arXiv subjects

Chunmei Wang

Publications and source records attributed to Chunmei Wang.

At least 19 recordsLinked to original sources

arXiv2026

A Least-Squares Weak Galerkin Method for the Biharmonic Cauchy Problem

We develop a least-squares weak Galerkin (LS-WG) finite element method for the Cauchy problem of the biharmonic equation. The proposed approach reformulates the fourth-order equation as a coupled system of two second-order equations, which are discretized using discrete weak Laplacian operators on weak finite element spaces. The resulting least-squares formulation yields a symmetric positive definite linear system, thereby eliminating the discrete inf-sup condition required by mixed finite element methods while avoiding the construction of globally $C^1$-conforming finite element spaces. Furthermore, the weak Galerkin framework naturally accommodates general polygonal meshes, offering considerable flexibility in mesh generation and approximation. Under the assumption that the continuous biharmonic Cauchy problem admits a unique solution, we establish the uniqueness of the discrete LS-WG solution and derive optimal-order error estimates in a discrete energy norm. Numerical experiments confirm the theoretical convergence rates and demonstrate the accuracy, robustness, and effectiveness of the proposed method.

math.NA↗

arXiv2026

Persistent Manifold Learning of Protein Properties

Predicting how tightly two biomolecules bind remains a major challenge, in part because different interaction classes present dissimilar interfaces, from compact metal-coordinated pockets to broad, featureless protein surfaces. We introduce persistent manifold learning (PML), a novel computational framework that describes a binding interface as a family of multiscale manifolds. Boundary-Induced Graph Laplacian, a discrete realization of de Rham-Hodge theory, then extracts topological invariants together with nonharmonic spectral information, capturing the geometry of an interface as well as its topology. These manifold embeddings are combined with protein and molecular language model representations and paired with gradient boosting decision trees. Our PML outperforms state-of-the-art methods on metalloprotein-ligand and protein-protein benchmarks.

q-bio.BM↗

arXiv2026

A Least Squares Weak Galerkin Framework for Linear Elasticity on Polytopal Meshes

This paper develops and analyzes a least-squares weak Galerkin (LS-WG) finite element method for linear elasticity. By employing weak differential operators, specifically the weak gradient, weak strain tensor, and weak divergence, defined on weak finite element spaces, the proposed framework facilitates the treatment of complex boundary conditions and internal interfaces while avoiding the restrictive discrete inf-sup condition. The resulting formulation is symmetric and positive definite and exhibits robust numerical performance in the nearly incompressible regime. In addition, the proposed method offers exceptional geometric flexibility, allowing implementation on general polytopal (polygonal and polyhedral) meshes. We establish the uniqueness of the numerical solution and derive optimal-order error estimates with respect to a tailored discrete energy norm. Extensive numerical experiments confirm the theoretical convergence rates and demonstrate the method's stability, efficiency, and locking-free performance for nearly incompressible materials.

math.NA↗

arXiv2026

A Least Squares Weak Galerkin Finite Element Method for Fokker-Planck Type Equations

This paper presents a least squares weak Galerkin (LS-WG) finite element method for a class of second order elliptic equations of Fokker-Planck type. To address the numerical challenges arising from non-smooth diffusion tensors, the proposed method utilizes a least-squares formulation that yields a symmetric positive definite (SPD) discrete system. The numerical scheme is designed by employing locally constructed weak second order partial derivatives and the weak divergence commonly used within the weak Galerkin framework. A rigorous theoretical foundation is provided, establishing the uniqueness of the discrete solution and deriving optimal-order error estimates in a discrete energy norm. Finally, extensive numerical experiments are reported to validate the theoretical findings and demonstrate the robustness and performance of the numerical scheme.

