Multi-fidelity learning in materials informatics: methodologies, applications, and outlook

Statistical and parametric models

A classical approach to incorporating MF information is to assume an explicit functional form grounded in physical or statistical reasoning. Data at different fidelity levels are represented with distinct noise variances, which are used to fit the functional form. HF data are accurate (small noise variance) but scarce, while LF data are abundant but noisy (large noise variance). Different data are therefore weighted according to their noise levels during the fitting process, typically via maximum likelihood under fidelity-dependent noise^[61,62].

A common strategy is to posit a parametric mapping y = F (x; θ) and estimate the parameters θ using maximum likelihood under fidelity-dependent noise. Concretely, one assumes

where σ²_{f (i)} denotes the noise variance associated with the fidelity level f (i) of sample i. The corresponding log-likelihood is

where the fidelity-dependent noise variance explicitly enters the likelihood. Maximizing this fidelity-weighted Gaussian log-likelihood with respect to θ is equivalent to solving a weighted least squares problem:

This formulation makes the role of fidelity explicit: higher-variance (lower-fidelity) data points are downweighted, while HF points with smaller variance exert a stronger influence on the parameter estimates^[61,62]. Thus, MF fusion is naturally achieved by adjusting the statistical weight of each fidelity level during parameter estimation.

The same variance-weighting principle also appears in GP and Kriging frameworks, where predictive means and variances from multiple fidelities are combined via inverse-variance weighting, and hyperparameters are fitted by maximum likelihood^[46,63]. In these approaches, fidelity is again treated statistically, with higher-fidelity data associated with smaller predictive variance, thereby exerting a stronger influence on the posterior model. This parallel highlights a unifying view: fidelity enters the modeling process through its impact on the variance structure.

The parametric and weighted least squares treatment is attractive because it preserves interpretability and transparency. The functional form F can be chosen to encode physical insights (e.g., linear or low-order polynomial response surfaces), while MF fusion is handled systematically through the likelihood-based weighting scheme. It is also computationally efficient and, when combined with projection-enabled linear regression, can accommodate non-overlapping MF datasets in which HF and LF data are observed at different inputs^[64]. When domain knowledge suggests a credible approximation form for F, parametric MF models are parsimonious, fast, and robust, providing a principled mechanism to prioritize HF data while still leveraging LF samples to capture global trends.

In practice, parametric MF models are most effective in applications where domain expertise provides guidance for a plausible functional form. For example, in hypersonic vehicle surface pressure modeling, MF linear regression improved predictive accuracy by up to 12% compared to SF regression, even with as few as 3-10 HF samples^[64]. In materials science, heteroscedastic Gaussian process regression (HGPR) has been applied to porous two-phase microstructures generated by finite element simulations. In this setting, input-dependent noise captures inherently noisy regimes, linking geometric descriptors (ellipse axes, porosity, spacing) to effective stress in representative volume element (RVE) (60%-40% split for training and validation sets)^[65]. This study illustrates that heteroscedasticity in materials datasets, traditionally regarded as a nuisance, can instead be leveraged as an informative signal to guide design and optimization.

These advantages, however, come with trade-offs. Parametric MF models often suffer from limited representational capacity: linear or polynomial structures may be inadequate for highly nonlinear, high-dimensional behaviors. They can also degrade when LF data introduce systematic bias or when fidelity relationships deviate significantly from simple mappings. Furthermore, mis-specified fidelity-dependent variances can undermine the intended emphasis on HF data. Such issues can be mitigated by re-estimating variance terms via maximum likelihood or by extending the framework to include fidelity-specific offsets and scaling factors^[46].

From a practical standpoint, for parametric MF models, a key design choice is the selection of the functional form. A practical strategy is to start from the simplest model justified by physical insight and progressively increase complexity as needed. Candidate models can be evaluated using cross-validation, and the model that exhibits the best generalization performance is selected.

Direct machine learning with fidelity features

A second strategy for MF learning is to directly encode fidelity information as an input feature to a unified model. Rather than building separate models at each fidelity or linking them hierarchically, all data are pooled and a single model is trained with an additional “fidelity feature”. For each sample (x, y) with fidelity f ∈ {1 (highest), …, K (lowest)}, a K-dimensional one-hot vector e _f is concatenated with the material descriptors x_i to form the augmented input x_i = [x_i ǀǀ e _{f (i)}]. At inference, setting e _f = [1, 0, …, 0] yields predictions at the highest fidelity, while other encodings correspond to lower fidelities [Figure 2A]. This approach is flexible because it does not require a one-to-one correspondence of data across fidelities or a fixed model hierarchy^[39,40].

