Surrogate-GP Emulation

• Surrogate-GP emulation is a Gaussian process-based approach that models expensive simulations with rigorous uncertainty quantification and interpolation guarantees.
• The method integrates boundary conditions and domain knowledge to enhance predictions and optimize design in complex scientific and engineering systems.
• Advanced variants such as linked, deep, and sparse GPs ensure computational scalability, high-fidelity uncertainty propagation, and efficient multi-fidelity emulation.

Surrogate-GP emulation refers to the construction and use of Gaussian process (GP)-based surrogate models (also called "emulators") for expensive computer simulations, with the aim of enabling efficient uncertainty quantification, optimization, sensitivity analysis, and design for scientific and engineering systems. GPs provide a nonparametric, Bayesian framework that is ideally suited for problems where simulator evaluations are computationally prohibitive and uncertainty quantification is essential. This article presents the mathematical foundation of surrogate-GP emulation, advanced extensions, and exemplar applications, with emphasis on rigor and recent developments.

1. Mathematical Foundations of Surrogate-GP Emulation

A surrogate-GP emulator models an unknown deterministic simulator $f:\mathbb{R}^d \to \mathbb{R}$ as a realization from a Gaussian process prior: $$f(x) \sim \mathcal{GP}\left(m(x),\, k(x,x')\right),$$ where $m(x)$ is the mean (often constant or low-order polynomial), and $k(x,x')$ is a positive-definite covariance kernel (commonly squared-exponential or Matérn). Given a set of expensive simulator evaluations $\mathcal{D} = \{(x_i, f(x_i))\}_{i=1}^n$, the predictive distribution at any $x_*$ is Gaussian with: $$\mu_*(x_*) = k_*^\top K^{-1}y,\qquad \sigma^2_*(x_*) = k(x_*,x_*) - k_*^\top K^{-1}k_*,$$ where $K$ is the kernel matrix on training points, $k_*$ is the vector of kernel values between $x_*$ and the training inputs, and $y$ is the vector of observed outputs.

The hyperparameters (kernel lengthscales, variance, noise) are typically estimated by marginal-likelihood maximization or Bayesian approaches (Schaechtle et al., 2015, Paul et al., 2024). Notably, the GP provides both interpolation and principled uncertainty quantification, enabling propagation of epistemic uncertainty in downstream UQ tasks.

2. Extensions: Incorporating Boundary Knowledge and Physics

A major advance in surrogate-GP emulation is the analytic enforcement of boundary conditions and the integration of domain knowledge:

• Known boundary incorporation: If analytic or ultra-cheap evaluations are available on boundaries or hyperplanes of the input space, the GP update can be performed with these boundary "observations" at essentially zero computational cost. The mean and covariance admit closed-form updates that collapse posterior variance on the known boundaries and propagate this information into the domain. For instance, for a known boundary $K = \{x_1 = 0\}$, the posterior mean and covariance simplify to:

$$m_K(x) = m(x) + r_1(a)\left[g(x_{2:d}) - m(0,x_{2:d})\right]$$

$$k_K(x,x') = \sigma^2 \left[r_1(a-a') - r_1(a) r_1(a')\right] \prod_{i=2}^d r_i(x_i-x_i'),$$

where $a = x_1$, and $r_i$ is the one-dimensional correlation (Vernon et al., 2018).

• Multiple, possibly intersecting boundaries are analytically integrated through successive closed-form rank-1 updates. This enables construction of emulators that encode exact simulator values or structure on analytic, physical, or geometric boundaries of the parameter space.
• Design in presence of known boundaries: Classical space-filling strategies (e.g., Latin hypercube sampling) are suboptimal when boundaries are known. Variance-optimal design strategies, minimizing global integrated posterior variance, or warping LHS to match posterior variance decay, are developed to optimally allocate expensive simulator runs away from well-known regimes (Vernon et al., 2018).
• Boundary-aware kernels on irregular domains: The BdryMatérn framework incorporates Dirichlet, Neumann, or Robin boundary information into a Matérn-type GP on an irregular, connected domain, using SPDEs and explicit Green's function/path-integral constructions, enabling sample paths to satisfy user-specified boundary physics exactly on domains with smooth or complex boundaries (Ding et al., 12 Jul 2025).

3. Surrogate-GP Emulation for Complex or Modular Simulators

For hierarchical or coupled systems, advanced constructions are deployed:

• Linked GP surrogates: Multi-stage, feed-forward coupled systems can be emulated by analytically linking GPs across submodules, propagating uncertainty and correlations through the system. For common kernel families (e.g., half-integer Matérn), closed-form linking formulas for the mean and variance are available, supporting fully analytic surrogates for multi-layered models (Ming et al., 2019, Ming et al., 2021).
• Deep GPs and stochastic imputation: To accommodate nonstationarity or regime changes, deep Gaussian processes (hierarchical compositions of GPs) are transformed via stochastic imputation into tractable linked GPs, making full Bayesian inference and uncertainty quantification feasible for complex simulators (Ming et al., 2021).
• Multi-output and multi-fidelity emulation: Cross-output covariance through co-kriging, latent process models, and hierarchical kriging enable emulation of vector-valued outputs, hierarchical fidelity codes, and transfer learning across related simulators, with advances such as adaptive local transfer (e.g., LOL-GP) to avoid negative transfer (Wang et al., 2024, Svendsen et al., 2019, 2206.12113).

