## 3 Construction and Training of the Multivariate Attention Neural Network
### Training of the Neural Network
\tag{41}\]  
The single simulation results are shown in Figures 6 and 7. In Figures 6 and 7, the blue curve is the OD error of the single simulation and the red curve is the standard deviation forecast by the MANN-UKF and UKF. As can be seen from Figures 6 and 7, the OD errors of both methods are within the \(\pm 3\sigma\) boundaries. In Figures 6 and 7, the symbol \(\sigma\) denotes the STD of estimated errors, which is calculated analytically using Equation (20). The values of the estimated STDs converge in the first 100 days. The position and velocity STDs at the initial epoch are 10,000 km and 0.01 km/s, respectively. The STDs of the position estimated errors converge to approximately 300 km at the ending epoch. Figure 6: Orbit determination errors of the single simulation of MANN-UKF. Figure 7: Orbit determination errors of the single simulation of UKF. A Monte Carlo (MC) simulation was further carried out, with the results shown below. For each method, 100 MC simulations were performed; for each MC simulation, the initial guess was obtained by the following equation:  
\[\hat{\mathbf{x}}=\mathcal{N}(x;x_{0},\mathbf{P}_{0}). \tag{43}\]  
The MC simulation results are shown in Figures 9 and 10. In Figures 9 and 10, the 3\(\sigma\) boundaries were statistically obtained based on the results of 100 MC simulations. The curve of one standard deviation for both methods is provided in Figure 11. As shown in Figure 11, the convergence tendency of estimation accuracy of MANN-UKF and UKF is very close. For example, during the epochs from 200 days to 600 days, the MANN-UKF is slightly more accurate than the UKF in terms of the estimated errors along the x-axis. Slight differences exist as initial estimations are randomly generated in the MC simulation. However, around the ending epoch the performance of MANN-UKF is slightly worse than that of UKF. This is because around the ending epoch the estimation error is smaller than that in the early stage, and the prediction accuracy of MANN decreases, resulting in the decline of the prediction accuracy. As a result, the performance of MANN-UKF decreases. Figure 8: MD results of the single simulation. Figure 9: MC results of MANN-UKF. Figures 12 and 13 are Box and Whisker Plots. The line splitting the box in two represents the median value. The upper edge and lower edge of the box represent the upper quartile and lower quartile respectively. The values at which the vertical lines stop at are the values of the upper and lower values of the data. The single points on the diagram show the outliers. As shown in Figure 12, the MDs of UKF are around 1, while the MC of MANN-UKF are between 1 and 5. This is again due to the decline in the prediction accuracy of the  
Figure 11: MC results comparing MANN-UKF and UKF. Figure 10: MC results of UKF. MANN when the deviation is small, resulting in inaccurate prediction of the covariance matrix. The primary purpose of the proposed MANN-UKF method is to reduce the computational cost in OD. Figure 13 and Table 4 compare the CPU times of the two methods. The CPU time of the UKF method is about 6.6 s, while the proposed MANN-UKF method only needs about 0.54 s. The proposed method saves on CPU time by more than 90%. Apart from the above case, a second case is presented to investigate the performance of the proposed method. The parameters of the second case are shown in Table 5. In this case, line-of-sight (LOS) measurements are employed instead of the measurement model in Equation (40). We assume that the spacecraft can obtain the LOS measurements (from the spacecraft to the Jupiter) using optical sensors. The LOS measurement model is represented as follows [42; 49; 50]:  
\[\mathbf{z}_{k+1}=\mathbf{h}(\mathbf{x}_{k+1})+\mathbf{\varepsilon}_{k+1}=\frac{\mathbf{r}_{l}-\bm {r}_{k+1}}{\left\|\mathbf{r}_{l}-\mathbf{r}_{k+1}\right\|}+\mathbf{\varepsilon}_{k+1}. \tag{44}\]