Figure 2. Examples of direct MF learning with fidelity features. (A) Schematic of the one-hot encoding fidelity approach: each sample is augmented with a fidelity indicator vector e _f, allowing a single unified model to learn across low- and high-fidelity data; (B) Interatomic potential for water: test MAEs of a mfi trained with varying fractions of HF (SCAN) data (left: energy; right: force). Dashed lines indicate the MAEs of 1fi baselines using the same fractions of HF data, highlighting the efficiency of MF learning^[39]; (C) Frequency-dependent optical spectra (reduced absorption coefficient α): radar charts show normalized median errors on the HF test set for three GNNs-SF, TL, and FE, demonstrating the performance gains from fidelity-aware embeddings^[66]; (D) Simulated inorganic materials discovery under sequential learning: the x-axis shows the experimental acquisition budget, and the y-axis shows the fraction of ideal materials (with visible-spectrum band gaps) discovered. MF agents with fidelity indicators accelerate discovery relative to SF agents^[47]. MF: Multi-fidelity; HF: high fidelity; LF: low fidelity; MAE: mean absolute error; mfi: multi-fidelity model; 1fi: single-fidelity model; SCAN: strongly constrained and appropriately normed (density functional); GNNs: graph neural networks; SF: single fidelity; TL: transfer learning; FE: fidelity embedding.

Formally, each observation can be written as

where x_i denotes the material descriptors, e _{f (i)} is the one-hot encoding of the fidelity level, and ε_i represents observation noise. Unlike the parametric formulations discussed in section “Statistical and parametric models”, the functional form F is not prescribed a priori. Instead, F is estimated directly from data using machine learning models such as neural networks, support vector machines, or GPs. The fidelity encoding, therefore, guides the model to integrate information across fidelities within a unified representation, rather than weighting data points through fidelity-dependent noise variances.

Direct MF learning with fidelity features has been successfully applied in several areas of materials modeling. Ko and Ong integrated fidelity information into the “global state” of a graph neural network (GNN; M3GNet) to build a MF interatomic potential. Training on combined datasets of LF and HF simulations (e.g., silicon and water), the MF M3GNet achieved the accuracy of a HF-only model using only ~10% of the HF data, representing an order-of-magnitude saving in computational cost for generating HF data [Figure 2B]^[39]. Kim et al. developed SevenNet-MF, an equivariant GNN trained on mixed generalized gradient approximation (GGA, LF) and meta-generalized gradient approximation (meta-GGA, HF) calculations using one-hot fidelity features. With just 10% HF data relative to the LF set, SevenNet-MF achieved < 10% error in Li-ion conductivity and R² ≈ 0.98 for mixing energies on validation sets (90%-10% split for training and validation sets), outperforming fine-tuning and ∆-learning baselines^[40]. Together, these studies demonstrate that fidelity features enable models to effectively expand HF datasets by leveraging LF trends.

The approach has also been extended beyond interatomic potentials. Ibrahim and Ataca developed a GNN for predicting frequency-dependent optical spectra by embedding fidelity information as a learnable vector. Joint training on LF and HF spectra (34,327 LF data and 14,560 HF data) significantly reduced errors on test sets (80%-5%-15% split for training-validation-test sets) in dielectric response predictions compared to HF-only training, and even outperformed transfer learning baselines[Figure 2C]^[66]. Similarly, in the inverse design of halide perovskites, MF surrogates with fidelity indicators guided genetic algorithms to yield experimentally stable compositions by combining generalized gradient approximation Perdew-Burke-Ernzerhof (GGA-PBE, LF, 570 data), Heyd-Scuseria-Ernzerhof hybrid functional (HSE06, HF, 347 data), and experimental measurements (alternative HF, 97 data)^[56]. In high-throughput screening, multi-output GPs incorporated fidelity information by concatenating fidelity tags with material descriptors, enabling the fusion of experimental and computational measurements within a single workflow and improving both efficiency and uncertainty quantification^[54]. Mora et al. encoded data sources as one-hot vectors and fed them into a Bayesian neural network that maps fidelity indicators onto a low-dimensional continuous fidelity manifold, through which uncertainty is propagated to the output layer. This formulation imposes no restriction on the number of fidelities and avoids a predefined hierarchy. The network outputs both predictive means and variances, enabling quantification of prediction uncertainty as well as fidelity-related uncertainty. Applied to MF damage simulations of porous metallic components and to binding energy prediction in hybrid organic-inorganic perovskites, the learned manifold disentangles heterogeneous data sources and yields well-calibrated uncertainty estimates^[67]. Beyond deterministic surrogates, Zanjani Foumani et al. introduced a cost-aware Bayesian optimization framework with a MF emulator that one-hot encodes fidelity as a categorical variable, learns across fidelities, and automatically detects when LF sources are too biased to be useful^[68]. Palizhati et al. further showed that MF agents encoding fidelity features (DFT as LF, experiments as HF) accelerated early discovery by 20%-60% compared to SF baselines [Figure 2D]^[47].