4. Computational Scalability and Approximate Methods

The classical cubic cost of GP regression in the number of training points is prohibitive for large datasets. The following scalable emulation approaches are used:

• Local approximate GPs (laGP): For each prediction location, a small, adaptively selected neighborhood is used to construct a local GP, dramatically reducing computational cost while preserving predictive accuracy (Sun et al., 2017, Hutchings et al., 2024).
• Sparse and structured GPs: Inducing-point methods, organized on structured grids, combined with Kronecker acceleration (E-SGP) or efficient variational inference, provide linear or near-linear scaling in data size, making emulation of large fluid-mechanics and multiphysics datasets feasible (Duan et al., 2023, Li et al., 2023).
• Product-of-experts and mini-batch variational methods: Nonstationary, massive-data settings are addressed by combining local sparse GPs, each tuned to a local regime, in a statistically consistent product-of-experts framework (ProSpar-GP), and optimized using GPU-accelerated stochastic variational inference with careful control for Kolmogorov consistency (Li et al., 2023).
• Fast functional-outcome emulation: For functional outputs (time/spatial curves or images), SVD/EOF-based dimensionality reduction combined with local GP emulation on latent coordinates enables emulation and modular calibration in high dimensions (Hutchings et al., 2024).

5. Adaptive Experimental Design and Active Learning

Effective surrogate emulation requires strategic allocation of computational budget. Methods include:

• Acquisition-function design: VIGF (Variance of Improvement for Global Fit) directly targets global emulation accuracy by balancing exploration (GP variance) and exploitation (squared deviation from nearest observed data), while batch and multi-fidelity extensions enable efficient parallelization and resource allocation across fidelity levels (2206.12113).
• Entropy/information-based criteria: Mutual information maximization (MICE) and related criteria seek input configurations that maximally reduce overall emulator uncertainty, supporting both sequential and parallel experimental designs (Mathikolonis et al., 2019, Svendsen et al., 2019).
• Uncertainty-aware acquisitions: Modern algorithms (e.g., Pareto-front acquisition for uncalibrated surrogates such as Epistemic Nearest Neighbors) enable principled exploration/exploitation when predictive uncertainty is not properly calibrated (Sweet et al., 15 Jun 2025).

6. Quantification and Assessment of Uncertainty

The quality of surrogate-GP predictions is critically tied to uncertainty quantification:

• Coverage guarantees and adaptivity: Cross-conformal and Jackknife+ methods adaptively rescale prediction intervals by the local posterior standard deviation, offering frequentist coverage guarantees, distribution-free properties, and improved correspondence between interval width and true surrogate error, even under kernel misspecification or model misspecification (Jaber et al., 2024).
• Calibration diagnostics: Tools such as conformity scores, interval width–error correlation, and empirical coverage are central to validating the fidelity and reliability of the surrogate's uncertainty quantification (Jaber et al., 2024).

7. Applications and Empirical Impact

Surrogate-GP emulation has become a critical enabler of scientific discovery and engineering design across domains:

• Geomechanical modeling: Emulators enable rapid, high-fidelity sensitivity analysis and Bayesian calibration for geological repositories, translating large-scale simulations into interactive, uncertainty-aware digital twins (Paul et al., 2024).
• Aero/astronautics: Surrogate-GP models deliver sub-percent-accuracy emulation of multi-dimensional satellite drag over millions of simulation runs, permitting real-time trajectory optimization and collision avoidance (Sun et al., 2017, Li et al., 2023).
• Fluid dynamics and multi-physics: Kronecker-accelerated sparse GPs, physics-informed co-kriging, and boundary-aware surrogates provide interpretable, uncertainty-quantified emulation and physical insight into turbulent flow, multi-field interactions, and nonlinear PDE-governed systems (Mak et al., 2016, Long et al., 2024, Ding et al., 12 Jul 2025).
• Complex hierarchical and modular codes: Linked and deep GP surrogates address multistage, multiscale computation and transfer learning across fidelity levels, workflow steps, and related systems (Ming et al., 2019, Ming et al., 2021, Wang et al., 2024).

8. Summary of Theoretical and Practical Guarantees

Surrogate-GP emulation frameworks provide:

• Provable interpolation and uncertainty quantification under regularity assumptions.
• Exact satisfaction of analytic boundary and physics constraints when these can be encoded as known boundaries or through SPDE/Green's function constructions (Ding et al., 12 Jul 2025, Vernon et al., 2018).
• Finite-sample frequentist coverage of intervals when using conformal/jackknife-based uncertainty quantification (Jaber et al., 2024).
• Explicit error and convergence bounds when using finite-element approximations or local GP surrogates (Ding et al., 12 Jul 2025, Sun et al., 2017).
• Statistically coherent and consistent prior/posterior structures for modular and product-of-experts surrogates (Li et al., 2023).

Surrogate-GP emulation thus represents a mathematically rigorous, computationally efficient, and highly extensible toolkit for scientific computing, enabling uncertainty-aware inference, design, and optimization on otherwise intractable simulation models.