Multi-Domain Intelligent State Estimation Network for Highly Maneuvering Target Tracking with Non-Gaussian Noise

Highlights

What are the main findings?

• We propose a Multi-domain Intelligent Estimation Network (MIENet) for tracking highly maneuvering targets with non-Gaussian noise.
• The MIENet consists of a fusion denoising model (FDM) and a parameter estimation model (PEM), which jointly enhance robustness and accuracy in state estimation for radar trajectory and remote sensing video data.

What are the implications of the main findings?

• The proposed MIENet achieves robust tracking performance across various noise intensities and distributions, significantly reducing observation noise and improving estimation stability.
• Our approach effectively generalizes from radar trajectory simulation to real satellite video tracking tasks, showing great potential for intelligent remote sensing target tracking in complex environments.

Abstract

In the field of remote sensing, tracking highly maneuvering targets is challenging due to its rapidly changing patterns and uncertainties, particularly under non-Gaussian noise conditions. In this paper, we consider the problem of tracking highly maneuvering targets without using preset parameters in non-Gaussian noise. We propose a multi-domain intelligent state estimation network (MIENet). It consists of two main models to estimate the key parameter for the Unscented Kalman Filter, enabling robust tracking of highly maneuvering targets under various intensities and distributions of observation noise. The first model, called a fusion denoising model (FDM), is designed to eliminate observation noise by enhancing multi-domain feature fusion. The second model, called a parameter estimation model (PEM), is designed to estimate key parameters of target motion by learning both global and local motion information. Additionally, we design a physically constrained loss function (PCLoss) that incorporates physics-informed constraints and prior knowledge. We evaluate our method on radar trajectory simulation and real remote sensing video datasets. Simulation results on the LAST dataset demonstrate that the proposed FDM can reduce the root mean square error (RMSE) of observation noise by more than 60%. Moreover, the proposed MIENet consistently outperforms the state-of-the-art state estimation algorithms across various highly maneuvering scenes, achieving this performance without requiring adjustment of noise parameters under non-Gaussian noise. Furthermore, experiments conducted on the real-world SV248S dataset confirm that MIENet effectively generalizes to satellite video object tracking tasks.

1. Introduction

In the context of Earth observation (EO), target tracking aims to estimate target states from noisy EO measurements acquired by satellite, radar, and UAV sensors. It supports UAV object tracking [1,2,3,4], remote sensing surveillance [5,6,7], and traffic control [8,9,10]. Within the domain of target tracking, the problem of tracking highly maneuvering targets is widely recognized as a fundamental challenge [11,12,13]. Specially, highly maneuvering targets are characterized by motion involving high speeds, frequent maneuvers, and long-duration maneuvers. Tracking such targets presents two main challenges. The first challenge is high-speed and high-frequency maneuvers [14]. Such maneuvers cause rapid motion changes and result in error accumulation. The second challenge arises from non-Gaussian noise with different intensities and distributions [12], causing instability in state estimation results and divergence in tracking performance. Currently, multiple paradigms have been proposed to solve these challenges. They can be classified into two types, which are model-based methods and data-driven methods.

For model-based methods [15,16,17,18,19,20,21], the interacting multiple model (IMM) filter [19] is a fundamental algorithm to solve rapidly changing patterns problem [22]. It combines several predefined motion models and works with the improved Kalman filter [15,17,23,24,25,26,27,28,29] to achieve reliable tracking in nonlinear systems. To enhance adaptability under non-Gaussian noise, Xu et al. [30] proposed an EM-based extended Kalman filter (EKF) tracker that decomposed non-Gaussian noise into Gaussian components and incorporated bias compensation, significantly improving angle-of-arrival target tracking under non-Gaussian noise conditions. Building on the idea of Gaussian decomposition, Chen et al. [31] proposed a joint probability data association based on noise-interaction Kalman filter (JPDA-IKF), which decomposed non-Gaussian noise into multiple Gaussian components and fused their results to enhance multitarget tracking performance in complex maritime environments. To further capture multimodal noise and motion characteristics, Wang et al. [32] introduced Gaussian mixture models (GMMs) into the IMM-KF, allowing accurate switching and fusion across multiple motion. Liu et al. [33] proposed an IMM-MCQKF algorithm that integrated the maximum correntropy criterion into the quadrature Kalman filter within an IMM framework, improving state estimation under non-Gaussian noise and maneuvering target scenarios. Xie et al. [34] proposed an adaptive TPM-based parallel IMM algorithm that integrated online transition probability adaptation with a threshold-controlled model-jumping mechanism within the IMM framework.

The above model-driven methods require accurate motion patterns and covariance noise parameters based on manual experiences. Indeed, constructing an accurate and well-matched model is challenging, and such prior information cannot be obtained promptly and reliably before tracking [22]. Additionally, these methods process trajectories only in the temporal domain and do not fully exploit multi-domain information. As a result, by neglecting the latent representational information in noisy observations, their sensitivity to maneuverability is reduced under non-Gaussian noise.

For data-driven methods, precise kinematic modeling of target motion is unnecessary. They leverage the strength of data mining to capture complex dynamics. They can be broadly categorized into two types, which are end-to-end architectures and hybrid architectures that integrate traditional algorithms. The former constructs an end-to-end model to directly map the input observations to the output states, eliminating the need for manually designed intermediate feature extraction. Cai et al. [35] proposed an adaptive LSTM-based tracking method that incorporated measurement–prediction errors to improve accuracy and adaptability for maneuvering targets. Liu et al. [11] learned the residual between the ground truth and estimation trajectories. This framework mainly consisted of BiLSTM layers [36,37,38] and enabled target state estimation without the need for a pre-defined model set. Building upon these, Zhang et al. [13,39] employed an enhanced transformer architecture to achieve further reductions in state estimation RMSE. Shen et al. [40] also proposed Transformer-based nonlinear target trackers, including a classical Transformer for smoothing and prediction and a recursive Transformer for filtering and prediction, achieving superior accuracy and efficiency in nonlinear radar target tracking. PCRLB [41] propose a physics-informed data-driven autoregressive nonlinear filter (DAF) based on the Transformer. Its end-to-end architecture embedded known dynamics and measurement models into the training process, ensuring physically consistent and efficient state estimation.

The above end-to-end data-driven methods primarily use a single end-to-end model to capture complex nonlinear functions often leads to convergence difficulties in practice. Moreover, a wide variation range of target motion may degrade method performance and generalization.

