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.