Probabilistic Neural Networks (PNNs) for Modeling Aleatoric Uncertainty in Scientific Machine Learning (2402.13945v1)

Abstract: 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.

• The paper introduces a novel probabilistic metric based on KL divergence to optimize neural network configurations for uncertainty modeling.
• It rigorously compares PNNs to Gaussian Process Regression, achieving near 0.97 R-squared in applications with heteroscedastic data.
• The study highlights practical benefits for scientific decision-making and proposes future integration with active learning frameworks.

Probabilistic Neural Networks for Aleatoric Uncertainty in Scientific Machine Learning

In the domain of scientific machine learning, capturing the inherent uncertainty within a system's input-output relationship is essential. This paper presents a focused paper on Probabilistic Neural Networks (PNNs) as tools for modeling aleatoric uncertainty, which represents this inherent variability. Unlike traditional deterministic neural networks, PNNs are engineered to predict probability distributions, thereby enabling an enriched understanding of both predicted means and prediction intervals. The detailed analysis within this paper explores the architecture of PNNs, their optimization strategies, and the practical implications of their application in modeling aleatoric uncertainty.

The paper outlines several significant contributions. It proposes an innovative probabilistic distance metric tailored for optimizing PNN architecture, thus substituting traditional deterministic scoring metrics. This metric, based on the Kullback-Leibler (KL) divergence, effectively evaluates the similarity between actual and predicted output distributions, serving as a critical tool for model selection and optimization. By minimizing KL divergence, the researchers can refine PNN configurations in terms of depth and width, ensuring enhanced model expressiveness without compromising generalization capabilities.

In rigorous comparative studies, the paper assesses the efficacy of PNNs against prevalent methods like Gaussian Process Regression (GPR). PNNs exhibit a superior capability to model aleatoric uncertainty across synthetic and real-world data sets, evidenced in a real-world materials science experiment involving fiber-reinforced composites. In particular, the paper demonstrates that PNNs achieve remarkably accurate output mean estimates with R-squared scores near 0.97 and strong correlation coefficients for predicted intervals, emphasizing their effectiveness in capturing heteroscedastic data variability. This empirical success highlights the limitations of GPR, a traditionally employed technique that struggles with the inherently heteroscedastic nature of model uncertainty.

The implications of this research are substantial, underscoring the potential of PNNs to more accurately characterize uncertainty in scientific and engineering contexts. Practically, such models can improve decision-making processes, particularly in scenarios where variability significantly impacts system performance. Theoretically, this research exemplifies how machine learning approaches can be refined to align with scientific computing needs by incorporating distinctive probabilistic mechanisms.

Future research directions could explore PNNs' integration into active learning frameworks, allowing models to not just adapt passively to data uncertainty but also guide data collection efforts proactively. Further advancements may also focus on simultaneously modeling aleatoric and epistemic uncertainties within PNNs, enhancing their utility across a broader spectrum of scientific inquiries. Overall, the findings presented in this paper contribute a well-founded evaluation of PNNs for modeling aleatoric uncertainty, bridging a significant gap in current scientific machine learning methodologies.