Xiaochuan Tian

I am an Associate Professor in the Department of Mathematics at UC San Diego. My research focuses on mathematical modeling, applied analysis, and numerical methods for partial differential and integro-differential equations, with a particular interest in nonlocal modeling and the emerging interface of machine learning and PDEs.

My current research topics include:

Calculus of variations, Theory of nonlocal function spaces, Asymptotic analysis, Γ-convergence
High order numerical methods, Meshfree methods, Petrov-Galerkin methods, Multiscale and Adaptive methods, Asymptotic compatible schemes
Modeling in continuum mechanics, complex physical and biological systems
Machine learning approaches for PDEs/IDEs, Reinforcement learning

My work has been recognized with the NSF CAREER award and the Alfred P. Sloan Fellowship. I am exicted to work with students who has a strong background in analysis and/or computational math.

Research

Mathematical Modeling and Applied Analysis

The study of PDE modeling and nonlocal modeling is a key area of interest, with a focus on developing mathematically well-posed models relevant to scientific and engineering disciplines. This field draws on a variety of mathematical techniques, including the calculus of variations, nonlocal vector calculus, function space theory, compactness techniques, variational methods and asymptotic analysis.

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An invitation to nonlocal models, slides.
Compactness results for a Dirichlet energy of nonlocal gradient with applications, Numer. Methods Partial Differ. Equ., 40, e23149, 2024.
Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications, Math. Models Methods Appl. Sci., 33(12),2507–2556, 2023.
Asymptotically compatible schemes for nonlinear variational models via Γ-convergence and applications to nonlocal problems, under review in Math. Comp..
Fractional Hardy-type and trace theorems for nonlocal function spaces with heterogeneous localizations, Anal. Appl., 20(3), 579-614, 2022.
Nonlocal trace spaces and extension results for nonlocal calculus, J. Funct. Anal., 282(12), 109453, 2022.

Numerical Methods for Partial/Integro-differential Equations

The design and analysis of innovative numerical methods for solving partial differential and integro-differential equations are central to our group's work, with applications spanning various scientific and engineering fields. The focus includes high-order numerical methods, meshfree approaches, Petrov-Galerkin methods, as well as multiscale and adaptive techniques, which offer significant advantages over traditional approaches. Special emphasis is placed on developing physics-preserving and robust methodologies that prevent inaccuracies or unphysical outcomes due to improper model or discretization choices. Additionally, theoretical analyses are pursued to further enhance the practical performance and functionality of these numerical methods.

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Numerical methods for nonlocal and fractional models, Acta Numerica, 29, 1-124, 2020.
Monotone meshfree methods for linear elliptic equations in non-divergence form via nonlocal relaxation , J. Sci. Comput., 96(3), 85, 2023.
Gaussian radial basis functions collocation for fractional PDEs: methodology and error analysis, preprint.
Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion, SIAM J. Numer. Anal., 59(1), 88-118, 2021.
Mathematics of Smoothed Particle Hydrodynamics: A Study via Nonlocal Stokes Equations , Found. Comput. Math., 20, 801-826, 2020.
The Double Adaptivity Paradigm (How to circumvent the discrete inf-sup conditions of Babuška and Brezzi) , Comput. Math. with Appl., 95, 41-66, 2021.
Asymptotically compatible schemes for robust discretization of parametrized problems with applications to nonlocal models, SIAM Rev., 62(1), 199–227, 2020.

Multiscale and Stochastic Modeling for Fracture and Beyond

Successful modeling of complex physical systems requires effective mathematical models and efficient model reduction techniques. Key areas of focus include local and nonlocal coupling, uncertainty quantification, and the development of well-posed and physically relevant nonlinear models.

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A meshfree peridynamic model for brittle fracture in randomly heterogeneous materials, Comput. Methods Appl. Mech. Eng., 399, 115340, 2022.
An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems, J. Comput. Phys., 111376, 2022.
A review of Local-to-Nonlocal coupling methods in nonlocal diffusion and nonlocal mechanics, J. Peridyn. Nonlocal Model., 4, 1–50, 2022.
Multiscale Modeling, homogenization and nonlocal effects: mathematical and computational issues , Contemporary Mathematics, 754, 115-140, 2020.
A peridynamic model of fracture mechanics with bond-breaking , J. Elast.,132 (2), 197-218, 2018.

Machine Learning meets PDEs

The growing influence of machine learning is reshaping how PDEs/IDEs are solved, while, conversely, PDE/IDE modeling offer valuable insights for advancing machine learning tasks. This interplay gives rise to two key areas of focus. First, how can machine learning approaches be designed and analyzed to solve PDEs/IDEs and their inverse problems in cases where traditional methods fall short? Second, how can PDE/IDE-based modeling contribute to the development of more robust machine learning algorithms? These efforts are inherently connected to classical subjects, including the calculus of variation, meshfree techniques, optimal control, probability, and stochastics. Additionally, adaptive strategies, sparse neural networks, and reinforcement learning present promising avenues for further exploration.

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A Model-Based Approach for Continuous-Time Policy Evaluation with Unknown Lévy Process Dynamics , slides.
A neural network kernel decomposition for learning multiple steady states in parameterized dynamical systems, under review in SIAM J. Numer. Anal..

Group

Zhaolong Han
PhD. Candidate
Research Interests: Nonlocal Vector Calculus, Bourgain-Bresiz-Mironescu type Compactness, Asymptotically Compatible Schemes, Petrov-Galerkin Methods, Optimal Control

Qihao Ye
PhD. Candidate
Research Interests: Meshfree Methods, Fast Algorithms, Data-driven Discovery of Stochastic Dynamics, Reinforcement Learning

Yimeng Zhang
PhD. Candidate
Research Interests: Machine Learning Approach in Parameterized Dynamical Systems, Biochemical Reaction Systems, Neural Network Approximation Theory

Teaching

"A unique human capacity, separating people from animals, is the ability to pass information and knowledge from one generation to the next. "

Fall 2024: MATH 272A Numerical Partial Differential Equations

Past courses