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Science China Mathematics 2025年第5期68卷 1051-1078页

作者： Rui Jin Yachun Li Liang Zhao School of Mathematical Sciences Shanghai Jiao Tong UniversityShanghai200240China CMA-Shanghai MOE-LSCand SHL-MACShanghai Jiao Tong UniversityShanghai200240China Mathematical Modelling&Data Analytics Center Oxford Suzhou Centre for Advanced ResearchSuzhou215123China

Due to extreme difficulties in numerical simulations of Euler-Maxwell equations,which are caused by the highly complicated structures of the equations,this paper concerns the simplification of the Euler-Maxwell system through the zero-relaxation limit towards drift-diffusion equations with non-constant doping *** carry out the global-in-time convergence analysis by establishing uniform estimates of solutions near nonconstant equilibrium regarding the relaxation parameter and passing to the limit by using classical compactness ***,we generalize the stream function method to the non-constant equilibrium case,and together with the anti-symmetric structure of the error system and an induction argument,we establish globalin-time error estimates between smooth solutions to the Euler-Maxwell system and those to the drift-diffusion system,which are bounded by some power of the relaxation parameter.

关键词： global convergence rate Euler-Maxwell system Euler-Poisson system non-constant equilibrium state zero-relaxation limit

XNet-Enhanced Deep BSDE Method and Numerical Analysis

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arXiv 2025年

作者： Zheng, Xiaotao Xia, Zhihong Li, Xin Yue, Xingye Center for Financial Engineering Soochow University Jiangsu Suzhou215008 China School of Natural Science Great Bay University Guangdong Dongguan523808 China Department of Mathematics Northwestern University EvanstonIL60208 United States Department of Computer Science Northwestern University EvanstonIL60208 United States Mathematical Modelling and Data Analytics Center Oxford Suzhou Centre for Advanced Research Jiangsu Suzhou215123 China

Solving high-dimensional semilinear parabolic partial differential equations (PDEs) challenges traditional numerical methods due to the"curse of dimensionality." Deep learning, particularly through the Deep BSDE method, offers a promising alternative by leveraging neural networks’ capability to approximate high-dimensional functions. This paper introduces a novel network architecture, XNet, which significantly enhances the computational efficiency and accuracy of the Deep BSDE method. XNet demonstrates superior approximation capabilities with fewer parameters, addressing the trade-off between approximation and optimization errors found in existing methods. We detail the implementation of XNet within the Deep BSDE framework and present results that show marked improvements in solving high-dimensional PDEs, potentially setting a new standard for such computations. © 2025, CC BY.

关键词： Numerical methods

Compositional data analysis for modeling and forecasting mortality with the α-transformation

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arXiv 2025年

作者： Lim, Han Ying Pathmanathan, Dharini Dabo-Niang, Sophie Institute of Mathematical Sciences Faculty of Science Universiti Malaya Kuala Lumpur50603 Malaysia Universiti Malaya Centre for Data Analytics Universiti Malaya Kuala Lumpur50603 Malaysia Center of Research for Statistical Modelling and Methodology Faculty of Science Universiti Malaya Kuala Lumpur50603 Malaysia UMR8524-Laboratoire Paul Painlevé Inria-MODAL University of Lille CNRS Lille59000 France CNRS-Université de Montréal CRM-CNRS Montréal Canada

Mortality forecasting is crucial for demographic planning and actuarial studies, particularly for predicting population ageing rates and future longevity risks. Traditional approaches largely rely on extrapolative methods, such as the Lee-Carter model and its variants which use mortality rates as inputs. In recent years, compositional data analysis (CoDA), which adheres to summability and non-negativity constraints, has gained increasing attention from researchers for its application in mortality forecasting. This study explores the use of the α-transformation as an alternative to the commonly applied centered log-ratio (CLR) transformation for converting compositional data from the Aitchison simplex to unconstrained real space. The α-transformation offers greater flexibility through the inclusion of the α parameter, enabling better adaptation to the underlying data structure and handling of zero values, which are the limitations inherent to the CLR transformation. Using age-specific life table death counts for males and females in 31 selected European countries/regions from 1983 to 2018, the proposed method demonstrates comparable performance to the CLR transformation in most countries with improved forecast accuracy in some cases. These findings highlight the potential of the α-transformation as a competitive alternative transformation technique for real-world mortality data within a non-functional CoDA framework. Copyright © 2025, The Authors. All rights reserved.