The latter hybrid methods [12,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57] integrate deep learning with model-driven approaches, thereby combining the strengths of both paradigms. These methods retain the framework of model-driven algorithms while incorporating deep learning modules to estimate variable parameters. By involving less functional learning and relying on simpler architectures, they achieve lower computational cost and enhanced interpretability. KalmanNet [42] integrated a structural state space model with dedicated recurrent neural networks (RNNs) to estimate the Kalman gain. It operated within the KF framework and effectively addressed non-linearity. Building on this idea, KalmanFormer [53] employed a Transformer to learn the Kalman Gain. To handle mismatches between state and observation models, Split-KalmanNet [43] calculated the Kalman gain through the Jacobian of the measurement function and leveraged two RNNs with a split architecture. Cholesky-KalmanNet [45] further enhances state estimation by providing and enforcing transiently precise error covariance estimation. Recursive KalmanNet [48] extended this line of research by employing the recursive Joseph’s formulation, thereby enabling accurate state estimation with consistent uncertainty quantification under non-Gaussian noise. Latent-KalmanNet [44] advanced the framework by jointly learning a latent representation and Kalman-based tracking from high-dimensional signals. More recently, MAML-KalmanNet [54] integrated model-agnostic meta-learning with artificially generated labeled data to achieve fast adaptation and accurate state estimation under partially unknown models. In addition, Liu et al. [12] introduced a digital twin system that combined the Unscented Kalman Filter with intelligent estimation models. Zhu et al. [58] developed an adaptive IMM algorithm that employed a neural network to estimate TPMs in real time, thereby reformulating IMM as a generalized recurrent neural network. Fu et al. [46] proposed deep learning-aided Bayesian filtering methods for guarded semi-Markov switching systems with soft constraints, addressing the challenge of state estimation under complex sojourn times and state-dependent transitions. Furthermore, Jia et al. [50] proposed a Multiple Variational Kalman-GRU model to capture multimodal ship dynamics and provide reliable trajectory prediction.

This paradigm endows the network with stronger generalization ability and interpretability. Considering these advantages, we adopt this paradigm in our work. However, existing hybrid approaches that integrate deep learning into traditional model-driven frameworks still have notable limitations. They typically rely solely on temporal-domain processing and fail to exploit complementary information from spatial and frequency domains, thereby restricting their effectiveness. In addition, when dealing with maneuvering targets, these methods often emphasize local temporal features while overlooking broader global motion patterns, which are critical for capturing abrupt changes in dynamics.

To overcome these issues, we propose a versatile and adaptive tracking framework called the Multi-Domain Intelligent State Estimation Network (MIENet) for highly maneuvering target tracking that can effectively handle non-Gaussian noise. The MIENet first applies a nonlinear transformation for coordinate system conversion and comprises two main models. Due to the nonlinear, heavy-tailed, and spatially correlated characteristics of non-Gaussian noise, standard Gaussian-based methods fail to remove it effectively. The independent module in the MIENet enables adaptive learning of these complex noise features, thereby improving convergence stability and denoising efficiency. Furthermore, leveraging multi-domain information helps capture the statistical characteristics of non-Gaussian noise. Therefore, the first model, termed the fusion denoising model (FDM), mitigates non-Gaussian observation noise by integrating temporal, spatial, and frequency-domain features.

As discussed in [12], bias in the state transition matrix ($\mathbf{F}$) causes rapidly growing deviations, while bias in the initial state $x_0$ results in linear growth. Hence, errors from an inaccurate transition matrix become much greater than those from an incorrect initial state as the trajectory extends. The main challenge in estimating the maneuvering target motion model lies in the uncertainty of $\mathbf{F}$, whose exact form is typically unknown; thus, accurately estimating its key parameters is critical to avoid computational bias. Motivated by this, the second model, termed the parameter estimation model (PEM), estimates the turn rate of target motion from temporal–spatial information. We also introduce a physically constrained loss function (PCLoss) to avoid motion estimation bias. The estimated parameters are then incorporated into the UKF to achieve accurate tracking of highly maneuvering targets. The entire tracking process is iteratively updated using a fixed sliding-window mechanism. The main contributions of this work are summarized as follows:

• We propose a MIENet to estimate target states and motion patterns from non-Gaussian observation noise. The multi-domain information of target trajectories can be well incorporated, thus helping to infer the latent state of targets.
• We design an FDM to handle noise with varying intensities and distributions. It employs a novel Inverted-UNet and FFT-based weighting to integrate temporal, spatial, and frequency-domain features within an encoder–decoder framework.
• We develop a PEM to estimate target motion patterns by fusing local and global spatial–temporal features. Considering that the state transition matrix characterizes motion dynamics and enables trajectory inference, we further introduce a PCLoss to help the PEM estimate the key turn rate parameter ($\omega$).
• Experiments on the LAST [11] and SV248S [59] datasets demonstrate the superior performance of our method in various highly maneuvering scenes. Compared with existing methods, our method is more robust to observation noise with varying intensities and distributions.

2. Materials and Methods

2.1. Problem Formulation

Our method is dedicated to maneuvering target tracking and shows limited dependence on the observation modality. Using the LAST dataset as an example, we analyzed a radar tracking system. As shown in Figure 1, the system structure is based on range and azimuth observations [12]. Initially, due to the long distance of the target and the limited measurement resolution, we ignored the shape of targets and treated them as points in space [14]. The relationship of target states and observations is represented by the state space model (SSM) as follows:

$$\begin{array}{cc}& \mathbf{State}\mathbf{Equation}\mathbf{\text{:}} {\mathit{x}}_k={\mathbf{F}}_k{\mathit{x}}_{k-1}+{\mathit{n}}_k,\hfill \end{array}$$

(1)

$$\begin{array}{cc}& \mathbf{Observation}\mathbf{Equation}\mathbf{\text{:}} {\mathit{z}}_k=h\left({\mathit{x}}_k\right)+{\mathit{m}}_k,\hfill \end{array}$$

(2)

where ${\mathit{x}}_k$ and ${\mathit{z}}_k$ are target states and observations at time step k, respectively. ${\mathit{x}}_k$ is defined as $[p_{x,k},p_{y,k},v_{x,k},v_{y,k}]$, where $[p_{x,k},p_{y,k}]$ is the two-dimensional position, and $[v_{x,k},v_{y,k}]$ is the corresponding velocity. ${\mathit{z}}_k$ is defined as $[d_k,{\theta }_k]$, where $d_k$ and ${\theta }_k$ are the range and azimuth of the target detected by the sensor. ${\mathit{n}}_k$ and ${\mathit{m}}_k$ are additive noise at time step k; ${\mathbf{F}}_k$ and $h(·)$ are the state transition matrix and observation function.

