## 3 Construction and Training of the Multivariate Attention Neural Network  
### Sample Construction  
As shown in Figure 1, the goal of MANN is to predict \(\mathbf{e}\). The variables that are related to \(\mathbf{e}\) are provided as follows:  
* The initial state of the nominal orbit: \(\mathbf{x}_{0}\in\mathbb{R}^{6}\). * The initial deviation: \(\delta\mathbf{x}_{0}\in\mathbb{R}^{6}\). * The orbital state of the nominal orbit at the epoch \(t\): \(\mathbf{x}\in\mathbb{R}^{6}\). * The orbit deviation propagated using the two-body dynamics: \(\delta\mathbf{x}^{\text{2 BP}}\in\mathbb{R}^{6}\). Apart from the four variables, there are some other variables that are available and can be potentially helpful in predicting \(\mathbf{e}\):  
* The deviation between the nominal orbits propagated using the high-fidelity and two-body dynamics: \(\delta\mathbf{x}_{\text{ref}}^{\text{2 BP}}=\mathbf{x}-\mathbf{x}^{\text{2 BP}}\in\mathbb{ R}^{6}\). This variable is expected to carry the information on the errors between the high-fidelity and two-body dynamics. * The state vectors of the Earth, Mars, and Jupiter in the heliocentric inertial coordinate at the initial epoch \(t_{0}\): \(\mathbf{x}_{E,0}\in\mathbb{R}^{6}\), \(\mathbf{x}_{M,0}\in\mathbb{R}^{6}\), and \(\mathbf{x}_{J,0}\in\mathbb{R}^{6}\). These variables contain information about the gravitational perturbation of the third body. Here, both the position of the perturbed object and the velocity of the perturbed bodies are considered, as the positions of the perturbed bodies in \([t_{0},t]\) have an effect on the orbital propagation and adding the velocities of the perturbed bodies to the sample helps to provide information about the position of the perturbed bodies at future times. * The propagated interval: \(\Delta t=t-t_{0}\in\mathbb{R}^{1}\). This is an important variable, as a longer propagated interval usually indicates more significant orbital deviations. * The variable related to the SRP: \(c\in\mathbb{R}^{1}\). Therefore, the input structure of the sample is designed as follows:  
\[\mathbf{s}=[\mathbf{x}_{0};\mathbf{x};\delta\mathbf{x}_{0};\delta\mathbf{x}^{\text{2 BP}};\delta\bm {x}_{\text{ref}}^{\text{2 BP}};\mathbf{x}_{c,0};\mathbf{x}_{M,0};\mathbf{x}_{J,0};\Delta t; c]\in\mathbb{R}^{50} \tag{21}\]  
and the structure of the sample is designed as  
\[S=\{\mathbf{s}\mapsto\mathbf{e}:\mathbb{R}^{50}\mapsto\mathbb{R}^{6}\}. \tag{22}\]  
Based on the above discussion of the input and output of the sample, a method is developed to quickly generate a large number of samples. Step 1 : Generate ten Earth-Jupiter transfer orbits between 1 January 2024 and 31 December 2033. Step 1.1: Divide the above periods (1 January 2024-31 December 2033.) with an interval of 1 year. For each time interval, optimize the launch window. The corresponding steps are as presented in Step 1.2-Step 1.3;  
Step 1.2: Two-impulse transfer is employed. The launch time JD0 and transfer time ToF (Time-of-flight, unit: day) are taken as the optimization variables. At the launch epoch, the C3 is constrained to be no more than 100 km\({}^{2}\)/s\({}^{2}\), and at the arrival epoch the spacecraft is required to enter an orbit with an eccentricity of 0.98 and a height of 10,000 km at the perigee of the Jupiter. Let \(r_{p}\) and \(r_{l}\) be the magnitudes of the position vectors of spacecraft and Jupiter at the arrival epoch heliocentric inertial coordinate, respectively; then, the velocity increment \(\Delta v\) that is required for the spacecraft to enter the Jupiter orbit can be expressed as shown below. \[ \begin{split}\Delta v&=v_{p}^{in}-v_{p}\\ v_{p}^{in}&=\sqrt{\left\|v_{2}-v_{l}\right\|^{2}+2 \mu_{I}/r_{p}}\\ r_{p}&=r_{I}+10^{4}\text{ km}\\ v_{p}&=\sqrt{\mu_{I}\Big{(}\frac{2}{r_{p}}-\frac{1 }{a_{Target}}\Big{)}}\\ a_{Target}&=\frac{r_{p}}{1-e_{Target}}\end{split}  \tag{23}\]  
Taking the magnitude of the velocity increment as the objective, the optimization problem can be provided as follows:  
\[