Overall, direct MF machine learning provides a powerful and flexible strategy for exploiting heterogeneous datasets under limited HF budgets. These approaches have already enabled the cost-effective development of interatomic potentials approaching coupled-cluster accuracy, predictors of complex optical properties, and efficient high-throughput surrogates. They can reduce HF data requirements by up to an order of magnitude while matching or even surpassing the accuracy of HF-only models^[39,40]. Compared to correction-based approaches introduced in section “Co-Kriging and correction-based methods”, direct methods do not require overlapping data across fidelities; instead, they integrate complementary information within a unified representation^[39,66].

Despite these strengths, several limitations remain. Encoding fidelity as a one-hot tag does not explicitly capture cross-fidelity correlations (LF→HF mapping), which reduces interpretability relative to autoregressive or correction-based models. The approach also does not inherently account for heteroscedasticity or calibration: noisy LF data can dilute the HF signal and compromise uncertainty quantification. In addition, severe class imbalance, where only a small number of HF samples are available, may lead to overfitting to LF trends or spurious correlations between fidelity and input features. Thus, while direct MF learning with fidelity features is highly flexible and data-efficient, it is most effective when LF data are reasonably correlated with HF targets and when a sufficient number of HF points are available to anchor the learning process.

From a practical standpoint, performance is also sensitive to model-specific hyperparameter choices. Hyperparameter tuning in direct MF learning depends on the underlying machine learning model. For neural networks, the primary hyperparameters include network architecture, learning rate, batch size, and regularization strength. For GPs, the dominant choices are the kernel (covariance function), its associated hyperparameters, and the noise level.

Co-kriging and correction-based methods

Correction-based methods, most prominently co-kriging, represent one of the most established strategies in MF learning^[48]. They extend GP regression by modeling correlations between LF and HF data, thereby 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

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.

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^[71]. In this approach, the HF function is expressed as^[71]

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_LF (x) and the discrepancy term δ (x) are typically modeled as stationary GPs, meaning 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 Ti-7Al against tensile data, markedly improving stress-strain predictions relative to the ODF-calibrated model [Figure 3B]^[77]. In precipitation modeling of Mg alloys, co-kriging identified low-error regions more efficiently than HF-only surrogates [Figure 3C]^[28]. In electronic structure applications, it has been used to 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]^[78]. For molecular crystal prediction, co-kriging enabled the re-ranking of candidate polymorphs with hybrid DFT accuracy using only a small subset of direct calculations^[33]. In alloy design, LF molecular dynamics simulations with a machine-learned potential have been combined with select HF DFT data to efficiently map composition-property relationships^[82]. Battery applications have likewise benefited, where LF data from related cells informed the extrapolation of long-cycle HF degradation profiles^[37]. Finally, in process 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 is a constant ρ, which can be generalized to an input-dependent ρ (x) (e.g., linear or polynomial in x) to capture more complex corrections. Common kernel choices include exponential, radial basis function (RBF), Matérn family, and periodic kernels. If the data exhibit repeating patterns, periodic kernels are appropriate; if a smooth response is expected, exponential or RBF kernels are often preferred; conversely, for rougher behavior, Matérn 3/2 or 5/2 kernels provide reasonable starting points. Kernel hyperparameters are typically tuned via grid search or gradient-based optimization combined with cross-validation. When data are noisy or heteroscedastic across fidelity levels, separate noise parameters should be retained and jointly optimized.

