## 1 Introduction  
An inertial navigation system (INS) offers benefits such as concealed, anti-electromagnetic interference, and reliable placement in autonomous scenarios. Nevertheless, its critical limitation lies in the degradation of positioning accuracy with time [1, 2]. Many studies have detailed the concept of multi-sensor information fusion, using a strapdown inertial navigation system (SINS), polarized light navigation system (PLNS), and global navigation satellite system (GNSS) to mitigate INS cumulative errors [3, 4, 5]. Due to the advantages of a geomagnetic navigation system (MNS), which include the lack of accumulation errors in time and distance, studying an INS/MNS navigation system that combines the advantages of both navigation systems is crucial from a strategic standpoint [6]. Research on INS/MNS information fusion has been conducted at Silsoe Institute [7], UK, and Florida Atlantic University [8], USA, with the aim of enhancing navigation accuracy. Due to the measurement errors of gyroscopes, accelerometers, and/or magnetometers inside the MIMU in various studies, sensor fusion algorithms (SFAs) have been proposed to accurately and robustly estimate the 3D orientation [9]. These studies offered important insights for INS/MNS navigation. The current studies focus on exploring the use of various Kalman algorithms to fuse INS/MNS information more efficiently, aiming to optimize the fusion algorithm and minimize errors. As a widely used optimization estimator, the Kalman filter (KF) is a customary filtering scheme to improve the accuracy of combinatorial navigation. However, it may not provide accurate estimates for nonlinear and non-Gaussian systems, and could lead to biased estimates in the presence of model errors [10]. Approximately 60 years ago, the extended KF (EKF) algorithm was proposed for nonlinear systems [11; 12]. The algorithm's fundamental approach involves linearizing nonlinear transformations; but this process results in significant truncation error and requires computationally intensive calculations of the Jacobi matrix. Therefore, a more stable and accurate nonlinear cubature filter (cubature KF, CKF) was proposed to address various limitations. It uses the multidimensional integrating sphere-radial volume rule to calculate multivariate moment integrals encountered in nonlinear Bayesian filters. The CKF can be applied to solve nonlinear filtering problems throughout a wide spectrum of dimensions, from low to high [13]. Additionally, since the covariance matrix of the measurement noise \(R\) and process noise \(Q\) has an important effect on the performance of the filter in estimating the dynamic state, an adaptive filtering method based on innovation and residuals adaptively estimates \(Q\) and \(R\), making its initial error more robust and improving the dynamic state estimation accuracy [10; 14]. In this paper, we propose an adaptive method for adjusting the CKF to have higher robustness and precision; however, in view of recent studies, it has become more important to solve navigation interruptions, such as out-of-lock problems when integrated navigation system-equipped ground vehicles or UAVs pass through environments with weak signals, which results in a scenario in which the INS single mode of operation deteriorates sharply in terms of navigation accuracy [15; 16]. There is an increasing interest in utilizing artificial neural network algorithms such as a radial basis function neural network (RBFNN), back-propagation neural network (BPNN), and Temporal Convolutional Network (TCN), to develop models that can effectively map vehicle dynamics (attitude, velocity, or position) based on inputs [17; 18]. Nevertheless, some of the aforementioned methods rely on static neural networks, which are challenging to identify with time-series data. In recent years, deep learning has achieved satisfactory performance in time-series prediction [19]. In [20], recurrent neural networks (RNNs) were integrated with UKF to real-time estimate and compensate for the random drift of gyroscopes.