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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 . In the Augmented Bayesian Treed Co-Kriging
[48]
(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 (n ) and memory as
3
O
2 [104]
O (n ) . 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., (B · P)]. The total training cost is therefore (E · n · P / B) for E
O
O
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 (P) per query and can be efficiently batched, enabling deployment
O
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.