## 2 Neural Network-Based Orbit Uncertainty Propagation Orbit Determination Framework  
### Orbit Uncertainty Propagation  
In this work, the dynamics of a spacecraft during the transfer from Earth to Jupiter considers the point-mass gravitational accelerations and the Solar Radiation Pressure (SRP) perturbations; thus, the high-fidelity dynamics model can be written as  
\[\dot{\mathbf{x}}= \begin{bmatrix}\dot{\mathbf{r}}\\ \dot{\mathbf{v}}\end{bmatrix} = \begin{bmatrix}-\frac{\mu_{S}}{\|\mathbf{r}\|^{3}}\mathbf{r }+\sum\limits_{k\in\{E,M,J\}}\mu_{k}\left(\frac{\tau_{k}-\tau}{\|\mathbf{r}_{k}-\tau \|^{3}}-\frac{\tau_{k}}{\|\mathbf{r}_{k}\|^{3}}\right)+C_{R}\frac{A}{m}P_{\odot} \text{AU}^{2}\frac{\mathbf{r}}{\|\mathbf{r}\|^{3}}\end{bmatrix} , \tag{1}\]  
where \(\mathbf{x}\in[\mathbf{r};\mathbf{v}]\in\mathbb{R}^{6}\) represents the state vector (with \(\mathbf{r}\in\mathbb{R}^{3}\) and \(\mathbf{v}\in\mathbb{R}^{3}\) being the position and velocity vectors, respectively) of the spacecraft in the heliocentric inertial coordinate, \(\mu_{S}\), \(\mu_{E}\), and \(\mu_{J}\) denote the gravitational constants of the Sun, Mars, Earth, and Jupiter, respectively, \(\mathbf{r}_{E}\in\mathbb{R}^{3}\), \(\mathbf{r}_{M}\in\mathbb{R}^{3}\), and \(\mathbf{r}_{J}\in\mathbb{R}^{3}\) are the position vectors of the Earth, Mars, and Jupiter in the heliocentric inertial coordinate, \(C_{R}\) is the reflection coefficient, \(A/m\) is the area-to-mass ratio, and \(P_{\odot}=4.56\times 10^{-6}\) N/m\({}^{2}\) is the reference value of the SRP at a distance of 1 AU (1 AU = 149,597,870 km, and AU\({}^{2}\) = 149,597,870 km \(\times\) 149,597,870 km) from the Sun [8; 11]. Note that in this work the ephemeris data of the Earth, Mars, and Jupiter come from the Jet Propulsion Laboratory's Development Ephemerides (DE432). In this work, the J2 term of the Earth and the propulsion term are not considered in order to simplify the problem (as in [11]). In addition, we define  
\[c=C_{R}\frac{A}{m} \tag{2}\]  
and in the remainder of this paper, given the initial epoch \(t_{0}\) and the corresponding initial orbital state \(\mathbf{x}_{0}\in\mathbb{R}^{6}\), the solution flow of the ordinary differential equation in Equation (1) can be represented as \(\mathbf{x}=\mathbf{\phi}(t;\mathbf{x}_{0},t_{0})\). Orbit deviation propagation is the premise of orbit determination [46]. This is because orbit deviation propagation, also called the orbit uncertainty propagation technique, can help predict the uncertainties, which is necessary to calculate the Kalman gain in a Kalman filter framework. In addition, it can help provide gradient information for the LSM. As shown in Figure 1, the black solid line and the black dashed line represent the nominal orbit and the neighboring orbit, respectively, under the high-fidelity dynamics model. Let \(\mathbf{\tilde{x}}_{0}=\mathbf{x}_{0}+\delta\mathbf{x}_{0}\in\mathbb{R}^{6}\) be the initial orbital state of the neighboring orbit, with \(\delta\mathbf{x}_{0}\in\mathbb{R}^{6}\) being the initial deviation between the nominal and neighboring orbits. By propagating the orbits under high-fidelity dynamics, the orbital states of the nominal and neighboring orbits at any given epoch \(t\) can be obtained as \(\mathbf{x}=\mathbf{\phi}(t;\mathbf{x}_{0},t_{0})\) and \(\mathbf{\tilde{x}}=\mathbf{\phi}(t;\mathbf{\tilde{x}}_{0},t_{0})\). The orbit deviation between the nominal and neighboring orbits at a given epoch \(t\) is defined as  
\[\delta\mathbf{x}=\mathbf{\tilde{x}}-\mathbf{x}\in\mathbb{R}^{6} \tag{3}\]The orbit deviation propagation is used to calculate or predict the deviation \(\delta\mathbf{x}\) given the initial state of the nominal orbit \(\mathbf{x}_{0}\) and the initial deviation \(\delta\mathbf{x}_{0}\). In general, there are two ways to calculate the deviation \(\delta\mathbf{x}\). The first is to propagate the neighboring orbit directly and calculate the deviation \(\delta\mathbf{x}\) using Equation (3).