2.1.1. State Equation

For most 2D maneuvering tracking scenes, Constant Turn (CT) models [14] are representative maneuvering patterns that can approximate most motion behaviors. The  ${\mathbf{F}}_k$ is defined as follows:

$$\begin{array}{c}{\mathbf{F}}_k=\left[\begin{array}{cccc}1& 0& {\displaystyle \frac{sin\left({\omega }_k{s}_{\tau }\right)}{{\omega }_k}}& {\displaystyle \frac{cos\left({\omega }_k{s}_{\tau }\right)-1}{{\omega }_k}}\\ 0& 1& {\displaystyle \frac{1-cos\left({\omega }_k{s}_{\tau }\right)}{{\omega }_k}}& {\displaystyle \frac{sin\left({\omega }_k{s}_{\tau }\right)}{{\omega }_k}}\\ 0& 0& cos\left({\omega }_k{s}_{\tau }\right)& -sin\left({\omega }_k{s}_{\tau }\right)\\ 0& 0& sin\left({\omega }_k{s}_{\tau }\right)& cos\left({\omega }_k{s}_{\tau }\right)\end{array}\right],\end{array}$$

(3)

where ${\omega }_k$ denotes the turn rate at time step k. It is a critical parameter whose accuracy significantly affects performance. The symbol $s_{\tau }$ represents the sampling interval of trajectories. If ${\omega }_k$ equals to zero, the motion model is simplified to Constant Velocity (CV) model, which could be defined as follows:

$$\begin{array}{c}{\mathbf{F}}_k=\left[\begin{array}{cccc}1& 0& s_{\tau }& 0\\ 0& 1& 0& s_{\tau }\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right].\end{array}$$

(4)

2.1.2. Observation Equation

(1) The Radar Tracking System

For the common radar tracking system, the detector at $(r_x,r_y)$ provides range and azimuth measurements contaminated by additive noise. The polar coordinate system of observations is given as follows:

$$\begin{array}{c}\left[\begin{array}{c}{d}_k\\ {\theta }_k\end{array}\right]=\left[\begin{array}{c}\sqrt{{(p_{x,k}-r_x)}^2+{(p_{y,k}-r_y)}^2}\\ \sqrt{arctan((p_{y,k}-r_y)/(p_{x,k}-r_x))}\end{array}\right]+\left[\begin{array}{c}{q}_{d,k}\\ q_{\theta ,k}\end{array}\right],\end{array}$$

(5)

where $q_{d,k}$ and $q_{\theta ,k}$ are additive noise in range and azimuth angle observations at time step k.

Subsequently, the polar coordinate system is transformed into the Cartesian system as the input of our method. In this step, additive noise is converted into non-Gaussian noise by nonlinear transformation. The process is formulated as follows:

$$\begin{array}{c}\left[\begin{array}{c}{p}_{x,k}\\ p_{y,k}\end{array}\right]=\left[\begin{array}{c}{d}_{k}cos\left({\theta }_k\right)\\ d_{k}sin\left({\theta }_k\right)\end{array}\right]+\left[\begin{array}{c}{r}_x\\ r_y\end{array}\right].\end{array}$$

(6)

(2) The Visual Remote Sensing Tracking System

For the visual remote sensing tracking system, the observation is linear and corresponds to the target’s position in the image. The observation matrix $\mathbf{H}$ is defined as follows:

$$\mathbf{H}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right].$$

The SV248S dataset used in our experiments was captured by the Jilin-1 video satellite, where platform-induced distortions—primarily blur and contrast variations—can be reasonably approximated as Gaussian noise [59].

2.1.3. Problem Addressed in This Work

The main task is to estimate the target states (${\mathit{x}}_k$) based on the nonlinear observation (${\mathit{z}}_k$). This work addresses two primary challenges. First, the observations are often corrupted by diverse types of noise. In practice, for nonlinear observations, the noise distributions we processed are rarely Gaussian; instead, they may vary in intensity, exhibit non-Gaussian characteristics, or change dynamically across different environments. Second, maneuvering targets introduce additional difficulties due to their highly dynamic and uncertain motion patterns. A key challenge lies in accurately estimating critical motion parameters in the state transition matrix (${\mathbf{F}}_k$). Errors in estimating such parameters can severely degrade trajectory inference, as the deviation induced by model mismatch often grows increasingly with trajectory length. This work aims to design a robust framework for stable and reliable tracking of highly maneuvering targets. The framework can effectively handle non-Gaussian noise and accurately estimate key motion parameters.

2.2. Multi-Domain Intelligent State Estimation Network

In this work, we introduce our MIENet in details. The overall architecture of the proposed method is shown in Figure 2.

2.2.1. Overall Architecture

As illustrated in Figure 2, we segmented the acquired trajectories using a sliding window, and our MIENet processed fixed-length noisy polar-coordinate trajectories as input. It first applied a nonlinear transformation (Equation (6)) to convert polar coordinates ($d_k,{\theta }_k$) into Cartesian coordinates ($p_{x,k},p_{y,k}$). In this process, additive noise was transformed into non-Gaussian noise. A fusion denoising model (FDM, Section 2.2.2) was then employed to suppress non-Gaussian observation noise by integrating temporal, spatial, and frequency-domain features. Subsequently, a parameter estimation model (PEM, Section 2.2.3) estimates the turn rate from the denoised trajectories output by the FDM. It leverages both local and global temporal–spatial information. The estimated parameter was then incorporated into a UKF for maneuvering target tracking. Notably, the FDM and PEM were trained independently and evaluated jointly with the UKF in testing. A summary of the overall tracking process is provided in Algorithm 1. Subsequently, the FDM and PEM are described in detail.

Algorithm 1 MIENet tracking process

Input: Noisy observations $\mathit{z}$ with N time steps. Output: Target states $\mathit{x}$ with N time steps.   1: Initialization: Independently trained FDM and PEM; Fixed model input length is n.   2: for $k=1$ in N do   3:     while $k\le N-n$ do   4:        Segment extraction: ${\mathit{z}}_{\mathrm{seg}\left(\mathrm{n}\right)}$   5:        Position recovery with Equation (5), to get noisy position sequences: ${\mathit{d}}_{\mathrm{seg}\left(\mathrm{n}\right)}$   6:        Noise elimination: ${\mathbf{D}}_{\mathrm{seg}\left(\mathrm{n}\right)}=\mathrm{FDM}\left({\mathit{d}}_{\mathrm{seg}\left(\mathrm{n}\right)}\right)$   7:        Turn rate estimation: ${\tilde{w}}_k=\mathrm{PEM}\left({\mathbf{D}}_{\mathrm{seg}}\left(\mathrm{n}\right)\right)$   8:        Calculate state transition matrix with Equation (3): $\tilde{\mathbf{F}}=\mathbf{F}\left({\tilde{w}}_k\right)$   9:     end while 10:     Prediction and update of UKF filter: ${\tilde{\mathit{x}}}_k\to {\mathit{x}}_k$ 11: end for

2.2.2. Fusion Denoising Model

(1) Motivation

In practice, the statistical characteristics of observation noise exist not only in the temporal domain but also in the spatial and frequency domains. Temporal features alone are insufficient to capture oscillatory energy distributions in the frequency domain or spatial structural correlations, both of which are essential for robust denoising and stable tracking under non-Gaussian noise.

As noted in [60], the encoder–decoder structure enables effective separation of noise and signal in the latent space, ensuring noise suppression during reconstruction. However, traditional encoder–decoder structure consists of an encoder, a decoder, and plain connections. The encoder is used to increase feature dimension while reducing the spatial resolution. Decoder helps to recover the spatial resolution to match the size of input trajectories. Such architectures mainly emphasize hierarchical semantic features derived from temporal information. Although deepening the network or modifying the encoder submodules can capture higher-level representations, they do not explicitly address the multi-domain characteristics of observation noise. This motivates the design of a dedicated module within the encoder–decoder framework to explicitly extract and fuse temporal, spatial, and frequency-domain features.

