fig3

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.