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Wang et al. J. Mater. Inf. 2026, 6, 16 Page 7 of 25

HF-only training, and even outperformed transfer learning baselines[Figure 2C] . Similarly, in the inverse
[66]
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) . In high-throughput screening,
[56]
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 . Mora et al. encoded data sources as one-hot
[54]
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 . Beyond deterministic
[67]
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 . Palizhati et al. further showed that MF agents
[68]
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.