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

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 . Recent innovations further embed these
[38]
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 . Dimensionality is
[97]
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 . Overall, MF active learning and adaptive sampling represent a critical step toward
[95]
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.