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An efficient spectral method for the fractional Schrödinger equation on the real line

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arXiv 2022年

作者： Shen, Mengxia Wang, Haiyong School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

The fractional Schrödinger equation (FSE) on the real line arises in a broad range of physical settings and their numerical simulation is challenging due to the nonlocal nature and the power law decay of the solution at infinity. In this paper, we propose a new spectral discretization scheme for the FSE in space based upon Malmquist-Takenaka functions. We show that this new discretization scheme achieves much better performance than existing discretization schemes in the case where the underlying FSE involves the square root of the Laplacian, while in other cases it also exhibits comparable or even better performance. Numerical experiments are provided to illustrate the effectiveness of the proposed method. © 2022, CC BY.

关键词： Laplace transforms

Benchmark modeling and 3D applications of solidification and macro-segregation based on an operator-splitting and fully decoupled scheme with term-wise matrix assembly

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arXiv 2023年

作者： Feng, Xiaoyu Chen, Huangxin Yu, Bo Sun, Shuyu Thuwal23955-6900 Saudi Arabia School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Xiamen361005 China School of Mechanical Engineering Beijing Institute of Petrochemical Technology Beijing102617 China

The solidification and macro-segregation problem involving unsteady multiphysics and multi-phase fields is typically a complex process with mass, momentum, heat, and species transfers among solid, mushy, and liquid phase regions. The quantitative prediction of phase change, chemical heterogeneities, and multi-phase and multi-component flows plays critical roles in many natural scenarios and industrial applications that involve many disciplines, like material, energy, and even planet science. In view of this, some scholars and research institutions have called for more contributors to join the benchmark analysis of solidification and segregation problems. Our work proposes an operator-splitting and matrix-based method to avoid non-linear systems. Also, the combination of vectorization and forward equation-based matrix assembly techniques enhances the implementability of extensions of 3D applications. Lastly, the novel scheme is well validated through a bunch of 2D and 3D benchmark cases. The numerical results also illustrate that this method can ensure accurate prediction and adequately capture the physical details of phenomena caused by the solutally and thermally driven flow, which include channel segregation, the formation of freckles, edge effect, aspect ratio effect, and 3D effect. Copyright © 2023, The Authors. All rights reserved.

关键词： Solidification

ON A CLASSIFICATION OF STEADY SOLUTIONS TO TWO-DIMENSIONAL EULER EQUATIONS

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arXiv 2024年

作者： Gui, Changfeng Xie, Chunjing Xu, Huan Department of Mathematics University of Macau China School of Mathematical Sciences Institute of Natural Sciences Ministry of Education Key Laboratory of Scientific and Engineering Computing CMA-Shanghai Shanghai Jiao Tong University Shanghai200240 China Department of Mathematics The University of Texas at San Antonio San AntonioTX78249 United States

In this paper, we provide a classification of steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the whole plane must be the whole circle unless the flow is a parallel shear flow. In an infinitely long horizontal strip or the upper half-plane supplemented with slip boundary conditions, besides the two types of flows appeared in the whole space case, there exists an additional class of steady flows for which the set of flow angles is either the upper or lower closed semicircles. This type of flows is proved to be the class of non-shear flows that have the least total curvature. As applications, we obtain Liouville-type theorems for two-dimensional semilinear elliptic equations with only bounded and measurable nonlinearity, and the structural stability of shear flows whose all stagnation points are not inflection points, including Poiseuille flow as a special case. Our proof relies on the analysis of some quantities related to the curvature of the streamlines. Copyright © 2024, The Authors. All rights reserved.

关键词： Stability

Over-the-air federated learning: Status quo, open challenges, and future directions

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Fundamental Research 2024年

作者： Xiao, Bingnan Yu, Xichen Ni, Wei Wang, Xin Poor, H. Vincent Key Laboratory for Information Science of Electromagnetic Waves (MoE) Department of Communication Science and Engineering Fudan University Shanghai 200433 China Data61 Commonwealth Scientific and Industrial Research Organisation (CSIRO) Sydney 2122 NSW Australia Department of Electrical and Computer Engineering Princeton University Princeton 08544 NJ United States

