Gaussian Process-Based Emulators

• Gaussian-process-based emulators are statistical surrogate models that use Gaussian processes to approximate computationally expensive simulations.
• They enable uncertainty quantification, interpolation, and prediction using a finite number of simulation runs, applied in fields such as engineering design and Bayesian calibration.
• Rigorous extensions address challenges like nonstationarity, piecewise behavior, and discontinuities, enhancing simulation-based inference.

A Gaussian-process-based emulator is a statistical surrogate model that uses the Gaussian process (GP) formulation to emulate, i.e., fast approximate, the output of expensive or analytically intractable computer codes. GP-based emulators are nonparametric models that place a joint Gaussian prior over the space of possible functions mapping from inputs (potentially high-dimensional vectors) to scalar or functional outputs. Such emulators support principled uncertainty quantification, interpolation, and prediction with a finite number of expensive computer simulations. Rigorous extensions handle vector-valued, functional, monotonic, nonstationary, piecewise, and discontinuous problems. GP-based emulators are widely implemented in scientific computation, engineering design, Bayesian calibration, and active learning for simulation-based inference.

1. Core Principles and Mathematical Framework

Let $f:\mathcal{X}\subset\mathbb{R}^d\to\mathbb{R}$ denote the function returned by a computationally expensive simulation at input $\mathbf{x}$. The GP-based emulator assumes

$$f(\mathbf{x}) \sim \mathrm{GP}\left(m(\mathbf{x}),\, k(\mathbf{x},\mathbf{x}')\right)$$

where $m(\cdot)$ is the mean function (often zero or low-order polynomial), and $k(\cdot,\cdot)$ is a positive-definite covariance kernel with hyperparameters $\theta$ (e.g., lengthscales, amplitude, nugget). With $n$ simulator runs at locations $X = [\mathbf{x}^{(1)},\dots,\mathbf{x}^{(n)}]$ and outputs $\mathbf{y}$, the predictive posterior at a new input $\mathbf{x}_*$ is Gaussian: [ \hat{f}(\mathbf{x}*) = m(\mathbf{x}_) = \mathbf{k}_*^\top [K + \sigma_n^2 I]^{-1}(\mathbf{y} - m(X