Learn Only Where You Can't Compute: Structured Corrections to Gaussian Process Posteriors

Abstract: Physics-informed Gaussian processes (GPs) provide a principled framework for scientific inference by expressing uncertainty over empirical measurements, boundary conditions, and governing equations. However, they struggle to scale to larger datasets and nonlinear information. Conversely, diffusion models excel at learning complex distributions over functions from massive datasets, but struggle in scientific contexts where training data are scarce and physical or observational information must be incorporated. We propose a structured posterior-correction framework that combines a tractable Gaussian reference with data-driven corrections via generative learning. After a mean correction and covariance calibration, a diffusion model learns a targeted correction in a low-dimensional subspace selected to capture discrepancies from the reference, while the complement retains its Gaussian law. Across three scientific inference tasks, these corrections recover hidden modes, improve sample structure, and reduce prediction error relative to the Gaussian reference. Comparisons with full-field generative models reveal task-dependent benefits and tradeoffs between predictive accuracy and computational cost.