The development of applications based on artificial intelligence and implemented over wireless networks is increasingly rapidly and is expected to grow dramatically in the future. The resulting demand for the aggregation of large amounts of data has caused serious communication bottlenecks in wireless networks and particularly at the network edge. Over-the-air federated learning (OTA-FL), leveraging the superposition feature of multi-access channels, enables users at the network edge to share spectrum resources and achieves efficient and low-latency global model aggregation. This paper provides a holistic review of progress in OTA-FL and points to potential future research directions. Specifically, we classify OTA-FL from the perspective of system settings, including single-antenna OTA-FL, multi-antenna OTA-FL, and OTA-FL with the aid of the emerging reconfigurable intelligent surface technology, and the contributions of existing works in these areas are summarized. Moreover, we discuss the trust, security and privacy aspects of OTA-FL, and highlight concerns arising from security and privacy. Finally, challenges and potential research directions are discussed to promote the future development of OTA-FL in terms of improving system performance, reliability, and trustworthiness. Specifical challenges to be addressed include model distortion under channel fading, the ineffective OTA aggregation of local models trained on substantially unbalanced data, and the limited accessibility and verifiability of individual local models. © 2024

关键词： Federated learning (FL) Machine learning (ML) Multiple-input multiple-out (MIMO) Over-the-air federated learning (OTA-FL) Privacy Reconfigurable intelligent surface (RIS) Security

Robust Graph Structure Learning with the Alignment of Features and Adjacency Matrix

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arXiv 2023年

作者： Lv, Shaogao Wen, Gang Liu, Shiyu Wei, Linsen Li, Ming Department of Statistics and Data Science Nanjing Audit University China University of Electronic Science and Technology of China China School of Astronautics Northwestern Polytechnical University China Key Laboratory of Intelligent Education Technology and Application of Zhejiang Province Zhejiang Normal University Jinhua China Key Laboratory of Scientific and Engineering Computing Ministry of Education Shanghai Jiao Tong University China

To improve the robustness of graph neural networks (GNN), graph structure learning (GSL) has attracted great interest due to the pervasiveness of noise in graph data. Many approaches have been proposed for GSL to jointly learn a clean graph structure and corresponding representations. To extend the previous work, this paper proposes a novel regularized GSL approach, particularly with an alignment of feature information and graph information, which is motivated mainly by our derived lower bound of node-level Rademacher complexity for GNNs. Additionally, our proposed approach incorporates sparse dimensional reduction to leverage low-dimensional node features that are relevant to the graph structure. To evaluate the effectiveness of our approach, we conduct experiments on real-world graphs. The results demonstrate that our proposed GSL method outperforms several competitive baselines, especially in scenarios where the graph structures are heavily affected by noise. Overall, our research highlights the importance of integrating feature and graph information alignment in GSL, as inspired by our derived theoretical result, and showcases the superiority of our approach in handling noisy graph structures through comprehensive experiments on real-world datasets. Copyright © 2023, The Authors. All rights reserved.

关键词： Graphic methods

Symplectic Stiefel manifold: tractable metrics, second-order geometry and Newton’s methods

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arXiv 2024年

作者： Gao, Bin Son, Nguyen Thanh Stykel, Tatjana State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Mathematics and Informatics Thai Nguyen University of Sciences Thai Nguyen24118 Viet Nam Universität Augsburg Universitätsstraße 12a Augsburg86159 Germany

Optimization under the symplecticity constraint is an approach for solving various problems in quantum physics and scientific computing. Building on the results that this optimization problem can be transformed into an unconstrained problem on the symplectic Stiefel manifold, we construct geometric ingredients for Riemannian optimization with a new family of Riemannian metrics called tractable metrics and develop Riemannian Newton schemes. The newly obtained ingredients do not only generalize several existing results but also provide us with freedom to choose a suitable metric for each problem. To the best of our knowledge, this is the first try to develop the explicit second-order geometry and Newton’s methods on the symplectic Stiefel manifold. For the Riemannian Newton method, we first consider novel operator-valued formulas for computing the Riemannian Hessian of a cost function, which further allows the manifold to be endowed with a weighted Euclidean metric that can provide a preconditioning effect. We then solve the resulting Newton equation, as the central step of Newton’s methods, directly via transforming it into a saddle point problem followed by vectorization, or iteratively via applying any matrix-free iterative method either to the operator Copyright © 2024, The Authors. All rights reserved.

