This paper presents a systematic analysis of inter-symbol and inter-carrier interference (ISCI) modeling in doubly dispersive channels for integrated sensing and communication (ISAC) systems. We propose a generalized OFDM (Weyl-Heisenberg) framework to evaluate four ISCI treatment approaches: (1) explicit estimation and compensation, (2) complete ignorance, (3) uncorrelated colored noise approximation, and (4) correlated colored noise modeling. Through continuous delay-Doppler channel characterization, we derive LMMSE channel estimators and corresponding estimation errors (as sensing metrics) for both pilot-assisted and fully-known symbol scenarios. The communication performance is quantified via ergodic capacity bounds under imperfect CSI. Our theoretical analysis and numerical results reveal fundamental performance-complexity trade-offs, providing insights for practical ISAC waveform and receiver design in doubly dispersive channels.

Differential equations are involved in modeling many engineering problems. Many efforts have been devoted to solving differential equations. Due to the flexibility of neural networks, Physics Informed Neural Networks (PINNs) have recently been proposed to solve complex differential equations and have demonstrated superior performance in many applications. While the L2 loss function is usually a default choice in PINNs, it has been shown that the corresponding numerical solution is incorrect and unstable for some complex equations. In this work, we propose a new PINNs framework named Kernel Packet accelerated PINNs (KP-PINNs), which gives a new expression of the loss function using the reproducing kernel Hilbert space (RKHS) norm and uses the Kernel Packet (KP) method to accelerate the computation. Theoretical results show that KP-PINNs can be stable across various differential equations. Numerical experiments illustrate that KP-PINNs can solve differential equations effectively and efficiently. This framework provides a promising direction for improving the stability and accuracy of PINNs-based solvers in scientific computing.

Partial Differential Equations (PDEs) underpin many scientific phenomena, yet traditional computational approaches often struggle with complex, nonlinear systems and irregular geometries. This paper introduces the AMG method, a Multi-Graph neural operator approach designed for efficiently solving PDEs on Arbitrary geometries. AMG leverages advanced graph-based techniques and dynamic attention mechanisms within a novel GraphFormer architecture, enabling precise management of diverse spatial domains and complex data interdependencies. By constructing multi-scale graphs to handle variable feature frequencies and a physics graph to encapsulate inherent physical properties, AMG significantly outperforms previous methods, which are typically limited to uniform grids. We present a comprehensive evaluation of AMG across six benchmarks, demonstrating its consistent superiority over existing state-of-the-art models. Our findings highlight the transformative potential of tailored graph neural operators in surmounting the challenges faced by conventional PDE solvers. Our code and datasets are available on https://github.com/lizhihao2022/AMG.

The metro train localization in an underground environment is a challenging task to provide the reliable and accurate train location for railway operation and maintenance. Nowadays, a user-friendly, high-precision, cost-effective, and convenient train localization method is needed for onboard railway inspection in the underground environment. In this research, we propose a train localization method by train motion and track geometry features fusion. A single microelectromechanical system (MEMS) inertial measurement unit (IMU) mounted onboard an in-service train, is used to sense the train motion, and the measurements are mapped into absolute location along the track by map-matching. Based on that, the train speed estimation is calibrated through train motion and track geometry features fusion to update the train location on the track. Then, the average localization error is estimated to evaluate the train localization performance in the absence of the real train trajectory data. The field experiment is carried out to verify the applicability and assess the train localization accuracy of the proposed method, the results of which show less than 1% average localization errors in an interval. Data from MEMS IMUs mounted on different cars are also tested, showing satisfactory repeatability and reliability performance.