## 1 Introduction
This adjustment is done by solving a system of linear equations, where the solution corresponds to the state vector that best fits the observations in a least-squares sense. LS-based OD is widely used in space-related applications such as satellite navigation and OD of space debris and asteroids. However, LS has limitations for onboard applications as it is a batch method that requires high computational cost. The Kalman filter and its variants belong to the category of sequential methods, requiring a lower computational cost than LS. Commonly used Kalman filters include (to name a few) the extended Kalman filter (EKF), unscented Kalman filter (UKF) [24; 25], cubature Kalman filter (CKF) [27], and Gaussian mixture filter (GMF) [44]. In order to effectively filter data on onboard applications, two fundamental prerequisites must be fulfilled, namely, achieving high accuracy while maintaining low complexity. Nevertheless, these requirements can sometimes be in conflict with one another [45]. For instance, the UKF is recognized for its superior accuracy compared to the EKF; however, its computational cost is significantly higher due to the integration of over ten sigma points [30; 46]. Several research studies have been carried out recently to improve the efficiency of the OUP and OD methods for onboard applications. [PERSON] and [PERSON] developed a surrogate-based particle filter in which a surrogate model replaces the orbit integrations in the OUP process of a conventional particle filter [47]. [PERSON] et al. developed an adaptive order-switching Kalman filter (AOSKF) for cislunar OD [45]. In AOSKF, a deep neural network (DNN) is employed to detect highly nonlinear points and the OUP's orders are adaptively adjusted according to the nonlinear detection results. More recently, [PERSON] et al. proposed a DNN-based orbital uncertainty propagation and estimation framework for low-Earth orbit (LEO) spacecraft in which the computational cost is significantly reduced [46]. In this method, the orbit propagation process in the conventional UKF is replaced by a DNN-based prediction process. However, such an OUP and OD method can be further improved, as the prediction capacity of the DNN is limited. Motivated by [PERSON] et al. 's work [46], this paper aims to improve the computational efficiency of the OD using a neural network. A deep space transfer case, the Earth-Jupiter transfer, is taken as an example to show the performance of such a method. Compared with [PERSON] et al. 's work, the innovations of this paper can be summarized as follows:* A multivariate attention-based neural network (MANN), rather than a simple DNN, is designed and utilized to replace the orbit integration process in the UT and UKF. It will be shown that the designed MANN has a much more advanced performance than the DNN in the investigated problem. * An Earth-Jupiter transfer case, rather than an LEO case, is considered in this work. The input structure of the sample is comprehensively discussed and carefully designed. * A method for fast generating a large number of samples for the Earth-Jupiter transfer case is presented. The remainder of this paper is organized as follows. Section 2 describes the neural network-based OD framework. Section 3 discusses the construction of the samples and introduces the structure and training process of the MANN. Numerical simulations are provided in Section 4 and conclusions in Section 5.