Analytical propagation of errors and uncertainties in deep neural networks in the domain of physical sciences

In recent years, neural networks have gained popularity for making predictions in various applications. However, these predictions come with inherent uncertainties due to factors such as data acquisition, parameter choices, network architecture, and the training process. These uncertainties, both epistemic and aleatory, require careful quantification and propagation for a comprehensive understanding of the predicted results. To address these requirements, we propose an algorithm called Analytical Propagation of Uncertainties (APU), which is capable of quantifying and assessing the uncertainties associated with predictions made by Multilayer Perceptron neural networks in the domain of physical sciences. APU, with its modular architecture, can provide model, data, and prediction uncertainties for pre-trained Multilayer Perceptron networks. While generally offering lower computational cost compared to Bayesian methods, the efficiency and effectiveness of APU may vary depending on the specific application and model complexity. To the best of our knowledge, this is the first method able to propagate measurement uncertainty, originating from the calibration of measuring instruments, in the training process. To evaluate the effectiveness of data uncertainty estimation, we compared our approach against Monte Carlo Dropout in the Airfoil Self-Noise dataset. The analysis revealed that APU is more realistic than Monte Carlo Dropout and satisfactorily matched Monte Carlo simulations. Thus, the APU method becomes promising under real-world conditions, especially in the field of physical sciences, where low computational cost and immediate response are essential. This is significant in edge computing scenarios, real-time monitoring, addressing concerns such as latency, bandwidth, and data privacy. The study contributes to advancing the application of deep learning in physical sciences. The source code is available at https://github.com/Geduardomello/Dissertation .