### Extreme Points Distribution of Typical Process Functions  
When fitting a measured signal using neural networks, noise often leads to the higher derivatives of the fitted signal to deviate from the true physical trend [37]. Generally, the relation between the physical quantities and time of a physical process can be modeled by a polynomial, Gaussian function, cosine function, or exponential function. The extreme points of those functions' higher derivatives on a specific interval are distributed in certain patterns. When extreme points distribution is involved in data fitting, the approximation between the higher derivatives of the measured signal and the higher derivatives of the real physical process could be improved. It is necessary to analyze the extreme points' distribution pattern of those typical process functions and their higher derivatives to involve the extreme points distribution in data fitting with neural networks. An \(n\)th order polynomial modeling a continuous process is shown in expression (7). \[p(x)=\sum_{i=0}^{n}a_{i}\cdot x^{i} \tag{7}\]  
Additionally, the \(k\)th order derivative of an \(n\)th order polynomial can be obtained in (8). \[p^{(k)}(x)=kla_{k}+\frac{(k+1)!}{1!}a_{(k+1)}x+\frac{(k+2)!}{2!}a_{(k+2)}x^{2}+ \cdots+\frac{n!}{(n-k)!}a_{n}x^{n-k} \tag{8}\]  
It can be seen that the \(k\)th order derivative of \(n\)th order polynomial is an \((n-k)\)th-order polynomial. Therefore, the number of the \(k\)th-order derivative's extreme points decreases with the increase of order \(k\). When fitting a measured signal modeled by an \(n\)th-order polynomial, the extreme point number could be applied to test the consistency between the fitted \(k\)th order derivative and that of the real physical process. An exponential function describing the growth or decline process is shown in (9). \[f(x)=a^{x} \tag{9}\]  
Additionally, the \(k\)th-order derivative of the exponential function is shown in (10). \[f^{(k)}(x)=[ln(a)]^{k}a^{x} \tag{10}\]  
According to (10), the number of extreme points of the \(0\)-\(k\)th-order derivatives of the exponential function is always zero. When fitting a measured signal modeled using the exponential function, the number of extreme points should always remain zero. Any extreme point appearing in the fitting results indicates a deviation from the physical trend. A Gaussian function describing a random process is shown in expression (11). \[g(x)=e^{ax^{2}} \tag{11}\]The \(k\)th-order derivative of the Gaussian function is shown in expression (12). \[\frac{d^{n}g(x)}{dx^{n}}=e^{nx^{2}}{\sum_{i=0}^{n}C_{it}^{n}(a)x^{m}} \tag{12}\]  
where \(C_{it}^{n}(a)=\frac{(2a)^{t}}{1+i\varepsilon_{n}^{-}}F_{i}(n)[(n-i)-\varepsilon _{n}^{+}]! !\), \(m=i+\varepsilon_{n}^{-}\), \(\varepsilon_{n}^{\pm}=\frac{1}{2}\big{[}1\pm(-1)^{n}\big{]}\), \(t=\frac{(n+i)}{2}+\varepsilon_{n}^{-}\) and \(F_{i}(n)=\frac{n!}{i!(n-i)! }\) is the binomial coefficient. The highest degree of \(x^{m}\) in expression (12) increases as the derivative order increases [38]. Then, the number of the derivative's extreme points increases as the derivative order increases. When fitting a measured signal modeled using the Gaussian function, the extreme point number could be applied to test the consistency between the fitted \(k\)th-order derivative and that of the real physical process. The cosine function describing a simple periodic process is shown in (13). \[h(x)=cos(x) \tag{13}\]  
The \(k\)th-order derivative of the cosine function is shown in expression (14). \[h^{(k)}(x)=cos(\frac{k\pi}{2}+x) \tag{14}\]  
Cosine function shifted left by \(k\pi/2\) yields its \(k\)th-order derivative. If the number of extreme points of the cosine function is \(N_{hp}\) on any arbitrary interval \([-a,+a]\), the number of extreme points of the cosine function's \(k\)th-order derivative is \(N_{hp}-1\), \(N_{hp}\) or \(N_{hp}+1\).