(2) The Overall Structure of FDM

As illustrated in Figure 3, the framework adopts a symmetric encoder–decoder structure with an additional multi-domain feature module (MDFM). The input trajectories were converted from polar to Cartesian coordinates and segmented into fixed-length sequences. These processed sequences were subsequently fed into the transformer encoder to extract temporal-domain features. The MDFM further captures frequency-domain and spatial-domain information and fuses them with temporal dynamics. Finally, the fused multi-domain features are passed into the decoder subnetworks for trajectory reconstruction and prediction.

(3) The Multi-Domain Feature Module

As shown in Figure 3, the multi-domain feature module (MDFM) is embedded within the encoder to enhance feature representation beyond the temporal domain. It first performs padding and reshaping on the input data and then passes it through the Inverted-UNet. The Inverted-UNet is specifically designed to capture both multi-channel spatial structures while maintaining computational efficiency, which is critical for robust trajectory representation. The extracted features are then normalized with Layer Normalization and activated by a LeakyReLU, with this block repeated p times according to experimental results.

In parallel, the Fast Fourier Transform (FFT) is applied to the temporal features to reveal frequency-domain amplitude distributions. These amplitudes are used as weights, allowing the model to emphasize oscillatory patterns that are often hidden in noisy observations. The denoised spatial features from the Inverted-UNet and the weighted frequency-domain information are fused through a softmax layer.

By combining the Inverted-UNet and FFT, MDFM explicitly leverages spatial structures and frequency-domain energy patterns, providing a more comprehensive feature representation that significantly improves noise suppression. The following sections present a detailed description of these two key components.

(a) The Inverted-UNet

For low-dimensional trajectory sequences, the traditional dense U-Net (Figure 4a) is unsuitable, as its initial downsampling step leads to an immediate loss of information. Based on this idea, we designed a novel Inverted-UNet (Figure 4b). It first upsamples the multi-channel trajectory features to preserve spatial structure and then downsamples them to restore the original trajectory shape. Skip connections promote gradient flow.

Assume $R^{i,j}$ denotes the output of each node, where i is the $i^{th}$ upsampling layer in the encoder and j is the $j^{th}$ neighboring layer. To balance network performance and computational efficiency, the Inverted-UNet was designed with a depth of four layers, where the indices i and j take values from 0 to 3. The computation of each node feature is illustrated as follows:

$$R^{i,0}=UP\left(R^{i-1,0}\right),$$

(7)

$$R^{i,1}=[UP\left(R^{i-1,1}\right),R^{i,0},DN\left(R^{i+1,0}\right)],$$

(8)

$$R^{i,2}=[SK\left(R^{i,0}\right),R^{i,1},DN\left(R^{i+1,1}\right)],$$

(9)

$$R^{i,3}=[SK\left(R^{i,0}\right),R^{i,2},DN\left(R^{i+1,2}\right)],$$

(10)

$$R^{out}=Conv\left(R^{i,3}\right),$$

(11)

where $UP$ and $DN$ denote upsampling and max-pooling with a stride of 2, respectively. $Conv$ denotes the convolution layer. $SK$ denotes skip connection. $[·,·]$ denotes the concatenation layer. The detail dimensions of each $R^{i,j}$ are shown in

Table 1. Channel dimensions of the inverted-UNet at each stage.

	${\mathit{R}}^{0,0}$	${\mathit{R}}^{1,0}$	${\mathit{R}}^{0,1}$	${\mathit{R}}^{2,0}$	${\mathit{R}}^{1,1}$	${\mathit{R}}^{0,2}$	${\mathit{R}}^{3,0}$	${\mathit{R}}^{2,1}$	${\mathit{R}}^{1,2}$	${\mathit{R}}^{0,3}$	${\mathit{R}}^{\mathit{out}}$
Dim 1	1	1	1	1	1	1	1	1	1	1	1
Dim 2	29	29	58	29	116	203	29	174	522	812	29
Dim 3	1	2	1	4	2	1	8	4	2	1	1
Dim 4	6	12	6	24	12	6	48	24	12	6	6

.

(b) FFT-Based Weighting

We applied Fast Fourier Transform (FFT) to extract their spectrum amplitude distribution. These serve as weights in the softmax layer to fuse different features. This approach utilizes temporal and frequency domains information to assess the importance of different signals. The process is illustrated as follows:

$$\begin{array}{c}IN=\sum _{n=0}^{N-1}I{N}_{\mathrm{input}}·e^{-j2\pi fn/N},\end{array}$$

(12)

$$\begin{array}{c}{A}_u=\left|IN\right|,\end{array}$$

(13)

$$\begin{array}{c}{W}_{\mathrm{normal}}=Softmax(A_u,{\tilde{R}}^{out})(u=1,2,\dots ,6),\end{array}$$

(14)

where $I{N}_{\mathrm{input}}$ represents the fixed-length input trajectory segment of the MDFM, generated through a sliding window mechanism. f denotes the frequency index, N is the length of each fixed trajectory segment, and n refers to the time-domain sampling index. The operator $|·|$ computes the magnitude in the frequency domain. ${\tilde{R}}^{out}$ denotes the output feature, which is subsequently processed by the Inverted-UNet, LayerNorm, and LeakyReLU modules in sequence.

2.2.3. Parameter Estimation Model

(1) Motivation

In most 2D maneuvering tracking scenes, Constant Turn (CT) models are important maneuvering patterns [11,12,14], where the motion pattern ${\mathbf{F}}_k$ and its turn rate ($w_k$) are defined in Equation (3). The changing turn rate of a target affects the direction of its trajectory. Short-term trajectory features reflect instantaneous velocity variations and local motion trends, whereas long-term features characterize the overall motion pattern and the magnitude of velocity. Therefore, parameter estimation based on historical trajectory information requires a special module to extract both global and local spatial features.

As noted in ref. [61], despite achieving accurate predictions with large-scale training data, networks still struggle to acquire inductive biases aligned with the underlying world model when adapting to new tasks. To address this limitation, the loss function becomes a crucial mechanism for embedding physical constraints into the learning process. Therefore, in our work, the PEM loss function should be designed to ensure that ${\mathbf{F}}_k$ complies with physical laws and avoids calculation bias in the network output.

