  This paper investigates the use of probabilistic neural networks (PNNs) to
model aleatoric uncertainty, which refers to the inherent variability in the
input-output relationships of a system, often characterized by unequal variance
or heteroscedasticity. Unlike traditional neural networks that produce
deterministic outputs, PNNs generate probability distributions for the target
variable, allowing the determination of both predicted means and intervals in
regression scenarios. Contributions of this paper include the development of a
probabilistic distance metric to optimize PNN architecture, and the deployment
of PNNs in controlled data sets as well as a practical material science case
involving fiber-reinforced composites. The findings confirm that PNNs
effectively model aleatoric uncertainty, proving to be more appropriate than
the commonly employed Gaussian process regression for this purpose.
Specifically, in a real-world scientific machine learning context, PNNs yield
remarkably accurate output mean estimates with R-squared scores approaching
0.97, and their predicted intervals exhibit a high correlation coefficient of
nearly 0.80, closely matching observed data intervals. Hence, this research
contributes to the ongoing exploration of leveraging the sophisticated
representational capacity of neural networks to delineate complex input-output
relationships in scientific problems.
