellite observations.