MF deep learning-based frameworks

Recent advances in MF modeling have increasingly turned to deep learning architectures as flexible tools for linking LF and HF datasets in materials science, as schematically shown in Figure 4A. Unlike co-kriging or other correction-based models, neural networks can capture highly nonlinear and non-smooth relationships, making them particularly attractive for property landscapes where traditional surrogates struggle^[49,85,86].

Figure 4. Deep learning-based MF frameworks for materials science. (A) Schematic overview of representative strategies: transfer learning, composite neural networks, correction learning, and MFGNN; (B) Transfer learning for formation energy prediction: performance comparison of SF models trained on the Open OQMD (LF, 341k entries) and EXP-TL. Incorporating LF knowledge reduces MAE across training sizes; results are obtained by 10-fold cross-validation, and error bars indicate the standard deviation (confidence interval) across folds^[35]; (C) Composite neural network for extracting mechanical properties from indentation: MAPE as a function of training dataset size for the HF-only model and MF model, in comparison with the LF baseline (noted as [5])^[30]. Reproduced from^[30], © (2020) National Academy of Sciences. Distributed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license; image reused without modification; (D) MF graph neural network for band-gap prediction: comparison of MAE distributions across 1-fi, correction (1-fi-stacked), 2-fi, and MF (4-fi, 5-fi) models. LF data comprise PBE band gaps, while HF datasets include GLLB-SC, SCAN, HSE, and experimental values. Increasing fidelity integration systematically lowers prediction error^[29]. Reproduced from^[29] under the Creative Commons Attribution-ShareAlike (CC BY-SA 4.0) license. Together, these examples highlight the versatility of deep learning architectures in capturing nonlinear correlations and leveraging heterogeneous datasets for improved accuracy in MF learning. MF: Multi-fidelity; HF: high fidelity; LF: low fidelity; SF: single fidelity; GNN: graph neural network; MFGNN: multi-fidelity graph neural network; OQMD: Open Quantum Materials Database; TL: transfer learning; EXP-TL: experimental data (HF) with a transfer-learned model; MAE: mean absolute error; MAPE: mean absolute percentage error; PBE: Perdew-Burke-Ernzerhof (functional); GLLB-SC: Gritsenko-van Leeuwen-van Lenthe-Baerends solid-correlation (functional); SCAN: strongly constrained and appropriately normed (functional); HSE: Heyd-Scuseria-Ernzerhof (hybrid functional); 1-fi: single-fidelity; 2-fi: two-fidelity; 4-fi: four-fidelity; 5-fi: five-fidelity.

One important strategy is transfer learning between simulation tiers or between simulations and experiments. A common paradigm involves pretraining a deep neural network (DNN) on abundant LF data, followed by fine-tuning selected layers using limited HF data, sometimes with physics-informed constraints^[87-89]. For example, Chakraborty proposed a MF physics-informed deep neural network (MF-PIDNN) that encodes approximate physical laws at the LF level and is subsequently updated using scarce HF data, demonstrating reliable predictions in reliability analysis where experiments are limited but governing physics is known^[88]. In materials applications, this approach has been used for mechanical property inference from small punch tests, polymer property prediction using heterogeneous data sources, and formation energy prediction, where DFT pretraining accelerates learning on scarce experimental labels [Figure 4B]^[35,43,57]. Similar paradigms have been applied to fatigue life prediction with small datasets^[90] and even cross-domain transfer from inorganic datasets to metal-organic frameworks, where pretraining on one materials domain accelerates discovery in another^[55]. The latter study provides concrete validation of the generalization capability of MF transfer learning, demonstrating that latent representations learned from one class of materials can be reused in chemically distinct domains. Such transfer substantially reduces HF data requirements when exploring new materials families and highlights the robustness of MF representations beyond narrowly defined training distributions.

