  Active learning optimizes the exploration of large parameter spaces by
strategically selecting which experiments or simulations to conduct, thus
reducing resource consumption and potentially accelerating scientific
discovery. A key component of this approach is a probabilistic surrogate model,
typically a Gaussian Process (GP), which approximates an unknown functional
relationship between control parameters and a target property. However,
conventional GPs often struggle when applied to systems with discontinuities
and non-stationarities, prompting the exploration of alternative models. This
limitation becomes particularly relevant in physical science problems, which
are often characterized by abrupt transitions between different system states
and rapid changes in physical property behavior. Fully Bayesian Neural Networks
(FBNNs) serve as a promising substitute, treating all neural network weights
probabilistically and leveraging advanced Markov Chain Monte Carlo techniques
for direct sampling from the posterior distribution. This approach enables
FBNNs to provide reliable predictive distributions, crucial for making informed
decisions under uncertainty in the active learning setting. Although
traditionally considered too computationally expensive for 'big data'
applications, many physical sciences problems involve small amounts of data in
relatively low-dimensional parameter spaces. Here, we assess the suitability
and performance of FBNNs with the No-U-Turn Sampler for active learning tasks
in the 'small data' regime, highlighting their potential to enhance predictive
accuracy and reliability on test functions relevant to problems in physical
sciences.