(2) The Overall Structure of PEM

As illustrated in Figure 5, PEM takes a denoised trajectory as input. It first uses a cross-and-dot-product (CADP) [12] to extract trajectory shape information by calculating the sine and cosine of the intersection angle. The equations are shown below:

$$s_k^{\sin }={\displaystyle \frac{\Delta p_{x,k}\Delta p_{y,k+1}-\Delta p_{y,k}\Delta p_{x,k+1}}{|\Delta {\mathbf{p}}_k\left|\right|\Delta {\mathbf{p}}_{k+1}|}}.$$

(15)

$$s_k^{\cos }={\displaystyle \frac{\Delta p_{x,k}\Delta p_{x,k+1}+\Delta p_{y,k}\Delta p_{y,k+1}}{|\Delta {\mathbf{p}}_k\left|\right|\Delta {\mathbf{p}}_{k+1}|}}.$$

(16)

where $\Delta p_{x,k}=p_{x,k}-p_{x,0}$, $\Delta p_{y,k}=p_{y,k}-p_{y,0}$, $\Delta {\mathbf{p}}_{\mathbf{k}}={[\Delta p_{x,k},\Delta p_{y,k}]}^{\mathrm{T}}$. Subsequently, the trajectory shape information was fed into multi-head attention with convolution neutral nodes (MHCN) to capture both global and local features from temporal and spatial domains. The attention output was fused with the original CADP features via a residual connection, and the result was further refined by convolutional neural nodes. Specially, a novel physically constrained loss function was applied during training to avoid bias in computing ${\mathbf{F}}_k$.

(3) Physics-Constrained Loss Function

Considering loss function design, we introduced a PCLoss to ensure that the estimated motion parameters and the resulting state transition matrix remain consistent with the underlying physical dynamics. Under both CV and CT motion models, the elements of the state transition matrix exhibit strict analytical dependencies dictated by the laws of motion. Constraining or estimating individual elements independently can violate these inherent relationships.

To avoid this issue, we jointly estimate the turn rate and impose analytical physical constraints on the entire transition matrix ${\mathbf{F}}_k$ via the proposed PCLoss. By assigning a larger weight to the ${\mathbf{F}}_k$-related term, the model is encouraged to maintain strict physical consistency, thereby stabilizing turn rate estimation and preventing implausible parameter updates. This design forms the foundation for the robustness and reliability of the proposed tracking framework.

Building on this physical consistency, the PEM is designed to estimate the key unknown motion parameter—the turn rate ${\omega }_k$—which dominates the behavior of targets under coordinated-turn (CT) dynamics. Estimating ${\omega }_k$ allows the model to capture its temporal variation, enabling reliable tracking even under high maneuvering levels (typically ${\omega }_k>{10}^{\circ }/\mathrm{s}$, e.g., transport aircraft [62]). However, estimating only a single parameter creates a large search space and may lead to unstable convergence. To mitigate this, we further incorporate physics-based constraints through PCLoss and formulate the loss as follows:

$$\begin{array}{c}{\tilde{\omega }}_k=\mathrm{PEM}\left(\mathbf{D}\mathrm{seg}\left(\mathrm{n}\right)\right),\end{array}$$

(17)

$$\begin{array}{c}{\tilde{\mathbf{F}}}_k={\mathbf{F}}_{\mathrm{CT}}\left({\tilde{\omega }}_k\right),\end{array}$$

(18)

$$\begin{array}{c}\mathrm{Loss}={\lambda }_1\mathrm{RMSE}\left({\tilde{\mathbf{F}}}_k\right)+{\lambda }_2\mathrm{RMSE}\left({\tilde{\omega }}_k\right),\end{array}$$

(19)

where $\mathbf{D}\mathrm{seg}\left(\mathrm{n}\right)$ denotes the noise-eliminated trajectory segment produced by the FDM and used as input to the PEM, ${\tilde{\omega }}_k$ is the predicted turn rate, and ${\mathbf{F}}_{\mathrm{CT}}(·)$ is the analytical CT model used to generate the corresponding transition matrix. The loss weights ${\lambda }_1$ and ${\lambda }_2$ are set to 0.8 and 0.2, respectively.

3. Results

In this section, we first introduce our training protocol. Then, we compare our MIENet to several state-of-the-art state estimation methods. Finally, we present ablation studies to investigate our work.

3.1. Implementation Details

Our MIENet is trained using the publicly available LAST dataset [11], and the implementation details are summarized in

Table 2. Parameter settings for the LAST datasets.

Contents	Ranges
Initial distance from radar	[1 km, 10 km]
Initial velocity of targets	[100, 200 m/s]
Initial distance azimuth	[−180°, 180°]
Initial velocity azimuth	[−180°, 180°]
Maneuvering turn rate	[−90°/s, 90°/s]
Variance of accelerated velocity noise	10 (m/s)2

. For testing, we primarily evaluate our method on the LAST dataset [11]. Additionally, experiments on the SV248S dataset [59] are included as supplementary validation to further demonstrate the applicability of our method to visual tracking.

For training on the LAST datasets [11], all trajectories involve both CV and CT kinetic patterns. The observed range in training data contains additive Gaussian noise with a mean of 0 and a standard deviation of 10, while the observed azimuth contains additive Gaussian noise with a mean of 0 and a standard deviation of 0.006. After nonlinear transformation (Equation (6)), the observation noise becomes non-Gaussian. We use the RMSE as the evaluation metric for the network performance. Our MIENet was trained with a batch size of 10, using the Adam optimizer with an initial learning rate of 0.01. To prevent the network from converging to a local minimum, the learning rate was adjusted using the Cosine Annealing method. All experiments were conducted on an NVIDIA 4090 GPU.

For testing on the LAST dataset [11], detailed parameter settings of testing scenes are provided in the caption of

Table 3. Parameter settings for tracking test trajectories.

Trajectories	Initial State	The First Part	The Second Part	The Third Part
1	[1000 m, −18,000 m, 150 m/s, 200 m/s]	27.4 s, $\omega$ = 50°/s	52.6 s, $\omega$ = 40°/s	27.4 s, $\omega$ = 30°/s
2	[1000 m, −18,000 m, 150 m/s, 200 m/s]	40 s, $\omega$ = −30°/s	27.4 s, $\omega$ = 70°/s	40 s, $\omega$ = 50°/s
3	[500 m, 2000 m, −300 m/s, 200 m/s]	25 s, $\omega$ = 60°/s	42.4 s, $\omega$ = 50°/s	40 s, $\omega$ = −20°/s
4	[−20,000 m, −5000 m, 250 m/s, 180 m/s]	30 s, $\omega$ = −30°/s	17.4 s, $\omega$ = 70°/s	20.9 s, $\omega$ = −10°/s

. We perform highly maneuvering scenes where ${\omega }_k$ is more than 10°/s (e.g., Transport Aircraft [62]), and each part has a maneuvering time exceeding 20 s. Specially, they also include abrupt turn rates (e.g., from −30°/s to 70°/s). Specifically, in the first test scene, the target is more than 20 km away from the radar sensor. The angular velocities gradually decrease, and the second segment lasts up to 52.6 s. This tracking scenario presents a greater challenge than those discussed in previous papers [11,12]. In the second scene, the initial state matches the first scene, but we change the direction of the turn rate and increase its value, with turns lasting up to 40 s in the first and third segments. In the third scene, the radar senor is closer to the target, within a range of less than 4.5 km. In the forth scene, the turn rate changes are more significant in both magnitude and direction (from −30°/s to 70°/s, and then from 70°/s to −10°/s) compared to the previous three trajectories. The four scenes vary in terms of target distance and maneuvering capability, which highlights the diversity and comprehensive coverage of the evaluation environments.