math.NA↗

arXiv2026

Solving the Stokes Equations via a Least Squares Weak Galerkin Method

We present a least-squares weak Galerkin (LS-WG) finite element method for solving the Stokes equations on arbitrary polygonal and polyhedral meshes. By utilizing discrete weak derivatives on discontinuous polynomial spaces, the proposed framework naturally accommodates complex domain geometries and general partitions. Crucially, this least-squares formulation bypasses the traditional inf-sup (LBB) compatibility condition, transforming the standard indefinite saddle-point problem into an inherently symmetric and positive definite (SPD) discrete linear system. We establish the well-posedness of the numerical scheme and rigorously derive optimal-order error estimates in a custom discrete energy norm. Specifically, we prove convergence rates of $\mathcal{O}(h^k)$ for the discrete projection error and $\mathcal{O}(h^{k-1})$ for the global approximation error when employing polynomials of degree $k \ge 1$ for the velocity field and $k-1$ for the pressure. Extensive numerical experiments confirm these theoretical convergence rates, demonstrating the method's robustness, geometric flexibility, and overall efficiency.

math.NA↗

arXiv2026

A Least-Squares Weak Galerkin Finite Element Scheme for Cauchy Problems in Helmholtz

This paper introduces and rigorously analyzes a least-squares weak Galerkin (LS-WG) finite element method for the severely ill-posed Cauchy problem associated with the Helmholtz equation. By utilizing a weak Laplacian operator defined on a space of discontinuous functions, the proposed framework facilitates the seamless treatment of complex boundary conditions and internal interfaces. We emphasize the geometric flexibility of the LS-WG scheme on general polygonal and polyhedral partitions. Furthermore, we prove the uniqueness of the numerical solution and derive optimal-order error estimates with respect to a specifically designed discrete energy norm. Extensive numerical experiments validate the theoretical convergence rates and demonstrate the algorithm's robustness and efficiency over traditional Galerkin approaches.

math.NA↗

arXiv2026

A Least-Squares Weak Galerkin Finite Element Scheme for Cauchy Problems in Convection--Diffusion

We introduce and rigorously analyze a least-squares weak Galerkin (LS-WG) finite element method for the severely ill-posed Cauchy problem of convection--diffusion equations. The proposed framework utilizes weak derivatives defined on a class of discontinuous weak functions, enabling the natural treatment of complex boundary conditions and internal interfaces. A key advantage of the least-squares formulation is that it transforms the underlying non-self-adjoint operator into a discrete linear system that is inherently symmetric and positive definite (SPD). We demonstrate the geometric flexibility of the method on arbitrary polygonal and polyhedral partitions. Furthermore, we establish the uniqueness of the numerical solution and derive optimal-order error estimates in a carefully defined discrete energy norm. Extensive numerical tests are presented to confirm the theoretical convergence rates and highlight the algorithm's robustness and efficiency compared to standard Galerkin approaches.

math.NA↗

arXiv2026

A Least-Squares Weak Galerkin Method for Second-Order Elliptic Equations in Non-Divergence Form

This article proposes a novel least-squares weak Galerkin (LS-WG) method for second-order elliptic equations in non-divergence form. The approach leverages a locally defined discrete weak Hessian operator constructed within the weak Galerkin framework. A key feature of the resulting algorithm is that it yields a symmetric and positive definite linear system while remaining applicable to general polygonal and polyhedral meshes. We establish optimal-order error estimates for the approximation in a discrete $H^2$-equivalent norm. Finally, comprehensive numerical experiments are presented to validate the theoretical analysis and demonstrate the efficiency and robustness of the method.

math.NA↗

arXiv2026

The finite expression method for turbulent dynamics with high-order moment recovery

Turbulent dynamical systems are characterized by nonlinear interactions and stochastic effects that generate coupled statistical quantities, such as non-zero higher-order moments, which are difficult to capture from data with accuracy. We propose a two-stage data-driven modeling framework that combines symbolic regression with generative models to jointly identify the governing dynamics and predict their key statistical quantities. In Stage I of the framework, the Finite Expression Method (FEX) is adopted to discover closed-form expressions of the deterministic dynamics, recovering nonlinear interaction terms and external forcing without predefined libraries. In Stage II, generative models are introduced to learn the residual stochastic components as a refined correction to the model error from the Stage I approximation, enabling accurate characterization of higher-order statistics. Theoretical analysis establishes the consistency of the symbolic estimator and quantifies the estimation error in terms of data size and numerical discretization. The model performance is verified through detailed numerical experiments on the stochastic triad models across multiple regimes, demonstrating that the framework successfully recovers interaction terms and forcing expressions, and accurately predicts statistical moments up to order five. These results highlight the potential of integrating interpretable symbolic discovery with data-driven stochastic modeling for complex turbulent systems.

