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Figure 3. Examples of correction-based MF learning using co-kriging. (A) Schematic illustration of the co-kriging approach, where a GP
model learns a correction that maps LF to HF responses (adapted from [71] ); (B) Crystal plasticity modeling of Ti-7Al alloy: the resulting
stress-strain responses indicate that the MF calculations perfectly reproduce the experimental results. The inset shows HF (CPFEM) and
LF (ODF) data [77] ; (C) Precipitate shape simulation in Mg alloys: heatmaps show predicted discrepancies between simulated and
experimental precipitate aspect ratios. White points mark observed samples, and black lines indicate the actual LER. Binary images are
predicted LERs from co-kriging (MF) and HF-only (SF) models, demonstrating that additional LF data improve efficiency [28] . The ε denotes
the crystal mismatch between the precipitate and matrix phases, while γ is the interface energy; (D) Dopant formation energies in hafnia
calculated using DFT: the prediction accuracy of co-kriging models improves as the number of HF (N hi ) and LF (N lo ) training points
increases [34] ; (E) Atomization energy benchmarks: the RMSE decreases as more fidelities are combined. Two-fidelity models use
CCSD(T)/HF or CCSD(T)/MP2, while 3Fi and 4Fi models show progressively better accuracy [78] . Overall, these case studies highlight the
versatility of correction-based surrogates in bridging simulations and experiments across diverse materials applications. MF: Multi-fidelity;
HF: high fidelity; LF: low fidelity; SF: single fidelity; GP: Gaussian process; DFT: density functional theory; CPFEM: crystal plasticity finite
element method; ODF: orientation distribution function; LER: lower-error region; RMSE: root-mean-square error; CCSD(T): coupled cluster
with single, double, and perturbative triple excitations; MP2: second-order Møller-Plesset perturbation theory; 3Fi: three-fidelity; 4Fi:
four-fidelity.

Co-kriging and correction-based methods
Correction-based methods, most prominently co-kriging, represent one of the most established strategies in
MF learning . They extend GP regression by modeling correlations between LF and HF data, thereby
[48]
leveraging the broad coverage of inexpensive LF data to complement the accuracy of HF data. The basic idea,
illustrated in Figure 3A, is to learn a statistical correction that maps LF outputs to HF outputs, effectively
“lifting” cheap predictions into the accuracy regime of expensive ones. Formally, one seeks a correction
operator C such that