For testing on the SV248S dataset [59], the observations are generated using the results of the STAR [63] method, since our approach does not rely on a detector or an acceleration module tailored for image processing. Under this formulation, the observed and estimated coordinates $(p_{x,k},p_{y,k})$ are directly comparable without any nonlinear transformation. We evaluate our method on all scenes included in the SV248S dataset, following the same evaluation protocol as STAR. The algorithm ranking is determined according to two criteria: the 5-pixel precision metric and the area under the curve (AUC) of the success plot, corresponding to the precision rate (PR) and success rate (SR), respectively [63].

3.2. Comparison to the State-of-the-Art (SOTA)

To demonstrate the superiority of our method, we compare our MIENet to several state-of-the-art (SOTA) methods, including traditional methods (UKF-A [64], IMM [19] combining multiple CT motion patterns) and deep learning-based methods (ISPM [12]).

3.2.1. Quantitative Results

(1) The LAST Dataset

To objectively evaluate the tracker performance, we record the momentary RMSEs over 100 Monte Carlo simulations.

Quantitative results are shown in

Table 4. Means RMSEs of observation position and velocity for trajectories with UKF-A, IMM, ISPM and MIENet. The best results are in red, and the second best results are in blue.

Trajectories p (m)/v (m/s)	Noise Standard Deviation (Azimuth, Range)		UKF-A [64]	IMM [19]	ISPM [12]	MIENet (Ours)
1	(1 × 10−3 rad, 2 m)	Part 1 Part 2 Part 3	6.667/11.38 6.329/7.463 6.457/7.963	9.193/32.57 8.689/28.72 8.366/25.87	3.770/7.920 3.370/8.030 2.490/7.040	1.800/4.170 1.500/3.100 1.350/3.100
		All	6.420/7.860	8.720/28.84	3.280 /7.770	1.580/3.470
	(2 × 10−3 rad, 4 m)	Part 1 Part 2 Part 3	39.73/64.94 13.23/13.67 10.86/10.17	17.01/47.30 15.92/39.23 15.52/36.66	7.950/9.570 7.360/9.630 5.500/8.410	4.650/6.150 2.650/4.380 3.310/4.870
		All	22.90/30.05	16.07/40.23	7.110/9.330	3.570/5.300
	(6 × 10−3 rad, 8 m)	Part 1 Part 2 Part 3	139.2/208.9 136.4/124.3 132.7/109.7	62.80/147.8 41.62/63.69 40.46/60.04	17.80/13.59 15.88/14.67 13.26/13.83	9.170/10.15 7.250/12.20 4.520/8.590
		All	133.5/125.8	42.90/67.37	15.80/14.21	7.780/10.64
2	(1 × 10−3 rad, 2 m)	Part 1 Part 2 Part 3	6.854/8.294 10.10/26.66 6.494/8.489	9.481/33.15 8.692/29.30 9.013/30.38	2.640/7.170 5.560/12.70 4.350/8.570	1.790/4.930 3.340/9.200 3.870/7.260
		All	6.920/10.36	9.090/31.10	4.170/9.290	2.840/6.190
	(2 × 10−3 rad, 4 m)	Part 1 Part 2 Part 3	12.04/12.22 94.06/172.2 83.27/118.6	17.95/45.88 15.82/44.09 16.33/42.03	5.600/8.870 11.74/10.53 9.970/10.07	3.440/6.190 10.06/9.740 11.53/10.58
		All	67.66/107.1	16.78/43.87	9.110/9.750	7.600/7.690
	(6 × 10−3 rad, 8 m)	Part 1 Part 2 Part 3	121.2/127.0 223.4/293.5 275.9/301.5	54.83/96.77 39.80/76.48 41.08/67.76	11.62/11.72 28.47/17.84 35.27/20.49	6.540/7.930 17.26/15.48 32.28/17.27
		All	244.4/272.4	44.05/73.98	24.30/16.07	18.65/12.64
3	(1 × 10−3 rad, 2 m)	Part 1 Part 2 Part 3	1.651/5.706 1.561/5.125 2.214/6.587	2.106/24.70 2.010/25.08 2.723/28.50	5.030/1.810 4.590/1.650 2.320/1.870	3.300/1.920 2.590/1.550 3.120/1.760
		All	1.810/5.640	2.290/26.23	4.000/1.770	2.710/1.670
	(2 × 10−3 rad, 4 m)	Part 1 Part 2 Part 3	2.953/8.071 2.758/6.747 4.097/9.735	3.882/32.96 3.730/32.38 5.018/36.97	10.31/2.440 9.540/2.150 4.760/2.470	6.400/2.670 5.690/2.270 4.090/1.770
		All	3.260/7.590	4.230/34.19	8.280/2.340	4.950/2.180
	(6 × 10−3 rad, 8 m)	Part 1 Part 2 Part 3	6.277/14.22 5.747/10.03 9.921/18.34	8.626/50.51 8.169/46.74 11.79/52.27	22.15/5.630 21.03/3.200 9.160/7.020	14.98/5.070 10.43/4.040 8.120/3.980
		All	7.210/12.06	9.620/49.53	17.92/5.450	10.49/4.100
4	(1 × 10−3 rad, 2 m)	Part 1 Part 2 Part 3	8.420/16.20 15.38/43.98 42.29/77.72	10.47/39.53 9.549/33.30 9.838/31.73	3.530/9.050 8.440/19.11 4.650/13.54	3.650/9.910 6.380/17.36 1.300/3.490
		All	18.13/40.22	10.01/35.40	5.430/13.49	4.180/11.38
	(2 × 10−3 rad, 4 m)	Part 1 Part 2 Part 3	28.23/43.75 99.56/175.2 114.5/124.9	20.41/55.48 17.93/50.32 18.37/44.73	7.150/11.13 18.42/13.35 10.81/12.76	8.440/6.140 16.01/18.94 2.370/6.230
		All	85.48/121.5	19.06/50.31	11.89/12.21	10.16/11.14
	(6 × 10−3 rad, 8 m)	Part 1 Part 2 Part 3	159.9/195.7 301.7/361.7 207.8/114.6	72.75/154.2 43.35/85.78 49.05/72.03	25.64/14.89 53.94/31.10 34.12/23.23	24.62/9.770 57.04/33.48 12.61/16.75
		All	235.5/244.0	53.04/87.63	36.33/22.10	34.57/20.50

. The improvements achieved by our MIENet over traditional methods are significant. That is because our MIENet can learn multi-domain features that are robust to scene variations and leverages both local and global information to capture latent motion patterns. In contrast, the traditional methods are usually designed for specific scenes (e.g., specific motion pattern and measurement environment).

