## 5 Conclusions  
The effect of the error on the higher derivatives of the measured signal was analyzed. The feasibility of using the extreme points distribution as constraints in data fitting was studied. Then, the extreme-points-distribution-based neural network (_EDNN_) was proposed. Finally, the superiority of the _EDNN_ on signal denoising was verified. The details are as follows:  
1. The error's deviation effect on the higher derivatives of the measured signal was analyzed and a possible way of applying the extreme points distribution as constraints on data fitting was studied.  
A mathematical model was established for the analysis of the error's effect on higher derivatives. It was found that even though the Taylor series of the noise only has the \(k\)th-order term and the coefficients of other terms are zero, the \(k\)th-order derivative of the measured signal could deviate from the real physical process greatly as long as the  
Figure 8: The first- through fourth-order derivatives. (**a**) First-order derivatives; (**b**) second-order derivatives; (**c**) third-order derivatives; (**d**) fourth-order derivatives.  
\(k\)th-order derivative of the noise is large enough. The necessity of finding a way to fit data with higher derivatives compatible to real process trends was clarified.  
The higher derivative's extreme points distribution of typical process functions were analyzed, and the pattern of the extreme points distribution of higher derivatives was investigated. The pattern provides a theoretical basis for applying the extreme points distribution as a constraint in denoising with data fitting techniques.  
The extreme points distribution pattern was adopted as a constraint and the _EDNN_ was established. The _EDNN_ consists of an input layer, hidden layers, an output layer, an automatic differentiation layer and an extreme points distribution feature layer. A recursive formulation was established for calculating the derivatives. Additionally, a novel loss function, embedded with the extreme feature error, was proposed.  
The effectiveness of the _EDNN_ on signal denoising was verified. The proposed _EDNN_ was applied to reduce the noise in the second-order damped free oscillation signal and the internal combustion engine cylinder pressure signal. Compared with shallow neural networks and smoothing splines, the _EDNN_ could obtain higher derivatives that are consistent with the real physical process in the absence of a detailed mathematical model. Therefore, data fitting for higher derivatives conforming to real physical process trends could be realized with _EDNN_, which provides a novel approach for analyzing physical processes with higher derivatives. Additionally, it could be used in understanding the real process with higher derivatives.  
In summary, the advantage of the _EDNN_ is that it could fit the measured signal with higher derivatives compatible to real physical process trends, which provides a novel tool to mine information of the real physical process through higher derivatives. However, it needs more computational resources and needs the knowledge of extreme distribution of the real process. The aim of future research is to study the computational efficiency of the _EDNN_ and apply it in other engineering fields.  
Conceptualization, X.W., F.M. and Y.G.; methodology, X.W., F.M., J.L. and Y.G.; software, X.W. and F.M.; validation, X.W., F.M. and J.L.; formal analysis, X.W.; investigation, X.W. and F.M.; resources, X.W.; data curation, X.W., F.M. and J.L.; writing--original draft preparation, X.W., F.M. and J.L.; writing--review and editing, X.W., F.M., J.L., Y.G. and C.Z.; visualization, F.M., J.L. and X.W.; supervision, X.W.; project administration, X.W.; funding acquisition, X.W. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.