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Page 4 of 25 Wang et al. J. Mater. Inf. 2026, 6, 16

STRATEGIES AND APPLICATIONS FOR MF LEARNING IN MATERIALS DESIGN
We classify MF learning methodologies into five broad categories[Figure 1]: statistical and parametric
models, direct machine learning with fidelity features, co-kriging and correction-based methods, MF deep
learning-based frameworks, and MF active learning and adaptive sampling. Each of these strategies
represents a distinct way of integrating information across fidelity levels. In the subsections below, we
describe their core ideas, illustrate their use with representative examples, and evaluate their strengths,
weaknesses, and appropriate domains of application.

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

2 (1)
H 8 = (G 8 ; \) + Y 8 , Y 8 ∼ N (0, f ),
5 (8)
where σ denotes the noise variance associated with the fidelity level f (i) of sample i. The corresponding
2
f (i)
log-likelihood is
Õ h 2 2 2 i
log ?(H | \) = − 1 (H 8 − (G 8 ; \)) /f + log f + , (2)
2 5 (8) 5 (8)
8
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:

Õ 2 2
ˆ
\ = arg min F 8 H 8 − (G 8 ; \) , F 8 = 1/f 5 (8) . (3)
\
8
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