Moreover, as shown in Figure 6, we set each scene with three different noise levels. With the increase in noise, traditional methods suffer a dramatic performance decrease, while our MIENet maintains accuracy. That is because the performance of traditional methods rely heavily on manually chosen parameters (e.g., the process noise covariance $\mathbf{Q}$ and the measurement noise covariance $\mathbf{R}$) and cannot adapt to various noise. It is worth noting that our method preserves error stability more effectively than traditional approaches when abrupt changes occur in turn rate direction or magnitude. Moreover, it remains robust as it does not require re-adjustment of the noise parameters ($\mathbf{Q}$ and $\mathbf{R}$) under changing environmental noise.

As shown in Table 4, the improvements achieved by the MIENet over deep learning-based methods (i.e., ISPM) are obvious. That is because we redesign the FDM, and the PEM is optimized for highly maneuvering target tracking. The Inverted-UNet combined with FFT-based softmax weighting enables fusion of spatial and frequency domain information. The physics-constrained loss function can avoid calculation bias. Consequently, multi-domain information of trajectories can be maintained and fully learned in the network.

In the tracking process, our MIENet takes 47.2ms per iteration, and ISPM takes 11.3ms. Both algorithms are suitable for real-time applications. The detailed memory requirements of the MIENet for the LAST dataset are shown in

Table 5. Detailed memory requirements of the MIENet for the LAST dataset.

Trajectories	1	2	3	4
Total Time Steps	1074	1074	1074	683
Allocated GPU Memory (MB)	8.05	8.36	7.99	8.07
Active GPU Memory (MB)	1316
FLOPs (MB)	FDM: 10    PEM: 0.02
# Params (MB)	FDM: 1239    PEM: 14.7

.

(2) The SV248S Dataset

Quantitative results are presented in

Table 6. Category-wise and difficulty-wise PR and SR evaluations on SV248S. The best results are in red.

Trackers	Category-Wise Evaluations								Difficulty-Wise Evaluations
	Vehicle		L-Vehicle		Airplane		Ship		Simple		Normal		Hard
	PR	SR	PR	SR	PR	SR	PR	SR	PR	SR	PR	SR	PR	SR
STAR [63]	0.7542	0.4911	0.8739	0.6555	0.8399	0.7517	1.0000	0.7562	0.8800	0.6228	0.7138	0.4653	0.6652	0.4177
Ours	0.7550	0.4944	0.8744	0.6598	0.8399	0.7530	1.0000	0.7611	0.8816	0.6278	0.7138	0.4675	0.6760	0.4440

and

Table 7. Attribute-wise PR/SR evaluations on SV248S. The best results are in red.

Trackers	STO	LTO	DS	IV	BCH	SM	ND	CO	BCL	IPR
STAR [63]	0.676/0.444	0.472/0.301	0.731/0.489	0.692/0.446	0.796/0.534	0.700/ 0.462	0.752/0.489	0.700/0.444	0.730/0.458	0.697/0.464
Ours	0.676/0.447	0.473/0.302	0.731/0.492	0.693/0.449	0.797/0.538	0.700/0.461	0.753/0.490	0.700/0.447	0.730/0.460	0.697/0.466

. The proposed MIENet demonstrates applicability to remote visual tracking, effectively reducing localization errors across various target classes, difficulty levels, and attribute categories without requiring retraining. These results further indicate that our method can be readily extended to two-dimensional visual object tracking tasks. The detailed memory requirements of the MIENet for the SV248S dataset are shown in

Table 8. The detailed memory requirements of the MIENet for the SV248S dataset.

Scenes	1	2	3	4	5	6
Total Time Steps	750	747	490	748	580	499
Allocated GPU memory (MB)	5.47
Active GPU Memory (MB)	1310
FLOPs (MB)	FDM: 10       PEM: 0.02
# Params (MB)	FDM: 1239       PEM: 14.7

.

3.2.2. Qualitative Results

(1) The LAST Dataset

Qualitative results are presented in Figure 7 and Figure 8. As shown in Figure 7b,c, the MIENet effectively reduces peak tracking errors. That is because traditional methods detect pattern switching only after accumulating errors. This causes delay and peak error accumulation. The deep learning-based methods (i.e., ISPM) perform much better than traditional methods. However, due to highly maneuvering scenes, ISPM lacks multi-domain information and physical constraints, which leads to large parameter estimation errors. Our MIENet is more robust compared to various highly maneuvering target tracking methods.

Moreover, as shown in the enlarged view of Figure 8a, our method achieves the lowest tracking error even when the target undergoes high-speed motion (turn rate of 40°/s). This is because our FFT-based weighting module captures the rich frequency-domain information induced by highly maneuvering motion, preventing the loss of high-frequency features and enabling more accurate estimation of motion parameters. In the enlarged view of Figure 8b, our method again attains the lowest tracking error at a lower turn rate of 20°/s. That is because the Inverted-UNet structure effectively captures both temporal and spatial cues of the target’s motion, maintaining reliable tracking performance when inter-frame motion changes are relatively small. Finally, Figure 8c shows that our method continues to track the target accurately even after it performs a maneuver from 70°/s to −10°/s. This is attributed to the joint use of convolution and the MHN module, which facilitates the fusion of local and global information, enabling the network to capture historical motion features and recognize motion-pattern transitions, thus adapting robustly to diverse motion scenarios.

(2) The SV248S Dataset

Qualitative results are shown in Figure 9. Our MIENet is capable of performing satellite localization in remote sensing images.

3.3. Ablation Study

In this section, we compare our MIENet with several variants. The objective is to investigate the potential benefits arising from our chosen modules and network design.

3.3.1. Ablation Study on the Magnitude and Distribution of Noise

As shown in

Table 9. The adaptation and generalization of FDM ablation study (Gaussian-injected noise). The best results are in red.

Methods RMSE (m)	Gaussian-Injected Noise $\mathit{\sigma }$ (10−3 rad, m)
	(2, 4)	(8, 10)	(11, 13)
Original Observation	35.310	140.94	193.77
Butterworth Filter	28.000	112.09	153.65
ISPM-NEN [12]	17.359	61.978	84.804
FDM (ours)	13.593 (↓61.5%)	43.690 (↓69.0%)	61.135 (↓68.4%)

and

Table 10. The adaptation and generalization of an FDM ablation study (uniform-injected noise). The best results are in red.

Methods RMSE (m)	Uniform-Injected Noise $\mathit{\sigma }$ (10−3 rad, m)
	(12, 21.3)	(40.3, 56.3)	(75, 133.3)
Original Observation	61.536	112.78	153.84
Butterworth Filter	47.836	87.673	119.59
ISPM-NEN [12]	28.994	52.348	71.290
FDM (ours)	23.295 (↓62.1%)	40.333 (↓64.2%)	54.277 (↓64.7%)

, we evaluate the denoising performance under injected Gaussian-distributed and Uniform-distributed additive noise at three intensity levels to assess adaptability and generalization. Importantly, although the injected noise follows Gaussian or Uniform distributions in the input space, the nonlinear observation transformation distorts these distributions, producing non-Gaussian noise in the observation space.

