## 3 Methodology  
### The Solution of a Nonlinear Problem for the Estimation of Helical Line Parameters  
The coordinate system of the SPD tracking detectors is arranged so that the direction of the beam of accelerated particles is the direction of the magnetic field of the SPD. The coordinates of event vertices, i.e., particle collision points, are located along the \(Oz\) axis near the origin of the coordinates. Assuming homogeneity of the magnetic field of the detector, the particle trajectories must be close to a helical line, with its axis directed along the \(Oz\) axis according to the magnetic field direction. The projection of the helical line onto the \(xOy\) plane forms a circle of radius R in the \(xOy\) plane, centered at \((x_{0},y_{0})\), while the helical line itself extends along the Oz axis with an inclination to the \(Oxy\) plane defined by an angle with tangent \(\lambda\). Because the circle touches the origin, we obtain \(y_{0}^{2}=R^{2}-x_{0}^{2}\), which reduces the number of parameters of the helical line to three. Thus, we obtain the helical equations in the form of (1):  
\[ \begin{cases}(x-x_{0})^{2}+(y-y_{0})^{2}=R^{2}\\ y_{0}^{2}=R^{2}-x_{0}^{2}\\ z=\lambda\arctan((y-y_{0})/(x-x_{0}))\end{cases}  \tag{1}\]  
To estimate the parameters of the helical line, we used a set of model data of particle trajectory measurements in the magnetic field of the SPD detector, simulated according to the pixel detector scheme without considering background measurements. For each track with n measured points, the values of the helical line parameters R, \((x_{0},y_{0})\), and \(\lambda\) had to be estimated, which is a highly nonlinear problem. Therefore, we first applied a variant of  
Figure 2: Main scheme of the TrackNET neuromodel. the maximum likelihood method with partial adoption of the approach from [12] by one of the authors to estimate the parameters of the circle \(R\), \((x_{0},y_{0})\) in the \(xOy\) plane, with the normalization of the minimizing functional in the form of (2):  
\[F(R,x_{0},y_{0})=\sum_{i-1}^{n}{((x_{i}-x_{0})^{2}+(y_{i}-y_{0})^{2}-R^{2})^{2}} \tag{2}\]  
by the approximated gradient of the circle \(R^{2}\). By dividing (2) by \(R^{2}\) and equating derivatives of \(R\), \((x_{0},y_{0})\) to zero, we obtain (3):  
\[ \begin{split} Fx_{0}+Hb-x_{0}\gamma&=P,\\ Hx_{0}+Gb-y_{0}\gamma&=Q,\\ 2 Px_{0}+2 Qy_{0}+\gamma^{2}&=T,\end{split}  \tag{3}\]  
where \(\gamma=R^{2}-x_{0}^{2}-y_{0}^{2}\). To simplify the equation, it will take the form (4):  
\[\sum_{i=1}^{n}{x_{i}^{p}y_{i}^{q}}=[x^{p}y^{q}] \tag{4}\]  
Then, the coefficients in (3) are denoted as (5):  
\[ \begin{split} F&=\frac{1}{n}[3x^{2}+y^{2}],\\ G&=\frac{1}{n}[x^{2}+3y^{2}],\\ H&=\frac{2}{n}[xy],\\ P&=\frac{1}{n}[x(x^{2}+y^{2})],\\ Q&=\frac{1}{n}[y(x^{2}+y^{2})],\\ T&=\frac{1}{n}[(x^{2}+y^{2})^{2}].\end{split}  \tag{5}\]  
Excluding \((x_{0},y_{0})\) in the system (3) and making the variable substitution \(\gamma=\gamma_{0}x\), where \(\gamma_{0}=1/n([x^{2}]+[y^{2}])\), we obtain the fourth-degree Equation (6):  
\[x^{4}+A_{0}x^{3}+B_{0}x^{2}+C_{0}x+D_{0}=0 \tag{6}\]  
with the coefficients (7)  
\[ \begin{split} A_{0}&=A/\gamma_{0},\\ B_{0}&=B/\gamma_{0}^{2},\\ C_{0}&=C/\gamma_{0}^{3},\\ D_{0}&=D/\gamma_{0}^{4},\end{split}