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Conditioning Gaussian Processes on Almost Anything

NeurIPS 2026 · Probabilistic and scientific modeling › Uncertainty and Gaussian processes · approach: PAC-Bayes and decoupled GP learning · 0 citations (OpenAlex)

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Shows GPs are an exact closed-form linear diffusion/flow-matching model and uses a Monte Carlo guided probability-flow ODE (FlowGP, with whitening) to condition GPs on non-linear, non-Gaussian constraints, PDEs and LLM-derived text.

Problem

Exact GP inference is limited to linear-Gaussian conditioning; non-Gaussian likelihoods and non-linear constraints (physics, text) require approximate inference schemes that are bespoke and hard to generalise.

Approach

Recast GP predictive sampling as a linear diffusion/probability-flow ODE with closed-form Gaussian dynamics, add a guidance term estimated by self-normalised importance sampling using only point-wise likelihood evaluations, and whiten the covariance to reduce stiffness and Wasserstein-2 transport cost.

Key results

• On the Allen-Cahn benchmark, FlowGP with whitening reaches RMSE 0.13 and NLPD -0.83 versus PHYSS 0.17/1.69 and AUTOIP 0.17/-0.29, at roughly 4 s versus 1e4 s runtime.
• On the non-linear pendulum benchmark, FlowGP-W matches PHYSS/AUTOIP RMSE of 0.05 with NLPD -1.05 (best) and ~4 s runtime (about 25x lower cost).
• FlowGP generates 100 samples for an LLM text prompt in under 2 seconds, with the paper stating runtimes from milliseconds to under 5 seconds.

Surprising findings

• The paper reports its simple Monte Carlo guidance estimator outperforms DPS and MPGD even at S=1 samples.
• Without whitening, FlowGP on Allen-Cahn has high RMSE variance (0.38 with std 1.39), whereas whitening stabilises it.

Limitations

• Cubic scaling in the number of grid points makes large-scale problems intractable.
• Theoretical guarantees are strong only for log-concave likelihoods; non-log-concave likelihoods admit only worst-case exponential bounds.
• Kernel hyperparameters are fitted only on the linear-Gaussian component, which may bias results when the conditioning information is highly informative; possible importance weight collapse in high dimensions.

Experimental setup

Benchmarks: monotone bounded regression benchmark non-linear pendulum Allen-Cahn equation Burgers equation

Metrics: RMSE NLPD runtime

Baselines: PHYSS AUTOIP DPS MPGD

Builds on: probability flow ODE diffusion guidance (DPS) stochastic interpolants pathwise conditioning probit likelihood

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