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Ti-7Al against tensile data, markedly improving stress-strain predictions relative to the ODF-calibrated
model [Figure 3B] . In precipitation modeling of Mg alloys, co-kriging identified low-error regions more
[77]
efficiently than HF-only surrogates [Figure 3C] . In electronic structure applications, it has been used to
[28]
correct bandgaps computed using semi-local DFT with the Perdew-Burke-Ernzerhof (PBE) functional to the
accuracy of the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional, enabling HF-quality predictions for
thousands of materials at a fraction of the computational cost [Figure 3D] [34,42,52] . In molecular benchmarks,
autoregressive GPs trained across multiple fidelities, including HF DFT, second-order Møller-Plesset
perturbation theory (MP2), and coupled-cluster with singles, doubles, and perturbative triples [CCSD(T)],
showed steadily decreasing errors as each fidelity level was incorporated [Figure 3E] . For molecular crystal
[78]
prediction, co-kriging enabled the re-ranking of candidate polymorphs with hybrid DFT accuracy using only
a small subset of direct calculations . In alloy design, LF molecular dynamics simulations with a
[33]
machine-learned potential have been combined with select HF DFT data to efficiently map
composition-property relationships . Battery applications have likewise benefited, where LF data from
[82]
related cells informed the extrapolation of long-cycle HF degradation profiles . Finally, in process
[37]
engineering, analytical melt-pool models have been corrected with finite-element simulations to predict
melt-pool dimensions in additive manufacturing, while in fracture mechanics, small sets of HF experimental
data (6 points) have been amplified by co-kriging with inexpensive theoretical predictions (22 points) [83,84] .
Together, these examples highlight the broad applicability of correction-based methods across length scales,
materials classes, and property types.

Beyond specific applications, co-kriging offers several general advantages. By jointly modeling LF and HF
data, it can deliver HF-level accuracy at a fraction of the computational cost, particularly when abundant LF
data are available to constrain global trends. Uncertainty quantification is inherent to the GP framework and
is especially valuable for downstream tasks such as active learning and Bayesian optimization. By contrast,
most deep learning-based MF approaches do not provide calibrated uncertainty estimates unless explicitly
augmented with probabilistic outputs, such as Bayesian neural networks or ensemble methods. Moreover,
co-kriging is versatile: whenever cross-fidelity correlations exist, it can fuse heterogeneous data sources,
including multiple levels of simulation, simplified analytical models, and experimental measurements.
Compared with multivariate probability distribution-based MF models, co-kriging explicitly learns
functional input-output relationships across fidelities, which is critical for capturing complex
structure-property mappings.

Nevertheless, several practical limitations remain. Reliable calibration of the LF→HF mapping often requires
overlapping (nested) data, i.e., the same inputs observed at multiple fidelity levels. When this condition
cannot be satisfied, more flexible alternatives, such as direct machine learning with fidelity features or
probabilistic deep learning models, may be better suited for non-nested or heterogeneous datasets. Beyond
data requirements, computational scalability poses a further challenge. The training cost scales cubically with
the number of samples, and the joint covariance structure grows with the number of fidelity levels, making
large-scale applications computationally expensive. Interpretability can also be limited, as the discrepancy
term δ (x) is a statistical adjustment that lacks direct physical meaning. Finally, simple linear autoregressive
assumptions may fail when LF→HF relationships are nonlinear or strongly state-dependent, thereby
motivating the development of deep or composite GP corrections. These trade-offs are important to consider
when deciding between correction-based and alternative MF strategies.

From a practical perspective, the successful application of co-kriging also depends on careful hyperparameter
selection and calibration. The dominant hyperparameters of co-kriging largely mirror those of GPs,
including the choice of kernel (covariance function), its associated hyperparameters, and the noise level. In
addition, for linear autoregressive formulations, the scaling factor ρ plays a critical role. The simplest choice