Structured corrections improve gaussian process predictions in science
Learn Only Where You Can't Compute: Structured Corrections to Gaussian Process Posteriors
Machine Learning
Summary
Gaussian processes are a way to predict based on imperfect scientific data but can struggle with lots of data or complex patterns. Diffusion models learn complicated patterns better but need much data and struggle with scientific rules. The authors created a method that combines a simple starting prediction with a focused correction learned by a diffusion model, improving predictions where the simple method falls short. This combined approach helps recover hidden details and reduces errors in scientific tasks compared to using just the simple method alone.
What this means in practice
- •For environmental modelers: Improve predictions in environmental systems by combining physics-based models with data-driven corrections that capture complex variability.
- •For engineering simulation teams: Enhance simulation accuracy for nonlinear or sparse data scenarios by applying targeted corrections to Gaussian process-based models.
Authors
Tobias Weber, Carla Sagebiel, Philipp Hennig
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.