关键词： Prediction models

Approximations to Isentropic Planar Magneto-Hydrodynamics Equations by Relaxed Euler-Type Systems

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Chinese Annals of Mathematics,Series B 2024年第3期45卷 413-440页

作者： Yachun LI Zhaoyang SHANG Chenmu WANG Liang ZHAO School of Mathematical Sciences CMA-ShanghaiMOE-LSC and SHL-MACShanghai Jiao Tong UniversityShanghai 200240China School of Finance Shanghai Lixin University of Accounting and FinanceShanghai 201209China School of Mathematical Sciences Shanghai Jiao Tong UniversityShanghai 200240China School of Mathematical Sciences Harbin Engineering UniversityHarbin 150001China Mathematical Modelling&Data Analytics Center Oxford Suzhou Centre for Advanced ResearchSuzhou 215123JiangsuChina

In this paper, the authors consider an approximation to the isentropic planar Magneto-hydrodynamics(MHD for short) equations by a kind of relaxed Euler-type system. The approximation is based on the generalization of the Maxwell law for nonNewtonian fluids together with the Maxwell correction for the Ampe`re law, hence the approximate system becomes a first-order quasilinear symmetrizable hyperbolic systems with partial dissipation. They establish the global-in-time smooth solutions to the approximate Euler-type equations in a small neighbourhood of constant equilibrium states and obtain the global-in-time convergence towards the isentropic planar MHD equations. In addition, they also establish the global-in-time error estimates of the limit based on stream function techniques and energy estimates for error variables.

关键词： Planar MHD equations Relaxation limits Global convergence Stream function

APPROXIMATIONS OF EULER-MAXWELL SYSTEMS BY DRIFT-DIFFUSION EQUATIONS THROUGH ZERO-RELAXATION LIMITS NEAR NON-CONSTANT EQUILIBRIUM

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arXiv 2023年

作者： Jin, Rui Li, Yachun Zhao, Liang School of Mathematical Sciences Shanghai Jiao Tong University Shanghai200240 China School of Mathematical Sciences CMA-Shanghai MOE-LSC SHL-MAC Shanghai Jiao Tong University Shanghai200240 China Mathematical Modelling & Data Analytics Center Oxford Suzhou Centre for Advanced Research Suzhou215123 China

Due to extreme difficulties in numerical simulations of Euler-Maxwell equations, which are caused by the highly complicated structures of the equations, this paper concerns the simplification of Euler-Maxwell system through the zero-relaxation limit towards the drift-diffusion equations with non-constant doping functions. We carry out the global-in-time convergence analysis by establishing uniform estimates of solutions near non-constant equilibrium regarding the relaxation parameter and passing to the limit by using classical compactness arguments. Furthermore, stream function methods are carefully generalized to the non-constant equilibrium case, with which as well as the anti-symmetric structure of the error system and an induction argument, we establish global-in-time error estimates between smooth solutions to the Euler-Maxwell system and those to drift-diffusion system, which are bounded by some power of relaxation *** Codes 35B25, 35L45, 35Q60, 35K45 Copyright © 2023, The Authors. All rights reserved.

