arXiv subjects

Fugui Ma

Publications and source records attributed to Fugui Ma.

6 recordsLinked to original sources

arXiv2026

iSMART: An Iterative Sampling-and-Regression Technique for Solving Martingale-Based PDEs

We propose the {\bf i}terative {\bf S}a{\bf M}pling-{\bf A}nd-{\bf R}egression {\bf T}echnique (iSMART) for high-dimensional martingale-based partial differential equations (PDEs) in this paper. By leveraging the $L^2$-projection property of conditional expectation and adopting the stop-gradient technique, iSMART reformulates the continuous martingale condition derived from PDEs into a sequence of tractable sampling-regression problems within an iterative framework. This approach relies solely on standard SDE path simulation and plain squared-error loss minimization, completely bypassing the need for adversarial optimization or nested expectation estimation in previous methods. iSMART accommodates linear, semi-linear, and fully nonlinear martingale-based PDEs within a unified iterative procedure. In particular, for fully nonlinear Hamilton-Jacobi-Bellman (HJB) equations, a freezing-and-compensating technique is introduced to strategically shift a portion of the nonlinearity into the SDE drift, thereby improving the convergence behavior of the iterations. Numerous numerical experiments on linear reaction-diffusion equations with sharp gradients, semilinear Burgers-type equations, and fully nonlinear HJB equations demonstrate the accuracy, efficiency, and robustness of the proposed approach in various high dimensions.

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arXiv2026

Dynamics of Chemotactic Gliding-Aggregation in Myxobacteria on Bounded Domains: Stochastic Modeling, Analysis, and Deep Neural Network Simulations

Bacterial chemotactic movement and collective aggregation have long attracted substantial interest in mathematical biology and applied modeling. Classical Keller--Segel-type systems, however, are typically formulated under idealized laboratory assumptions, such as smooth agar substrates, and thus cannot adequately capture the gliding dynamics of myxobacteria in naturally rough environments like soil. In this paper, we propose a unified framework that integrates stochastic modeling, rigorous analysis, and deep neural network-based simulation of chemotactic gliding--diffusion and aggregation processes on bounded domains. Starting from a lattice-based discrete agent description and a subordinated Langevin equation driven by an inverse stable subordinator at the microscopic level, we characterize anomalous gliding dynamics on rough surfaces and derive a macroscopic time-nonlocal Keller--Segel-type chemotaxis model with logarithmic sensitivity. We then establish a comprehensive solution theory for the resulting model, covering mass conservation, novel regularity results, local well-posedness in any spatial dimension, and global well-posedness in two and three. The analysis relies on several newly developed ingredients, including a fractional Lyapunov functional, a variational inequality adapted to the time-nonlocal structure, logarithmic Sobolev-type estimates, Bregman distance techniques, and a weighted bootstrap mechanism adapted to the singular sensitivity and time-nonlocal memory. Finally, we design a mesh-free, positivity-preserving, multi-objective, time-marching physics-informed neural network method with separate architectures and tailored variable transformations. Numerical experiments on complex geometries, including a butterfly-shaped domain, demonstrate the robustness, accuracy, and flexibility of the proposed computational framework across a range of Keller--Segel-type systems.

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arXiv2025

Scaling crossover of the generalized Jeffreys-type law

The generalized Jeffreys-type law is formulated as a multi-term time-fractional Jeffreys-type equation, whose dynamics exhibit rich scaling crossover phenomena entailing different diffusion mechanisms. In this work, we provide a novel physical explanation for the equation from first principles, beginning with a microscopic description based on the continuous-time random walk framework with a generalized waiting time distribution and further deriving the equation from an overdamped Langevin equation subject to a stochastic time-change (subordination). Employing the Laplace transform method, we conduct a rigorous analysis of the equation, establishing its well-posedness and providing a detailed Sobolev regularity analysis. We also develop a novel numerical scheme, termed the CIM-CLG algorithm, which achieves spectral accuracy in both time and space while substantially relaxing the temporal regularity requirements on the solution. The algorithm reduces the computational complexity to $\mathcal{O}(N)$ in time and $\mathcal{O}(M\log M)$ in space and is fully parallelizable. Detailed implementation guidelines and new technical error estimates are provided. Extensive numerical experiments in 1D and 2D settings validate the efficiency, robustness, and accuracy of the proposed method. By integrating stochastic modeling, mathematical analysis, and numerical computation, this work advances the understanding of the generalized Jeffreys-type law and offers a mathematically rigorous and computationally efficient framework for tackling complex nonlocal problems.

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arXiv2024

Mathematical modeling and analysis for the chemotactic diffusion in porous media with incompressible Navier-Stokes equations over bounded domain

Myxobacteria aggregate and generate fruiting bodies in the soil to survive under starvation conditions. Considering soil as a porous medium, the biological mechanism and dynamic behavior of myxobacteria and slime (chemoattractants) affected by favorable environments in the soil can not be well characterized by the classical full parabolic Keller-Segel system combined with the incompressible Navier-Stokes equations. In this work, we employ the continuous time random walk (CTRW) approach to characterize the diffusion behavior of myxobacteria and slime in porous media at the microscale, and develop a new macroscopic model named as the time-fractional Keller-Segel system. Then it is coupled with the incompressible Navier-Stokes equations through transport and buoyancy, resulting in the TF-KSNS system, which reveals the biological mechanism from micro to macro and then appropriately describes the dynamic behavior of the chemotactic diffusion of myxobacteria and slime in the soil. In addition, we demonstrate that the TF-KSNS system associated with initial and no-flux/no-flux/Dirichlet boundary conditions over smoothly bounded domain in $\mathbb{R}^{d}$ ($d\geq2$) admits a local well-posed mild solution, which continuously depends on the initial data with proper regularity under a small initial condition. Moreover, the blow-up of the mild solution is rigorously investigated.

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arXiv2023

The Contour integral method for Feynman-Kac equation with two internal states

We develop the contour integral method for numerically solving the Feynman-Kac equation with two internal states [P. B. Xu and W. H. Deng, Math. Model. Nat. Phenom., 13 (2018), 10], describing the functional distribution of particle's internal states. The striking benefits are obtained, including spectral accuracy, low computational complexity, small memory requirement, etc. We perform the error estimates and stability analyses, which are confirmed by numerical experiments.

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arXiv2022

Analyses of the contour integral method for time fractional subdiffusion-normal transport equation

In this work, we theoretically and numerically discuss the time fractional subdiffusion-normal transport equation, which depicts a crossover from sub-diffusion (as $t\rightarrow 0$) to normal diffusion (as $t\rightarrow \infty$). Firstly, the well-posedness and regularities of the model are studied by using the bivariate Mittag-Leffler function. Theoretical results show that after introducing the first-order derivative operator, the regularity of the solution can be improved in substance. Then, a numerical scheme with high-precision is developed no matter the initial value is smooth or non-smooth. More specifically, we use the contour integral method (CIM) with parameterized hyperbolic contour to approximate the temporal local and non-local operators, and employ the standard Galerkin finite element method for spacial discretization. Rigorous error estimates show that the proposed numerical scheme has spectral accuracy in time and optimal convergence order in space. Besides, we further improve the algorithm and reduce the computational cost by using the barycentric Lagrange interpolation. Finally, the obtained theoretical results as well as the acceleration algorithm are verified by several 1-D and 2-D numerical experiments, which also show that the numerical scheme developed in this paper is effective and robust.

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