FunctionalGPs.jl

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FunctionalGPs.jlLinear Functionals for Gaussian Processes

Linear transforms of GPs. Mix & match point observations, integrals and derivatives.

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API Reference
FunctionalGPs.jl

🔗 Composable Functionals

Chain evaluation, differentiation, and integration with intuitive ∘, +, ⊗ operators.

⚡ Automatic Optimization

Kernel trait dispatch selects the fastest algorithm - Toeplitz, sparse, or dense - automatically.

📐 Derivative Observations

Condition GPs on derivative data. Perfect for physics-informed learning and slope constraints.

∫ Integral Observations

Bayesian quadrature and cell-averaged measurements made simple with VectorizedLebesgueIntegral.

🎯 Physics-Informed Learning

Encode physical constraints, boundary conditions, and conservation laws directly into your GP.

🧩 AbstractGPs Integration

Seamlessly extends the JuliaGaussianProcesses ecosystem. Use your favorite kernels.

What is FunctionalGPs.jl? ​

FunctionalGPs.jl provides a composable framework for applying linear functionals (evaluation, integration, differentiation) to Gaussian processes. Use it for GP regression with derivative or integral observations, Bayesian quadrature, physics-informed learning, and more.

Quick Start ​

julia
using Pkg
Pkg.add("FunctionalGPs")

GP regression with mixed observations ​

julia
using FunctionalGPs
using AbstractGPs

k = WendlandKernel(1, 2, 1.0)
f = GP(k)

# Condition on function values
f1 = condition_on_observation(f, [0.0, 1.0], [0.0, 0.84]; noise = 1.0e-8)

# Condition further on a derivative observation
∂x = PartialDerivative((1,))
ℒ = EvaluationFunctional([0.5]) ∘ ∂x
f2 = condition_on_observation(f1, ℒ, [1.0]; noise = 1.0e-8)

Joint functional Gaussians ​

For models that bundle several linear functionals of the same GP — function values, derivatives, integrals — and need their cross-covariances preserved (e.g. for hyperparameter inference in Turing), use FunctionalGaussian:

julia
using FunctionalGPs, FunctionalGPs.Notation

fg = FunctionalGaussian(f;
    y  = δ(X_obs),
    dy = δ(X_pred) ∘ ∂(1),
    q  = ∫([Interval(0.0, 1.0)]),
)

# Marginal log-likelihood for hyperparameter optimisation / sampling
ℓ = loglikelihood(fg, (; y = y_obs); noise = (; y = σ²))

# Posterior over the latent (unobserved) blocks
post = posterior(fg, (; y = y_obs); noise = (; y = σ²))
post.dy   # LazyMvNormal over derivative locations

See the Joint Functional Gaussians and Notation pages in the API Reference for details.