Gradient-Enhanced Kriging

When gradients are available, gradient-enhanced kriging uses them as extra training data to build accurate surrogates from fewer function evaluations.

Using derivatives as data

A Gaussian process is defined by a covariance function, and differentiation is a linear operation, so the derivative of a GP is itself a GP with a covariance obtained by differentiating the kernel. Gradient-enhanced kriging (also called cokriging with derivatives) augments the training set with observed gradients, tightening the fit especially between sample points.

Why gradients help

Kronos motion — training from sim

Each gradient at a point supplies d additional pieces of information in d dimensions, so a single expensive run that also returns its adjoint gradient is worth many function-only samples. This is powerful when adjoint solvers or automatic differentiation make gradients nearly free relative to the primal solve.

The augmented system

The covariance matrix is expanded with blocks for function-function, function-derivative, and derivative-derivative covariances, all derived from the same kernel. Prediction and its variance follow the standard kriging equations on this larger system. A smooth, differentiable kernel such as the squared-exponential or a high-order Matern is required.

Practical issues

The covariance matrix grows to size N(1+d), raising storage and conditioning concerns
Noisy or inconsistent gradients degrade the fit and can make the matrix ill-conditioned
Kernel choice must guarantee differentiability of the required order

Where it fits

Gradient-enhanced surrogates pair naturally with adjoint-based simulations common in design optimization and UQ. In fusion design workflows that already compute sensitivities of a figure of merit to operating parameters, feeding those sensitivities into the emulator sharpens the response surface and reduces the number of costly full runs needed near a candidate operating point.