While transfer learning leverages sequential training, an alternative approach is to integrate multiple fidelities simultaneously within a composite neural network^[49]. Islam et al.^[91] demonstrated this concept for molecular dynamics, where coarse time-step data serve as LF inputs and fine time-step results provide HF corrections, yielding accurate pressure predictions with reduced computational time cost. To assess not only accuracy but also reliability, the authors evaluated model stability under repeated training. For each training-set size, the networks were retrained multiple times using different random selections of HF data while keeping the LF dataset fixed (10,000 samples for E* or 100,000 samples for σ _y). The error bars in Figure 4C report the standard deviation of the resulting mean absolute percentage error (MAPE) across these repeated trainings, thereby quantifying variability in generalization performance. The substantially narrower error bars observed for MF models indicate more stable predictions and higher reliability under changes in the HF training subset. This example illustrates that, beyond improving mean accuracy, MF learning can significantly enhance robustness and reduce sensitivity to data selection^[30]. Similar frameworks have been applied to predict the residual strength of graded composites and to recover stress-strain fields from digital image correlation in welds, all by coupling simulated and experimental data within a unified deep network^[36,92]. These studies show how composite neural networks exploit correlations between fidelities to deliver robust surrogates that outperform HF-only training at much lower cost.

A related approach is correction learning, where a network is trained directly on the discrepancy between LF and HF predictions. In band-gap benchmark studies, this strategy achieved lower errors than both transfer learning and composite neural networks, as the model concentrates its capacity on systematic LF bias^[93]. However, correction learning requires nested LF/HF pairs for training and typically demands an LF evaluation for every new query, which can limit its applicability when LF evaluations are computationally expensive.

Building on these approaches, the most ambitious efforts extend deep learning into MF GNNs. Chen et al. incorporated fidelity tags into the Materials Graph Network (MEGNet) to learn across band-gap datasets ranging from the PBE (LF) functional to the HSE hybrid functional and experimental measurements (HF)^[29]. By embedding fidelity as a global state, their model integrates heterogeneous datasets and captures correlations across fidelity levels. As shown in Figure 4D, MF GNNs consistently outperform SF and stacking models. Performance is evaluated on randomly segmented test sets (80%-10%-10% split for training-validation-test sets, repeating 6 runs), where the error bars show one standard deviation across runs (confidence intervals) and the dots denote individual model errors. Beyond the common two-level low- and HF setting, the effect of introducing multiple intermediate fidelity levels has been systematically examined. As additional fidelities are incorporated, prediction errors generally decrease and uncertainty bands narrow, reflecting improved regularization of the shared latent representation. However, adding a fidelity level with very limited data or weak correlation can degrade performance, as observed when a small SCAN dataset was included. This finding illustrates that increasing the number of fidelities is beneficial only when the added levels contribute sufficient data volume and maintain meaningful correlation with existing fidelities. These graph-based frameworks are particularly suited to materials discovery, as they naturally represent atomic topology while allowing fidelity-specific corrections.

Deep learning-based MF frameworks therefore provide remarkable flexibility in handling heterogeneous and non-nested data, while capturing strongly nonlinear trends that challenge traditional surrogates. At the same time, they are computationally intensive, sensitive to hyperparameter choices, and often lack interpretability, which motivates the continued integration of physics-based constraints to mitigate small-data challenges^[94]. Overall, these methods are emerging as powerful tools in materials science, particularly in scenarios involving strong nonlinearity, heterogeneous datasets, and the need for rapid discovery. By coupling the adaptability of deep learning with domain knowledge, MF neural frameworks are poised to become indispensable for next-generation materials design.

From a practical standpoint, the effectiveness of deep learning-based MF frameworks depends strongly on careful hyperparameter selection and training strategies. In practice, their effective deployment requires strategy-specific hyperparameter tuning. For transfer learning, a common approach is to stabilize the shared representation using abundant LF data, followed by limited fine-tuning with HF labels. When HF data are scarce, freezing early layers is often beneficial. For composite neural networks, loss weighting should prioritize HF accuracy while still retaining LF trend learning; a practical strategy is to begin with HF-weighted objectives and gradually relax toward a balanced multi-task loss. Throughout training, both accuracy and probabilistic metrics (e.g., negative log-likelihood and prediction interval coverage) should be monitored to ensure that uncertainty calibration does not degrade as LF information is incorporated. For correction learning, residual blocks should be kept lightweight, with sufficient capacity to capture systematic LF→HF discrepancies without overfitting. For MF GNNs, fidelity embeddings should remain compact and be jointly tuned with the shared trunk so that they modulate predictions without overwhelming structural features. In general, modest and budgeted hyperparameter searches focused on learning rate, weight decay, and loss weights (guided by HF validation and uncertainty calibration) tend to yield larger gains than aggressive architectural changes, particularly when HF data are limited.

