## 2 Neural Network-Based Orbit Uncertainty Propagation Orbit Determination Framework
### Orbit Determination Method  
The framework of the MANN-UT is illustrated in Figure 2. Given the mean of the initial state \(\mathbf{x}_{0}\in\mathbb{R}^{6}\) and its associated covariance \(\mathbf{P}_{0}\in\mathbb{R}^{6\times 6}\), thirteen sigma points \(\mathbf{\chi}_{i}(t_{0})\in\mathbb{R}^{6}\) (\(i\in\{0,1,\cdots,12\}\)) can be obtained under the UT framework as follows:  
\[ \begin{array}{ll}\mathbf{\chi}_{i}(t_{0})=\mathbf{x}_{0}&i=0\\ \mathbf{\chi}_{i}(t_{0})=\mathbf{x}_{0}+\left(\sqrt{(n+\lambda)\mathbf{P}_{0}}\right)_{i}& i=1,\cdots,n\\ \mathbf{\chi}_{i}(t_{0})=\mathbf{x}_{0}-\left(\sqrt{(n+\lambda)\mathbf{P}_{0}}\right)_{i}& i=n+1,\cdots,2n\end{array}  \tag{5}\]  
where \(n=6\) is the dimension of the orbit state, \(\lambda=\mathfrak{a}^{2}(n+\kappa)-n\) is an adjusted factor, \(\left(\sqrt{(n+\lambda)\mathbf{P}_{0}}\right)_{i}\) is the i-th column of the matrix \(\sqrt{(n+\lambda)\mathbf{P}_{0}}\), \(\mathbf{P}_{0}=\sqrt{\mathbf{P}_{0}}\left(\sqrt{\mathbf{P}_{0}}\right)^{T}\), and \(\sqrt{\mathbf{P}_{0}}\) is computed using the Cholesky decomposition. In this work, \(\kappa=3-n\) and \(\alpha=1\). In the conventional UT framework, the thirteen sigma points in Equation (5) are propagated from the initial epoch \(t_{0}\) to a given future epoch \(t\) using high-fidelity dynamics; thus, we have \(\mathbf{\chi}_{i}(t)=\phi[t;\mathbf{\chi}_{i}(t_{0}),t_{0}]\) (\(i\in\{0,1,\cdots,12\}\)). After weighing these propagated sigma points, the mean and covariance can be computed as  
\[ \begin{array}{l}\mathbf{\bar{x}}(t)=\sum\limits_{i=0}^{2n}w_{i}^{(m)}\mathbf{\chi}_ {i}(t)\\ \mathbf{P}(t)=\sum\limits_{i=0}^{2n}w_{i}^{(c)}[\mathbf{\chi}_{i}(t)-\mathbf{\bar{x}}(t)] [\mathbf{\chi}_{i}(t)-\mathbf{\bar{x}}(t)]^{T}\end{array}  \tag{6}\]  
where \(\mathbf{\bar{x}}(t)\in\mathbb{R}^{6}\) and \(\mathbf{P}(t)\in\mathbb{R}^{6\times 6}\) represent the mean and covariance at epoch \(t\) and \(w_{i}^{(m)}\) and \(w_{i}^{(c)}\) are the weights of the mean \(\mathbf{\bar{x}}(t)\) and covariance \(\mathbf{P}(t)\), with their expressions provided by  
\[\left\{ \begin{array}{ll}w_{i}^{(m)}=\frac{\lambda_{UT}}{\lambda_{UT}+n},\,w _{i}^{(c)}=w_{i}^{(m)}+(1-\mathfrak{a}^{2}+\beta)&i=0\\ w_{i}^{(m)}=w_{i}^{(c)}=\frac{1}{2(\lambda_{UT}+n)}&\text{elsewise}\end{array} \right. \tag{7}\]  
where \(\beta=2\) is used in this work. Using Equation (4), the propagated sigma points can be written as  
\[ \begin{array}{l}\mathbf{\chi}_{i}(t)=\mathbf{x}(t)+\delta\mathbf{\chi}_{i}^{\text{2 BP}} +\mathbf{\hat{e}}_{\text{MANN},j}\\ =\phi(t;\mathbf{x}_{0},t_{0})+\delta\mathbf{\chi}_{i}^{\text{2 BP}}+\mathbf{\hat{e}}_{\text {MANN},j}\end{array}  \tag{8}\]  
Figure 2: An illustration of the framework of MANN-UT. where \(\delta\mathbf{\chi}_{i}^{\rm 2 BP}\) denotes the two-body orbital deviation of the i-th sigma point. Substituting Equation (8) into Equation (6), the mean can be approximated as  
\[ \begin{split}&\mathbf{\tilde{x}}(t)\approx\sum_{i=0}^{2n}w_{i}^{(m)} \{\phi[t;\mathbf{\chi}_{i}(t_{0}),t_{0}]+\delta\mathbf{\chi}_{i}^{\rm 2 BP}+\mathbf{\hat{e}}_{ \rm MANN,i}\}\\ &=\phi(t;\mathbf{\chi}_{i}(t_{0}),t_{0})+\sum_{i=0}^{2n}w_{i}^{(m)}( \delta\mathbf{\chi}_{i}^{\rm 2 BP}+\mathbf{\hat{e}}_{\rm MANN,i}).\end{split}  \tag{9}\]  
In addition, the covariance \(\mathbf{P}(t)\) can be rewritten as  
\[ \begin{split}&\mathbf{P}(t)\approx\sum_{i=0}^{2n}w_{i}^{(c)}\Big{[} \delta\mathbf{\chi}_{i}^{\rm 2 BP}+\mathbf{\hat{e}}_{\rm MANN,i}-\sum_{j=0}^{2n}w_{j}^{(m)}( \delta\mathbf{\chi}_{j}^{\rm 2 BP}+\mathbf{\hat{e}}_{\rm MANN,j})\Big{]}\\ &\times\Big{[}\delta\mathbf{\chi}_{i}^{\rm 2 BP}+\mathbf{\hat{e}}_{\rm MANN,i}-\sum_{j=0}^{2n}w_{j}^{(m)}(\delta\mathbf{\chi}_{j}^{\rm 2 BP}+\mathbf{\hat{e}}_{\rm MANN,j})\Big{]}^{T}.\end{split}  \tag{10}\]  
Motivated by [46], by applying such a MANN-UT method into a UKF framework, a MANN-assisted UKF (denoted as MANN-UKF) is presented herein. First, the OD system of the Earth-Jupiter transfer orbit can be written as  
\[\left\{