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Researcher profile

Chunmei Wang

Chunmei Wang contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
math.NANumerical AnalysisMachine LearningComputer Vision

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Published work

13 published item(s)

preprint2026arXiv

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.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

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.

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preprint2026arXiv

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.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

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.

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preprint2022arXiv

A parallel iterative procedure for weak Galerkin methods for second order elliptic problems

A parallelizable iterative procedure based on domain decomposition is presented and analyzed for weak Galerkin finite element methods for second order elliptic equations. The convergence analysis is established for the decomposition of the domain into individual elements associated to the weak Galerkin methods or into larger subdomains. A series of numerical tests are illustrated to verify the theory developed in this paper.

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preprint2022arXiv

An $L^p$- Primal-Dual Weak Galerkin method for div-curl Systems

This paper presents a new $L^p$-primal-dual weak Galerkin (PDWG) finite element method for the div-curl system with the normal boundary condition for $p>1$. Two crucial features for the proposed $L^p$-PDWG finite element scheme are as follows: (1) it offers an accurate and reliable numerical solution to the div-curl system under the low $W^{α, p}$-regularity ($α>0$) assumption for the exact solution; (2) it offers an effective approximation of the normal harmonic vector fields on domains with complex topology. An optimal order error estimate is established in the $L^q$-norm for the primal variable where $\frac{1}{p}+\frac{1}{q}=1$. A series of numerical experiments are presented to demonstrate the performance of the proposed $L^p$-PDWG algorithm.

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preprint2022arXiv

Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning

This paper proposes Friedrichs learning as a novel deep learning methodology that can learn the weak solutions of PDEs via a minmax formulation, which transforms the PDE problem into a minimax optimization problem to identify weak solutions. The name "Friedrichs learning" is for highlighting the close relationship between our learning strategy and Friedrichs theory on symmetric systems of PDEs. The weak solution and the test function in the weak formulation are parameterized as deep neural networks in a mesh-free manner, which are alternately updated to approach the optimal solution networks approximating the weak solution and the optimal test function, respectively. Extensive numerical results indicate that our mesh-free method can provide reasonably good solutions to a wide range of PDEs defined on regular and irregular domains in various dimensions, where classical numerical methods such as finite difference methods and finite element methods may be tedious or difficult to be applied.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

A New Numerical Method for Div-Curl Systems with Low Regularity Assumptions

This paper presents a numerical method for div-curl systems with normal boundary conditions by using a finite element technique known as primal-dual weak Galerkin (PDWG). The PDWG finite element scheme for the div-curl system has two prominent features in that it offers not only an accurate and reliable numerical solution to the div-curl system under the low $H^α$-regularity ($α>0$) assumption for the true solution, but also an effective approximation of normal harmonic vector fields regardless the topology of the domain. Results of seven numerical experiments are presented to demonstrate the performance of the PDWG algorithm, including one example on the computation of discrete normal harmonic vector fields.

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preprint2021arXiv

Reproducing Activation Function for Deep Learning

We propose reproducing activation functions (RAFs) to improve deep learning accuracy for various applications ranging from computer vision to scientific computing. The idea is to employ several basic functions and their learnable linear combination to construct neuron-wise data-driven activation functions for each neuron. Armed with RAFs, neural networks (NNs) can reproduce traditional approximation tools and, therefore, approximate target functions with a smaller number of parameters than traditional NNs. In NN training, RAFs can generate neural tangent kernels (NTKs) with a better condition number than traditional activation functions lessening the spectral bias of deep learning. As demonstrated by extensive numerical tests, the proposed RAFs can facilitate the convergence of deep learning optimization for a solution with higher accuracy than existing deep learning solvers for audio/image/video reconstruction, PDEs, and eigenvalue problems. With RAFs, the errors of audio/video reconstruction, PDEs, and eigenvalue problems are decreased by over 14%, 73%, 99%, respectively, compared with baseline, while the performance of image reconstruction increases by 58%.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A simplified primal-dual weak Galerkin finite element method for Fokker-Planck type equations

A simplified primal-dual weak Galerkin (S-PDWG) finite element method is designed for the Fokker-Planck type equation with non-smooth diffusion tensor and drift vector. The discrete system resulting from S-PDWG method has significantly fewer degrees of freedom compared with the one resulting from the PDWG method proposed by Wang-Wang \cite{WW-fp-2018}. Furthermore, the condition number of the S-PDWG method is smaller than the PDWG method \cite{WW-fp-2018} due to the introduction of a new stabilizer, which provides a potential for designing fast algorithms. Optimal order error estimates for the S-PDWG approximation are established in the $L^2$ norm. A series of numerical results are demonstrated to validate the effectiveness of the S-PDWG method.

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preprint2020arXiv

De Rham Complexes for Weak Galerkin Finite Element Spaces

Two de Rham complex sequences of the finite element spaces are introduced for weak finite element functions and weak derivatives developed in the weak Galerkin (WG) finite element methods on general polyhedral elements. One of the sequences uses polynomials of equal order for all the finite element spaces involved in the sequence and the other one uses polynomials of naturally decending orders. It is shown that the diagrams in both de Rham complexes commute for general polyhedral elements. The exactness of one of the complexes is established for the lowest order element.

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preprint2020arXiv

New Primal-Dual Weak Galerkin Finite Element Methods for Convection-Diffusion Problems

This article devises a new primal-dual weak Galerkin finite element method for the convection-diffusion equation. Optimal order error estimates are established for the primal-dual weak Galerkin approximations in various discrete norms and the standard $L^2$ norms. A series of numerical experiments are conducted and reported to verify the theoretical findings.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

A primal-dual finite element method for first-order transport problems

This article devises a new numerical method for first-order transport problems by using the primal-dual weak Galerkin (PD-WG) finite element method recently developed in scientific computing. The PD-WG method is based on a variational formulation of the modeling equation for which the differential operator is applied to the test function so that low regularity for the exact solution of the original equation is sufficient for computation. The PD-WG finite element method indeed yields a symmetric system involving both the original equation for the primal variable and its dual for the dual variable (also known as Lagrangian multiplier). For the linear transport problem, it is shown that the PD-WG method offers numerical solutions that conserve mass locally on each element. Optimal order error estimates in various norms are derived for the numerical solutions arising from the PD-WG method with weak regularity assumptions on the modelling equations. A variety of numerical results are presented to demonstrate the accuracy and stability of the new method.

0 citations0 reviewsNo code linkedOpen access