MF active learning and adaptive sampling

Active learning and adaptive sampling extend MF modeling into a dynamic, decision-making context. Rather than passively integrating heterogeneous datasets, these approaches sequentially decide not only which candidate material to evaluate next, but also at which fidelity level [Figure 5A]. The central objective is to maximize the expected information gain per unit cost, thereby accelerating discovery while respecting finite computational and experimental budgets.

Figure 5. MF active learning and adaptive sampling in materials science. (A) Active-learning loop: schematic of candidate selection and fidelity allocation, where promising candidates are evaluated with costly HF methods and less valuable candidates with inexpensive LF approximations; (B) COF optimization: MF active learning achieves lower simple regret for xenon-krypton selectivity compared to SF and random baselines^[60]; (C) Molecular discovery: MF-GFN accelerate convergence toward diverse molecules with desirable electron affinity compared with SF and random-fidelity approaches^[96]. Together, these examples illustrate how MF active learning dynamically balances exploration, exploitation, and cost to accelerate discovery across molecular and porous materials. MF: Multi-fidelity; HF: high fidelity; LF: low fidelity; SF: single-fidelity; MF-BO: multi-fidelity Bayesian optimization; COF: covalent organic framework; GFN: generative flow network; MF-GFN: multi-fidelity generative flow network; SF-GFN: single-fidelity generative flow network; PPO: proximal policy optimization; MF-PPO: multi-fidelity proximal policy optimization; MES: max-value entropy search; MF-MES: multi-fidelity max-value entropy search; EI: expected improvement; SFEI: single-fidelity expected improvement.

A central paradigm is multi-fidelity Bayesian optimization (MF-BO), in which GP surrogates or deep-kernel variants model correlations between fidelity levels [Table 1]^[50]. Acquisition functions are adapted to weigh both evaluation cost and expected improvement (EI), enabling the algorithm to alternate between inexpensive LF approximations and costly HF evaluations^[50]. For example, MUlti-task Max-value Bayesian Optimization (MUMBO) employs a cost-aware acquisition function that selects the candidate-fidelity pair expected to deliver the largest reduction in uncertainty about the global optimum per unit cost, scaling effectively to multi-task and MF settings^[50]. Let (x, z) denote a material candidate x evaluated at fidelity level z. The next query (x_n+1, z_n+1) is chosen according to^[50]

where α_n (x, z) is an acquisition function quantifying the EI of the target property at iteration n, and c (x, z) is a cost function associated with fidelity z^[50]. This explicit balance between information gain and evaluation cost allows the algorithm to dynamically allocate resources across fidelities, alternating between LF approximations and HF evaluations^[50].

Applications in materials science highlight the versatility of these strategies. In materials screening, multi-output GPs have been used to fuse experimental and computational data on-the-fly, replacing rigid stagewise funnels with progressive, budget-aware exploration and cutting optimization costs by nearly threefold in benchmark studies^[54]. In covalent organic framework (COF) optimization, MF-BO achieved lower regret at a given budget compared to SF baselines, as shown in Figure 5B. To assess both average performance and reliability, MF and SF active learning strategies were evaluated using 20 independent random seeds, corresponding to different initial samples and acquisition trajectories. In Figure 5B, the shaded regions represent the standard deviation of simple regret at each budget level, capturing the spread of optimization outcomes across repeated campaigns. The visibly narrower shaded bands for MF strategies compared to their SF counterparts indicate reduced run-to-run variability and therefore higher reliability. These results highlight that MF active learning not only accelerates convergence but also yields more reproducible and dependable discovery trajectories^[60].

Sequential learners extend this idea by combining regressors such as GPs, random forests, and support vector machines into active loops that adaptively decide whether to query a fast DFT simulation (LF) or a costly experiment (HF), with successful demonstrations in two-fidelity band-gap discovery agents^[47]. In structural materials, MF-BO has been applied to dual-phase steel design, where LF micromechanical models (isostrain, isostress, isowork, secant, elastic-constraint) were queried far more often than costly finite element method (FEM) simulations (with RVE), especially in early iterations^[53,95]. Similarly, in interatomic potential development under extreme conditions, adaptive MF sampling efficiently identified promising parameter regimes that would have been infeasible to explore exhaustively^[38]. Recent innovations further embed these principles into generative frameworks. In the multi-fidelity generative flow network (MF-GFN), the acquisition function is treated as a reward signal that guides stochastic generation of candidate-fidelity tuples, enabling discovery of diverse, high-performing materials while maintaining cost efficiency and outperforming random or SF baselines [Figure 5C]^[96].

