Conditioning Gaussian Processes on Almost Anything

2026-05-21 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

A new general-purpose Gaussian Process (GP) inference scheme, FlowGP, establishes an explicit equivalence between GPs and linear diffusion models. This framework recasts GP predictive sampling as an Ordinary Differential Equation (ODE) with closed-form Gaussian dynamics and a likelihood-dependent guidance term. It allows conditioning beyond the traditional linear-Gaussian regime. FlowGP handles any conditioning statement permitting point-wise likelihood evaluation, including complex non-linear physics and natural language via large language models. The method employs whitening to minimize Wasserstein-2 transport cost and enhance numerical stability by eliminating stiffness. FlowGP demonstrates computational efficiency, with runtimes from milliseconds for monotonic bounded regression to under 5 seconds for LLM- and physics-constrained generation, typically using 1000 ODE steps and 5 Monte Carlo samples.

Key takeaway

For Machine Learning Engineers or Research Scientists developing GP models with complex real-world constraints, you can now incorporate non-linear physics, monotonicity, or even natural language descriptions directly into your GP inference. This eliminates the need for bespoke derivations and approximate inference schemes, enabling more flexible and accurate probabilistic modeling. Consider FlowGP to extend GP capabilities beyond traditional linear-Gaussian assumptions.

Key insights

Gaussian Processes can be exactly recast as linear diffusion models, allowing arbitrary conditioning through ODE-based sampling and Monte Carlo guidance.

Principles

• GPs are exact linear diffusion models.
• Whitening minimizes transport cost.
• Conditioning can be applied at test-time.

Method

Integrate a whitened probability-flow ODE backwards from white noise (t=1) to the target GP distribution (t=0). A Monte Carlo approximated guidance term modifies the ODE drift for arbitrary conditioning.

In practice

• Constrain GP regression with non-linear rules.
• Incorporate non-linear physics into GP models.
• Guide GP samples using natural language prompts.

Topics

• Gaussian Processes
• Diffusion Models
• Probabilistic Inference
• Physics-informed AI
• Large Language Models
• Bayesian Optimization

Code references

• HanyangHenry-Wang/BOPE-MoNNE

Best for: AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.