cs.LG↗

arXiv2026

A Neural-Enhanced Weak Galerkin Method for Second-Order Elliptic Problems with Low-Regularity Solutions

We propose a neural-enhanced weak Galerkin (WG) finite element method for second-order elliptic problems with low-regularity solutions. The method augments the classical WG approximation space with neural network functions constructed via a residual-driven Galerkin enrichment procedure. This approach preserves the variational structure, symmetry, and stability of the WG formulation while enhancing its ability to approximate non-smooth and singular solution components. We establish a quasi-optimal error estimate in a discrete WG energy norm, incorporating both projection and consistency errors. In particular, the method retains optimal convergence rates for smooth solutions. For problems admitting a regular--singular decomposition, we further show that the neural enrichment effectively captures the singular component, yielding improved accuracy over standard WG methods.

math.NA↗

arXiv2026

Auto-Stabilized Weak Galerkin Finite Element Methods for Biot's consolidation model on Non-Convex Polytopal Meshes

This paper presents an auto-stabilized weak Galerkin (WG) finite element method for the Biot's consolidation model within the classical displacement-pressure two-field formulation. Unlike traditional WG approaches, the proposed scheme achieves numerical stability without the requirement of traditional stabilizers. Spatial discretization is performed using weak Galerkin finite elements for both displacement and pressure approximations, while a backward Euler scheme is employed for temporal discretization to ensure a fully implicit and stable formulation. We establish the well-posedness of the resulting linear system at each time step and provide a rigorous error analysis, deriving optimal-order convergence. A significant merit of this WG scheme is its flexibility on general shape-regular polytopal meshes, including those with non-convex geometries. By utilizing bubble functions as a primary analytical tool, the method produces stable, oscillation-free pressure approximations without specialized treatment. Numerical experiments are presented to validate the theoretical convergence rates and demonstrate the computational efficiency and robustness of the auto-stabilized formulation.

math.NA↗

arXiv2026

Finite Expression Methods for Discovering Physical Laws from Data

Nonlinear dynamics is a pervasive phenomenon observed in scientific and engineering disciplines. However, the task of deriving analytical expressions to describe nonlinear dynamics from limited data remains challenging. In this paper, we shall present a novel deep symbolic learning method called the "finite expression method" (FEX) to discover governing equations within a function space containing a finite set of analytic expressions, based on observed dynamic data. The key concept is to employ FEX to generate analytical expressions of the governing equations by learning the derivatives of partial differential equation (PDE) solutions through convolutions. Our numerical results demonstrate that our FEX surpasses other existing methods (such as PDE-Net, SINDy, GP, and SPL) in terms of numerical performance across a range of problems, including time-dependent PDE problems and nonlinear dynamical systems with time-varying coefficients. Moreover, the results highlight FEX's flexibility and expressive power in accurately approximating symbolic governing equations.

cs.LG↗

arXiv2026

A Fast Algorithm for the Finite Expression Method in Learning Dynamics on Complex Networks

Complex network data is prevalent in various real-world domains, including physical, technological, and biological systems. Despite this prevalence, predicting trends and understanding behavioral patterns in complex systems remain challenging due to poorly understood underlying mechanisms. While data-driven methods have advanced in uncovering governing equations from time series data, efforts to extract physical laws from network data are limited and often struggle with incomplete or noisy data. Additionally, they suffer from computational costs on network data, making it difficult to scale to real-world networks. To address these challenges, we introduce a novel approach called the Finite Expression Method (FEX) and its fast algorithm for learning dynamics on complex networks. FEX represents dynamics on complex networks using binary trees composed of finite mathematical operators. The nodes within these trees are trained through a combinatorial optimization process guided by reinforcement learning techniques. This unique configuration allows FEX to capture complex dynamics with minimal prior knowledge of the system and a small dictionary of mathematical operators. We also integrate a fast, stochastic algorithm into FEX, reducing the computational complexity from $O(N^2)$ to $O(N)$. Our extensive numerical experiments demonstrate that FEX excels in accurately identifying dynamics across diverse network topologies and dynamic behaviors.