The strengths of MF active learning lie in its ability to allocate resources efficiently. By dynamically balancing exploration of uncertain regions with exploitation of promising candidates, these methods accelerate discovery trajectories compared to static or fixed-fidelity sampling. They are particularly well-suited for autonomous experimental laboratories and adaptive simulation campaigns, where real-time decisions about the next measurement are essential.

The main challenges arise from implementation complexity and the need for robust uncertainty quantification. Acquisition functions must be carefully calibrated to avoid over-exploitation of biased LF data and to maintain trustworthy uncertainty estimates in high-dimensional, structured search spaces. Surrogate models that combine diverse fidelities can also become computationally demanding when embedded in sequential loops. Studies have shown that while approximate LF solvers can accelerate discovery, they may also mislead active learning if correlations with HF targets are not explicitly modeled^[97]. Dimensionality is another bottleneck. Recent advances such as adaptive active-subspace methods embedded within MF-BO have improved efficiency by reducing the search to informative subspaces in process-structure-property optimization^[95]. Overall, MF active learning and adaptive sampling represent a critical step toward autonomous, closed-loop materials discovery, where algorithms not only predict outcomes but also decide the most efficient way to generate new data.

From a practical perspective, effective deployment of MF active learning benefits from staged hyperparameter tuning. A recommended workflow is to first stabilize the MF surrogate and cost models to ensure reliable predictions and uncertainty estimates. Next, the acquisition function should be calibrated to balance expected information gain against evaluation expense. Finally, scheduling parameters such as stopping criteria, fidelity budgets, or query ratios can be defined. In practice, a useful starting point is to pair a well-calibrated MF surrogate with a simple cost model, for example, based on computational time or dataset size.

The choice of acquisition strategy should align with the problem constraints and fidelity structure. Information-gain-per-cost criteria (e.g., entropy-based methods or MF knowledge-gradient) are robust when fidelities differ strongly in accuracy and cost, whereas EI-per-cost or Upper Confidence Bound (UCB)-per-cost strategies are effective when costs are relatively stationary across the domain. Stopping criteria can be organized into three practical tiers. Budget-based termination (e.g., fixed wall-time, total cost, or number of HF evaluations) is the default in autonomous campaigns with predefined resource limits. Progress-based criteria monitor the learning signal, such as halting when the maximum EI (or EI-per-cost) remains below a threshold for several iterations, or when posterior uncertainty in the region of interest falls below a prescribed margin. Finally, value-of-information (VoI)-based stopping rules compare the expected knowledge gain per unit cost against a minimum acceptable return and terminate when further evaluations are no longer justified. The VoI-based strategies, including cost-aware knowledge-gradient variants, are particularly suitable when cost, risk, and uncertainty must be explicitly balanced.

Bias vs. variance

Compared with HF data, LF sources typically exhibit larger errors, which can be decomposed into bias and variance. Bias refers to systematic deviations introduced by differences in data-generation pipelines, such as experiments vs. simulations, simplified physical models, alternative algorithms (e.g., HSE vs. PBE), or choices of simulation hyperparameters^{[29,32-34,52,82,91,95,98,99]}. Variance, by contrast, reflects random measurement noise and is generally lower in HF data. Examples include measurements made under the same protocol but carried out by novice vs. experienced researchers, or using worn vs. newly calibrated instruments^[100].

The relative contributions of bias and variance strongly influence the choice of MF methodology, as shown in Table 2. When systematic bias dominates, correction-based methods are particularly effective because they explicitly model cross-fidelity discrepancies. Deep learning approaches or models with explicit fidelity indicators (e.g., one-hot encoding) can play a similar role by learning fidelity-specific corrections. When variance is the primary concern, statistical approaches with heteroscedastic modeling or inverse-variance weighting provide a principled treatment. In practice, especially in experimental materials data, bias and variance almost always coexist^[101]. In such mixed settings, kriging-based methods (e.g., co-kriging), which simultaneously capture cross-fidelity correlations and account for heteroscedastic errors, are attractive options^[48]. Active learning strategies also provide robust solutions, as they can adaptively downweight LF inputs in non-critical regions without requiring explicit assumptions about error type.

