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Wang et al. J. Mater. Inf. 2026, 6, 16 Page 9 of 25

(5)
5 HF (G) = ( 5 LF (G)),
where C is learned from data. In co-kriging, this mapping is embedded within a GP framework so that LF
and HF functions are modeled as correlated random processes, and the learned correction enables the
surrogate to follow HF trends while exploiting LF coverage. The GP framework naturally provides predictive
uncertainty across the input space [41,48,69-71] . In GP regression, the posterior prediction at a new input x* is
Gaussian with a closed-form mean and variance, which naturally provides calibrated predictive uncertainty.
Here, uncertainty reflects both data noise (aleatoric uncertainty) and limited knowledge of the underlying
function (epistemic uncertainty). In materials informatics, such uncertainty-aware predictions are essential
for risk-sensitive decision making and adaptive sampling, where uncertainty estimates guide the selection of
new simulations or experiments. GP-based co-kriging, therefore, enables principled quantification and
propagation of uncertainty across fidelity levels.

A particularly common instantiation of this correction framework is the first-order autoregressive (AR1)
formulation introduced by Kennedy-O’Hagan and later popularized by Forrester . In this approach, the HF
[71]
function is expressed as [71]

(6)
5 HF (G) = d 5 LF (G) + X(G),

where ρ is a scaling factor and δ (x) is an independent GP that captures the systematic LF→HF discrepancy.
This formulation can be viewed as a specific parametric choice for the correction operator C (·), with LF
predictions rescaled and then adjusted by an additive discrepancy term. The resulting joint GP induces a
block covariance structure linking LF and HF observations, thereby enabling interpolation of HF data while
rigorously quantifying predictive uncertainty. Despite its popularity, the AR1 co-kriging formulation relies
on several implicit assumptions. First, the mapping between LF and HF responses is assumed to be
approximately linear, such that a single scaling factor captures the dominant cross-fidelity relationship.
When the LF→HF relationship is strongly nonlinear or state-dependent, this linear autoregressive
assumption may break down, motivating the use of more flexible nonlinear correction operators. Second,
both the LF process f (x) and the discrepancy term δ (x) are typically modeled as stationary GPs, meaning
LF
that their mean functions are constant and their covariance structures depend only on the separation
between inputs. This co-stationarity assumption can be restrictive in systems where cross-fidelity
correlations vary significantly across the input space [41,48,70,72-75] . When these linearity or stationarity
assumptions are violated, alternative MF strategies may be more appropriate. Examples include nonlinear
correction-based formulations, composite or deep GP models, and direct machine learning approaches with
fidelity features that do not impose explicit parametric structure on LF-HF mappings. In practice, diagnosing
the degree of linearity and stationarity in cross-fidelity relationships provides useful guidance for choosing
among classical co-kriging, nonlinear GP variants, and deep learning-based frameworks.

There are also alternative forms of the general correction operator C (·). Recursive co-kriging extends the
autoregressive framework to more than two fidelity levels by chaining GPs across coarse, intermediate, and
fine datasets [48,70,72] . Composite and deep GP models enrich the operator C (·) to capture nonlinear mappings
between fidelities that cannot be represented by a simple affine relation [41,73-75] . Other formulations, such as
those based on radial basis functions (RBFs) or multi-output GP surrogates, have likewise been introduced,
further broadening the family of correction-based strategies .
[76]

Co-kriging has already enabled significant advances in materials design [32,51,77-81] . In crystal plasticity, a
co-kriging model combining 100 LF orientation distribution function (ODF) data with 10 HF crystal
plasticity finite element method (CPFEM) simulations was used to calibrate slip and twinning parameters of