关键词： Maxwell equations

Global Convergence and Error Estimates in Infinity-ion-mass Limits for Bipolar Euler-Poisson System

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arXiv 2022年

作者： Li, Yachun Wang, Shihao Zhao, Liang School of Mathematical Sciences CMA-Shanghai MOE-LSC SHL-MAC Shanghai Jiao Tong University Shanghai200240 China School of Mathematical Sciences Shanghai Jiao Tong University Shanghai200240 China Department of Mathematics University of Wisconsin-Madison Madison 53706 United States Mathematical Modelling & Data Analytics Center Oxford Suzhou Centre for Advanced Research Suzhou215123 China

This paper is concerned with the global-in-time convergence from bipolar Euler–Poisson system (BEP) to unipolar one (UEP) through the infinity-ion-mass limit by letting the ratio of the mass of ion mi over that of electron me goes to infinity. The global convergence of the limit is obtained for smooth solutions sufficiently close to constant equilibrium states. Furthermore, by applying the stream function method and taking advantage of the anti-symmetric structure of the error system, one obtains the corresponding global-in-time error estimates between smooth solutions of (BEP) and (UEP). It is worth mentioning that due to the strong coupling through the Poisson equation in bipolar system, stream functions for ions and electrons equations should be constructed separately based on asymptotic expansions of solutions, which is very different from the case of unipolar *** Codes 35B25, 35L45, 35L60, 35Q35, 35Q60 © 2022, CC BY.

关键词： Ions

Model Comparisons: XNet Outperforms KAN

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arXiv 2024年

作者： Li, Xin Xia, Zhihong Zheng, Xiaotao Department of Computer Science Northwestern University EvanstonIL United States Mathematical Modelling and Data Analytics Center Oxford Suzhou Centre for Advanced Research Suzhou China School of Natural Science Great Bay University Guangdong China Department of Mathematics Northwestern University EvanstonIL United States Department of Mathematics Soochow University Suzhou China

In the fields of computational mathematics and artificial intelligence, the need for precise data modeling is crucial, especially for predictive machine learning tasks. This paper explores further XNet, a novel algorithm that employs the complex-valued Cauchy integral formula, offering a superior network architecture that surpasses traditional Multi-Layer Perceptrons (MLPs) and Kolmogorov-Arnold Networks (KANs). XNet significant improves speed and accuracy across various tasks in both low and high-dimensional spaces, redefining the scope of data-driven model development and providing substantial improvements over established time series models like LSTMs. © 2024, CC BY.

关键词： Choquet integral

Cauchy activation function and XNet

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arXiv 2024年

作者： Li, Xin Xia, Zhihong Zhang, Hongkun Department of Computer Science Northwestern University EvanstonIL United States Mathematical Modelling and Data Analytics Center Oxford Suzhou Centre for Advanced Research Suzhou China School of Natural Science Great Bay University Guangdong China Department of Mathematics Northwestern University EvanstonIL United States Department of Mathematics and Statistics University of Massachusetts Amherst AmherstMA United States

We have developed a novel activation function, named the Cauchy Activation Function. This function is derived from the Cauchy Integral Theorem in complex analysis and is specifically tailored for problems requiring high precision. This innovation has led to the creation of a new class of neural networks, which we call (Comple)XNet, or simply XNet. We will demonstrate that XNet is particularly effective for high-dimensional challenges such as image classification and solving Partial Differential Equations (PDEs). Our evaluations show that XNet significantly outperforms established benchmarks like MNIST and CIFAR-10 in computer vision, and offers substantial advantages over Physics-Informed Neural Networks (PINNs) in both low-dimensional and high-dimensional PDE scenarios. © 2024, CC BY.

关键词： Image classification

Forecasting mortality rates with functional signatures

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arXiv 2024年

作者： Jing, Yap Zhong Pathmanathan, Dharini Dabo-Niang, Sophie Institute of Mathematical Sciences Faculty of Science Universiti Malaya Kuala Lumpur50603 Malaysia Universiti Malaya Centre for Data Analytics Universiti Malaya Kuala Lumpur50603 Malaysia Center of Research for Statistical Modelling and Methodology Faculty of Science Universiti Malaya Kuala Lumpur50603 Malaysia CNRS UMR 8524-Laboratoire Paul Painlevé INRIA-MODAL Université Lille LilleF-59000 France CNRS Université de Montréal CRM—CNRS Montréal Canada

