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Wang et al. J. Mater. Inf. 2026, 6, 16 Page 13 of 25

(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 .
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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.

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] . Acquisition functions are adapted to weigh
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both evaluation cost and expected improvement (EI), enabling the algorithm to alternate between
inexpensive LF approximations and costly HF evaluations . For example, MUlti-task Max-value Bayesian
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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