cs.SC↗

arXiv2026

Neural Correction Operator: A Reliable and Fast Approach for Electrical Impedance Tomography

Electrical Impedance Tomography (EIT) is a non-invasive medical imaging method that reconstructs electrical conductivity mediums from boundary voltage-current measurements, but its severe ill-posedness renders direct operator learning with neural networks unreliable. We propose the neural correction operator framework, which learns the inverse map as a composition of two operators: a reconstruction operator using L-BFGS optimization with limited iterations to obtain an initial estimate from measurement data and a correction operator implemented with deep learning models to reconstruct the true media from this initial guess. We explore convolutional neural network architectures and conditional diffusion models as alternative choices for the correction operator. We evaluate the neural correction operator by comparing with L-BFGS methods as well as neural operators and conditional diffusion models that directly learn the inverse map over several benchmark datasets. Our numerical experiments demonstrate that our approach achieves significantly better reconstruction quality compared to both iterative methods and direct neural operator learning methods with the same architecture. The proposed framework also exhibits robustness to measurement noise while achieving substantial computational speedup compared to conventional methods. The neural correction operator provides a general paradigm for approaching neural operator learning in severely ill-posed inverse problems.

math.NA↗

arXiv2026

A Simple Weak Galerkin Finite Element Method for Convection-Diffusion-Reaction Equations on Nonconvex Polytopal Meshes

This article introduces a simple weak Galerkin (WG) finite element method for solving convection-diffusion-reaction equation. The proposed method offers significant flexibility by supporting discontinuous approximating functions on general nonconvex polytopal meshes. We establish rigorous error estimates within a suitable norm. Finally, numerical experiments are presented to validate the theoretical convergence rates and demonstrate the computational efficiency of the approach.

math.NA↗

arXiv2026

A weak Galerkin least squares finite element method for linear convection equations in non-divergence form

This article develops a weak Galerkin least-squares (WG--LS) finite element method for first-order linear convection equations in non-divergence form. The method is formulated using discontinuous finite element functions and does not require any coercivity assumption on the convection vector or reaction coefficient. The resulting discrete problem leads to a symmetric and positive definite linear system and is applicable to general polygonal and polyhedral meshes. Under minimal regularity assumptions on the coefficients, optimal-order error estimates are established for the WG--LS approximation in a suitable energy norm. Numerical experiments are presented to confirm the theoretical convergence results and to demonstrate the accuracy and efficiency of the proposed method.

math.NA↗

arXiv2025

Weak Galerkin finite element methods for elliptic interface problems on nonconvex polygonal partitions

This paper proposes a weak Galerkin (WG) finite element method for elliptic interface problems defined on nonconvex polygonal partitions. The method features a built-in stabilizer and retains a simple, symmetric, and positive definite formulation. An optimal-order error estimate is rigorously derived in the discrete $H^1$ norm. Furthermore, a series of numerical experiments are provided to verify the theoretical results and to demonstrate the robustness and effectiveness of the proposed WG method for elliptic interface problems.

math.NA↗

arXiv2025

A Simple Weak Galerkin Finite Element Method for the Reissner-Mindlin Plate Model on Non-Convex Polytopal Meshes

This paper presents a simple weak Galerkin (WG) finite element method for the Reissner-Mindlin plate model that partially eliminates the need for traditionally employed stabilizers. The proposed approach accommodates general, including non-convex, polytopal meshes, thereby offering greater geometric flexibility. It utilizes bubble functions without imposing the restrictive conditions required by existing stabilizer-free WG methods, which simplifies implementation and broadens applicability to a wide range of partial differential equations (PDEs). Moreover, the method allows for flexible choices of polynomial degrees in the discretization and can be applied in any spatial dimension. We establish optimal-order error estimates for the WG approximation in a discrete H^1 norm, and present numerical experiments that validate the theoretical results.

math.NA↗