This study introduces an innovative methodology for mortality forecasting, which integrates signature-based methods within the functional data framework of the Hyndman-Ullah (HU) model. This new approach, termed the Hyndman-Ullah with truncated signatures (HUts) model, aims to enhance the accuracy and robustness of mortality predictions. By utilizing signature regression, the HUts model is able to capture complex, nonlinear dependencies in mortality data which enhances forecasting accuracy across various demographic conditions. The model is applied to mortality data from 12 countries, comparing its forecasting performance against variants of the HU models across multiple forecast horizons. Our findings indicate that overall the HUts model not only provides more precise point forecasts but also shows robustness against data irregularities, such as those observed in countries with historical outliers. The integration of signature-based methods enables the HUts model to capture complex patterns in mortality data, making it a powerful tool for actuaries and demographers. Prediction intervals are also constructed with bootstrapping methods. Copyright © 2024, The Authors. All rights reserved.

关键词： Prediction models

Best practices for estimating and reporting epidemiological delay distributions of infectious diseases using public health surveillance and healthcare data

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arXiv 2024年

作者： Charniga, Kelly Park, Sang Woo Akhmetzhanov, Andrei R. Cori, Anne Dushoff, Jonathan Funk, Sebastian Gostic, Katelyn M. Linton, Natalie M. Lison, Adrian Overton, Christopher E. Pulliam, Juliet R.C. Ward, Thomas Cauchemez, Simon Abbott, Sam Mathematical Modelling of Infectious Diseases Unit Institut Pasteur Université Paris Cité CNRS UMR 2000 Paris France Department of Ecology and Evolutionary Biology Princeton University PrincetonNJ United States College of Public Health National Taiwan University Taipei Taiwan MRC Centre for Global Infectious Disease Analysis School of Public Health Imperial College London London United Kingdom Departments of Mathematics & Statistics and Biology McMaster University HamiltonON Canada Department of Biology McMaster University HamiltonON Canada M. G. DeGroote Institute for Infectious Disease Research McMaster University HamiltonON Canada Department of Infectious Disease Epidemiology and Dynamics London School of Hygiene & Tropical Medicine London United Kingdom Centre for Mathematical Modelling of Infectious Diseases London School of Hygiene & Tropical Medicine London United Kingdom Center for Forecasting and Outbreak Analytics U.S. Centers for Disease Control and Prevention AtlantaGA United States Graduate School of Medicine Hokkaido University Sapporo-shi Hokkaido Japan Department of Biosystems Science and Engineering ETH Zurich Zurich Switzerland Department of Mathematical Sciences University of Liverpool Liverpool United Kingdom All Hazards Intelligence Infectious Disease Modelling Team Data Analytics and Surveillance UK Health Security Agency United Kingdom Department of Mathematics University of Manchester Manchester United Kingdom

Epidemiological delays, such as incubation periods, serial intervals, and hospital lengths of stay, are among key quantities in infectious disease epidemiology that inform public health policy and clinical practice. This information is used to inform mathematical and statistical models, which in turn can inform control strategies. There are three main challenges that make delay distributions difficult to estimate. First, the data are commonly censored (e.g., symptom onset may only be reported by date instead of the exact time of day). Second, delays are often right truncated when being estimated in real time (not all events that have occurred have been observed yet). Third, during a rapidly growing or declining outbreak, overrepresentation or underrepresentation, respectively, of recently infected cases in the data can lead to bias in estimates. Studies that estimate delays rarely address all these factors and sometimes report several estimates using different combinations of adjustments, which can lead to conflicting answers and confusion about which estimates are most accurate. In this work, we formulate a checklist of best practices for estimating and reporting epidemiological delays with a focus on the incubation period and serial interval. We also propose strategies for handling common biases and identify areas where more work is needed. Our recommendations can help improve the robustness and utility of reported estimates and provide guidance for the evaluation of estimates for downstream use in transmission models or other analyses. Copyright © 2024, The Authors. All rights reserved.

关键词： Diseases