Overall, recognizing whether bias, variance, or both dominate a MF dataset is essential for selecting an appropriate strategy, and remains a central cross-cutting theme in the design of effective materials informatics workflows.

Data requirements: nested vs. non-nested

In addition to error characteristics, the feasibility of MF methods is strongly influenced by their data requirements, particularly the distinction between nested and non-nested designs. In a nested design, the same input x is observed at multiple fidelity levels, allowing direct comparison and correction between low- and HF outputs. By contrast, in a non-nested design, most inputs are observed at only one fidelity level, leaving little or no overlap across fidelities.

Correction-based methods such as co-kriging are especially sensitive to this distinction, as they typically assume a nested structure to learn systematic deviations between fidelities. In practice, however, materials datasets are often non-nested, arising from disparate sources such as separate DFT databases, experimental measurements, or high-throughput simulations conducted with different parameter sets. This lack of overlap makes direct discrepancy modeling challenging and can reduce the effectiveness of classical co-kriging.

To address these challenges, several extensions have been proposed that relax the nesting requirement. Specifically, recursive variants of classical co-kriging can propagate correlations across multiple fidelity levels without requiring strict nesting of input locations^[48]. In the Augmented Bayesian Treed Co-Kriging (ABTCK) framework, missing cross-fidelity pairings are statistically imputed through hierarchical partitioning of the input space, creating an augmented dataset in which the joint posterior can be factorized. This approach enables fully Bayesian predictive inference under non-nested designs without enforcing explicit co-location^[102]. Beyond ABTCK, the Generalized Co-Kriging (GCK) framework aggregates a calibrated LF Kriging model with a stochastic discrepancy Kriging term, such that the LF predictive distribution (mean and variance) is propagated to locations where only HF observations are available. This probabilistic coupling permits separate parameter estimation for each fidelity and enables robust fusion under both nested and non-nested conditions, improving accuracy and stability relative to classical AR1 formulations when LF-HF correlations are imperfect^[103]. These methods are particularly useful in materials science, where integrating heterogeneous datasets, such as semi-local DFT with hybrid-functional calculations, or simulations with experiments, rarely yields perfectly aligned input sets.

It is important to note, however, that not all MF strategies depend on nested data. For example, statistical weighting and direct machine learning with fidelity features can operate on non-nested datasets. Deep learning frameworks provide additional flexibility, as architectures such as hierarchical neural operators or graph-based encoders can learn shared latent spaces even when input sets across fidelities do not overlap; however, such approaches typically require larger training datasets to generalize reliably. In contrast, active learning and adaptive sampling are particularly suitable for few-sample scenarios, as they strategically select HF evaluations to maximize information gain and can dynamically create nested datasets when needed. As summarized in Table 2, the degree of nesting required varies widely across methodologies, underscoring the need for approaches that can exploit both nested and non-nested data in real-world materials design.

Computational cost and scalability

The computational demands of MF algorithms vary widely, and scalability is often a decisive factor in method selection. Kriging-based approaches, including co-kriging, require inversion of an n × n covariance matrix, where n is the number of data points. As a result, training time scales as $$ \mathcal{O} $$ (n³) and memory as $$ \mathcal{O} $$ (n²)^[104]. In practice, this cubic scaling quickly becomes prohibitive, making covariance-matrix inversion the dominant computational bottleneck and limiting the applicability of co-kriging once MF datasets reach thousands of entries^[48].

By contrast, deep learning-based MF models can be trained using mini-batch stochastic gradient descent, for which the per-iteration computational cost scales approximately linearly with the batch size B and the number of trainable parameters P [i.e., $$ \mathcal{O} $$ (B · P)]. The total training cost is therefore $$ \mathcal{O} $$ (E · n · P / B) for E epochs over n samples. This structure maps naturally onto modern hardware accelerators: data-parallel training across multiple GPUs can achieve near-linear speedup (subject to communication overhead), while mixed-precision training and gradient checkpointing substantially reduce memory footprint and wall time. At inference, prediction cost scales as $$ \mathcal{O} $$ (P) per query and can be efficiently batched, enabling deployment on datasets that are orders of magnitude larger than those tractable with GP or co-kriging models.

In summary, while kriging-based methods provide strong uncertainty quantification and interpretability, their poor scalability necessitates caution when applied to large datasets. Neural networks and other scalable surrogates offer practical alternatives in data-rich regimes, making computational cost a central criterion when choosing among MF methodologies.