A Novel Neural Network with the Ability to Express the Extreme Points Distribution Features of Higher Derivatives of Physical Processes

Higher derivatives are important to interpret the physical process. However, higher derivatives calculated from measured data often deviate from the real ones because of measurement errors. A novel method for data fitting without higher derivatives violating the real physical process is developed in this paper. Firstly, the research on errors’ influence on higher derivatives and the typical functions’ extreme points distribution were conducted, which demonstrates the necessity and feasibility of adopting extreme points distribution features in neural networks. Then, we proposed a new neural network considering the extreme points distribution features, namely, the extreme-points-distribution-based neural network (EDNN), which contains a sample error calculator (SEC) and extreme points distribution error calculator (EDEC). With recursive automatic differentiation, a model calculating the higher derivatives of the EDNN was established. Additionally, a loss function, embedded with the extreme points distribution features, was introduced. Finally, the EDNN was applied to two specific cases to reduce the noise in a second-order damped free oscillation signal and an internal combustion engine cylinder pressure trace signal. It was found that the EDNN could obtain higher derivatives that are more compatible with physical trends without detailed differentiation equations. The standard deviation of derivatives’ error of the EDNN is less than 62.5 percent of that of traditional neural networks. The EDNN provides a novel method for the analysis of physical processes with higher derivatives compatible with real physical trends.