Knowledge Informed Machine Learning using a Weibull-based Loss Function

Abstract

Machine learning can be enhanced through the integration of external knowledge. This method, called knowledge informed machine learning, is also applicable within the field of Prognostics and Health Management (PHM). In this paper, the various methods of knowledge informed machine learning, from a PHM context, are reviewed with the goal of helping the reader understand the domain. In addition, a knowledge informed machine learning technique is demonstrated, using the common IMS and PRONOSTIA bearing data sets, for remaining useful life (RUL) prediction. Specifically, knowledge is garnered from the field of reliability engineering which is represented through the Weibull distribution. The knowledge is then integrated into a neural network through a novel Weibull-based loss function. A thorough statistical analysis of the Weibull-based loss function is conducted, demonstrating the effectiveness of the method on the PRONOSTIA data set. However, the Weibull-based loss function is less effective on the IMS data set. The results, shortcomings, and benefits of the approach are discussed in length. Finally, all the code is publicly available for the benefit of other researchers.

Multiform Informed Machine Learning Based on Piecewise and Weibull for Engine Remaining Useful Life Prediction

“…Consequently, using the Weibull function to express the external knowledge regarding aero-engine failure in this research is both rational and scientific. According to the practice of von Hahn et al [ 29 ], Weibull’s cumulative distribution function is used in the specific implementation, in the form of Equation ( 1 ): where t is the time, is the shape parameter, and is the characteristic lifetime. The specific values of and need to be determined by combining the equipment failure time data and the existing knowledge of the device in reliability engineering.…”

“…As seen in the table, the ranges of parameters and are and . Based on the knowledge about aero-engine failures [ 39 , 41 , 42 ], particularly the experience of von Hahn et al [ 29 ], the model parameter values and are taken as 2.0 and 90, respectively, in this paper.…”

“…Combining the Weibull CDF with a traditional label-based loss function such as the mean square error loss function constitutes a hybrid loss function incorporating Weibull [ 29 ], which takes the form of Equation ( 5 ): where n is the number of units in the dataset, is the actual remaining life, is the predicted remaining life, is the lifetime of the unit, is the failure probability of the unit at the time , calculated from the Equation ( 1 ), and is the estimated failure probability.…”

“…An algebraic equation representing prior information is typically used by the learning algorithm to attach a loss term to the model’s loss function, placing restrictions on the model. Inspired by Rueden’s paper, von Hahn et al [ 29 ] represented the knowledge of bearings in reliability engineering by a Weibull cumulative distribution function and integrated the knowledge into ML models by training models based on Weibull’s loss function for the first time. Hahn used the simplest feature processing and model architecture to experiment on the IMS [ 30 ] and PRONOSTIA [ 31 ] datasets, and the Weibull-combined loss function performed well on most of the datasets, especially the IMS dataset, which significantly increased the accuracy of the RUL prediction and confirmed the viability and effectiveness of the Weibull-based loss function.…”

Label-free timing analysis of SiPM-based modularized detectors with physics-constrained deep learning

“…The idea of training NNs without explicit labels was originally invented to locate or detect particular objects in images in the domain of computer vision [17]. Researches in multiple disciplines combined the idea (i) to formulate loss functions to include a physics-constrained term [18][19][20][21][22][23], (ii) to eliminate the canonical loss term and totally rely on physical relations [24,25], or (iii) to solve partial differential equations [26][27][28].…”