Our method reduces RMSEs by at least 60% (with an average reduction of 66.4%) for cases with Gaussian-injected noise and at least 55% (with an average reduction of 63.28%) for cases with Uniform-injected noise. This consistent improvement demonstrates that the FDM effectively exploits multi-domain information in observations, making it applicable to denoising tasks under varying noise levels and across different non-Gaussian noise distributions induced by nonlinear transformations.

3.3.2. Ablation Study on MDFM

The MDFM is used for temporal, spatial and frequency-domain features enhancement to achieve better feature fusion. To investigate the benefits introduced by this module, we compare its effects on the FDM and MIENet.

• FDM w/o MDFM: As shown in

  Table 11. MDFM to FDM ablation study. The best results are in red.

  Methods RMSE (m)			Noise
  			(2 × 10−3 rad, 4 m)	(8 × 10−3 rad, 10 m)	(11 × 10−3 rad, 13 m)
  FDM w/o MDFM	pos (m)	Part 1 Part 2 Part 3	15.53 16.34 15.69	49.34 42.67 44.04	74.20 58.67 62.43
  	All		15.97 (↓36.3%)	44.81 (↓55.2%)	63.91 (↓53.5%)
  Ours	pos (m)	Part 1 Part 2 Part 3	11.63 10.25 9.960	39.28 33.58 38.76	55.11 46.97 57.77
  	All		10.55 (↓57.9%)	36.45 (↓63.5%)	52.02 (↓62.1%)

  , the original FDM achieves a 61.2% reduction in the RMSE, while removing the MDFM module decreases the reduction to 48.3%. The results are averaged over three noise intensity levels. This performance gap demonstrates that the Inverted-UNet is essential for denoising, as it extracts and fuses spatial-domain features more effectively from noisy observations.
• MIENet w/o MDFM: As shown in

  Table 12. Results of the ablation study.

  Tracking Variations	MDFM	MHN	PCLoss	Distance (m)/ Velocity (m/s)
  Ablation 1	✓			25.88/21.61
  Ablation 2		✓		19.39/28.87
  Ablation 3			✓	33.29/38.21
  Ablation 4		✓	✓	15.83/29.61
  Ablation 5	✓		✓	16.90/14.07
  Ablation 6	✓	✓		9.900/13.55
  Ours	✓	✓	✓	7.780/10.64

  , our MIENet achieves position and velocity RMSEs of 7.78 m and 10.64 m/s, respectively. In Ablation 4, removing the MDFM module increases RMSEs to 15.83 m and 29.61 m/s. That is because the FDM, as a component of the MIENet, effectively suppresses noise and thereby contributes to its tracking performance.

3.3.3. Ablation Study on Multi-Head Attention

We evaluate the contribution of multi-head attention (MHN) to the MIENet and present the results in Table 12. In Ablation 5, removing the MHN module increases RMSEs to 16.9 m and 14.07 m/s. That is because our proposed MHN can capture historical global information about trajectories to enhance temporal domain features.

3.3.4. Ablation Study on Loss Function Design

The PCLoss function for the PEM replaces the single parameter with eight unknown parameters. Results are shown in Table 12. Both Ablation 6 and our method demonstrate that the PCLoss function reduces the distance RMSE by 21.41% and the velocity RMSE by 21.47%. That is because the PCLoss function can incorporate physical constrains and prevent bias in the calculations.

4. Discussion

Through extensive experiments on both radar-based trajectory simulations and real satellite video datasets, the MIENet has demonstrated strong robustness and adaptability in tracking highly maneuvering targets under non-Gaussian noise conditions. Across all evaluation scenarios, the MIENet consistently improves estimation accuracy and stability compared with traditional state-estimation methods such as UKF, PF, and their variants. These performance gains arise from the complementary roles of the fusion denoising model (FDM) and the parameter estimation model (PEM), which jointly enhance the reliability of the entire estimation pipeline. First, the FDM effectively suppresses observation noise through multi-domain feature extraction and fusion. Unlike conventional denoising filters or handcrafted feature-based approaches that are sensitive to noise intensity and distribution, the FDM learns noise-resilient representations directly from data. This allows it to handle a wide range of non-Gaussian noise patterns without relying on explicit assumptions or manual parameter tuning. Quantitative evaluations on the LAST dataset show that the FDM reduces observation noise by more than 60%, directly improving the stability of subsequent estimation. Second, the PEM accurately predicts key motion parameters by integrating global motion trends with local trajectory variations. This dual-level modeling enables the PEM to capture complex dynamics of highly maneuvering targets, especially in scenarios involving abrupt motion changes or large accelerations. In addition, the PCLoss function enforces consistency with motion laws and prior knowledge, preventing unrealistic predictions and improving model interpretability. Ablation studies confirm that each component—FDM, PEM, and PCLoss—contributes substantively to the overall performance, and their combined effect yields a more robust and accurate estimation process.

Another important observation is the strong generalization ability of the MIENet. Although trained only on radar trajectory simulations, the MIENet focuses on learning domain-invariant motion patterns rather than modality-dependent appearance cues. Its multi-domain learning framework extracts stable geometric and temporal features of target motion that are governed by physical laws and therefore shared across radar and satellite video data. In addition, the PCLoss function further regularize the model to produce physically consistent motion predictions, reducing reliance on dataset-specific noise characteristics. Together, these factors enable the MIENet to maintain robust performance when transferred to the real-world SV248S dataset despite substantial differences in sensing modality and noise distribution. Finally, the MIENet exhibits promising computational efficiency. With an inference time of approximately 47.2 ms per iteration and a stable memory footprint of only 5–8 MB, the MIENet is well suited for real-time remote sensing tasks such as maritime surveillance, UAV monitoring, or on-orbit object tracking. Its minimal dependence on sequence length further enhances applicability in long-duration tracking scenarios.

5. Conclusions

In this paper, we propose a versatile and adaptive multi-domain intelligent state estimation network to enhance highly maneuvering target tracking that can handle non-Gaussian noise. To address the challenges of converting noisy observation trajectories into clean inputs, we design the FDM with an encoder–decoder architecture. This model enables reduction in RMSEs by at least 60% through exploiting multi-domain features. To achieve a more accurate motion model for state estimation, we propose the PEM to capture temporal domain information across both global and local spatial scales. We also redesign the loss function to avoid calculation bias. Finally, experimental results demonstrate that our method achieves the state-of-the-art performance and maintains robustness across various noise levels and distributions.

There are two promising directions for future work. First, this paper considers only range and azimuth observations, as incorporating Doppler measurements may further enhance maneuvering target tracking. Second, this work is restricted to tracking in the two-dimensional X–Y plane; extending the framework to three-dimensional scenarios is a valuable direction.