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.

翻译：本文研究了利用概率神经网络（PNNs）建模偶然不确定性——即系统输入-输出关系中固有的变异性，通常表现为不等方差或异方差性。与传统神经网络产生确定性输出不同，PNNs 为目标变量生成概率分布，从而能够确定回归场景中的预测均值与区间。本文的贡献包括：开发了一种概率距离度量以优化PNN架构，并在受控数据集以及涉及纤维增强复合材料的实际材料科学案例中部署PNNs。研究结果证实，PNNs 能有效建模偶然不确定性，相较于常用的高斯过程回归，该方法在此场景下更具优势。具体而言，在真实世界的科学机器学习应用中，PNNs 可产生异常精确的输出均值估计（R² 分数接近0.97），其预测区间与观测数据区间高度吻合（相关系数接近0.80）。因此，本研究为探索如何利用神经网络强大的表征能力来刻画科学问题中复杂的输入-输出关系提供了持续性贡献。