关键词： Geometry

Learning Robust Travel Preferences Via Check-In Masking for Next Poi Recommendation

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SSRN

SSRN 2023年

作者： Duan, Chenghua Zhou, Wei Zeng, Jun Zhang, Yihao Wen, Junhao School of Bigdata and Software Engineering Chongqing University Daxuecheng South Rd. 55 Chognqing400044 China Key Laboratory of Dependable Service Computing in Cyber Physical Society Moe Chognqing400044 China School of Artificial Intelligence Chongqing University of Technology Chognqing401135 China

Next point-of-interest (POI) recommendation is a crucial part of many location-based service applications. It predicts users’ next visit based on their historical trajectories and current *** solutions primarily focus on sequential modeling of users’ check-in behavior. Unfortunately, most current research overlooks the fact that check-ins are implicit feedback, and inherent noise exists within them. To this end, we propose an approach to cope with noise-learning robust user travel preferences. Specifically, we design a masking strategy for check-ins, which masks the spatio-temporal information of some check-ins in users’ trajectories and forces our model to learn visit preferences without relying on individual inputs, thereby mitigating the bias caused by noisy check-ins. Then, we utilize multi-level geographical information to recommend the next POI more accurately. Additionally, we propose another masking strategy that considers users’ living area and automatically selects the proper strategy based on trajectory features. To the best of our knowledge, this is the first work that investigates the characteristics of noisy check-ins and performs denoising in next POI recommendation. Through extensive experiments, we demonstrate that our model outperforms state-of-the-art approaches and effectively handles data sparsity and noise issues. © 2023, The Authors. All rights reserved.

关键词： Trajectories

Benchmark Modeling and 3d Applications of Solidification and Macro-Segregation Based on an Operator-Splitting and Fully Decoupled Scheme with Term-Wise Matrix Assembly

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SSRN

SSRN 2023年

The solidification and macro-segregation problem involving unsteady multi-physics and multi-phase fields is typically a complex process with mass, momentum, heat, and constituent transfers among solid, mushy, and liquid phase regions. The quantitative prediction of phase change, chemical heterogeneities, and multi-phase and multi-component flows plays critical roles in many natural scenarios and industrial applications that involve many disciplines, like material, energy, and even planet science. In view of this, some scholars and research institutions have called for more contributors to join the benchmark analysis of solidification and segregation problems. Our work proposes fully decoupled and iterative schemes based on operator-splitting and matrix-based ideas to avoid solving non-linear systems directly. Also, the combination of direct vectorization and equation-based forward matrix assembly techniques enhances the implementability and efficiency of extensions to 3D applications. Lastly, the novel scheme is well validated through a bunch of 2D and 3D benchmark cases. The numerical results also illustrate that this method can ensure accurate prediction and efficient computation. It adequately captures the physical details of phenomena caused by the solutally and thermally driven flow, which include channel segregation, the formation of freckles, edge effect, aspect ratio effect, and 3D effect. © 2023, The Authors. All rights reserved.

关键词： Solidification

Task-oriented Self-supervised Learning for Anomaly Detection in Electroencephalography

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arXiv 2022年

作者： Zheng, Yaojia Liu, Zhouwu Mo, Rong Chen, Ziyi Zheng, Wei-Shi Wang, Ruixuan School of Computer Science and Engineering Sun Yat-sen University China Key Laboratory of Machine Intelligence and Advanced Computing MOE China The First Affiliated Hospital Sun Yat-sen University Guangzhou China

Accurate automated analysis of electroencephalography (EEG) would largely help clinicians effectively monitor and diagnose patients with various brain diseases. Compared to supervised learning with labelled disease EEG data which can train a model to analyze specific diseases but would fail to monitor previously unseen statuses, anomaly detection based on only normal EEGs can detect any potential anomaly in new EEGs. Different from existing anomaly detection strategies which do not consider any property of unavailable abnormal data during model development, a task-oriented self-supervised learning approach is proposed here which makes use of available normal EEGs and expert knowledge about abnormal EEGs to train a more effective feature extractor for the subsequent development of anomaly detector. In addition, a specific two-branch convolutional neural network with larger kernels is designed as the feature extractor such that it can more easily extract both larger-scale and small-scale features which often appear in unavailable abnormal EEGs. The effectively designed and trained feature extractor has shown to be able to extract better feature representations from EEGs for development of anomaly detector based on normal data and future anomaly detection for new EEGs, as demonstrated on three EEG datasets. The code is available at https://***/ironing/EEG-AD. © 2022, CC BY.

关键词： Electroencephalography