Chunmei Wang

Chunmei Wang, Junping Wang

In this paper, the authors devise a new discretization scheme for div-curl systems defined in connected domains with heterogeneous media by using the weak Galerkin finite element method. Two types of boundary value problems are considered in the algorithm development: (1) normal boundary condition, and (2) tangential boundary condition. A new variational formulation is developed for the normal boundary value problem by using the Helmholtz decomposition which avoids the computation of functions in the harmonic fields. Both boundary value problems are reduced to a general saddle-point problem involving the curl and divergence operators, for which the weak Galerkin finite element method is devised and analyzed. The novelty of the technique lies in the discretization of the divergence operator applied to vector fields with heterogeneous media. Error estimates of optimal order are established for the corresponding finite element approximations in various discrete Sobolev norms.

Chunmei Wang, Junping Wang

This paper presents a primal-dual weak Galerkin (PD-WG) finite element method for a class of second order elliptic equations of Fokker-Planck type. The method is based on a variational form where all the derivatives are applied to the test functions so that no regularity is necessary for the exact solution of the model equation. The numerical scheme is designed by using locally constructed weak second order partial derivatives and the weak gradient commonly used in the weak Galerkin context. Optimal order of convergence is derived for the resulting numerical solutions. Numerical results are reported to demonstrate the performance of the numerical scheme.

Chunmei Wang, Junping Wang

The authors propose and analyze a well-posed numerical scheme for a type of ill-posed elliptic Cauchy problem by using a constrained minimization approach combined with the weak Galerkin finite element method. The resulting Euler-Lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable, and is thus an example of the primal-dual weak Galerkin finite element method. This new primal-dual weak Galerkin algorithm is consistent in the sense that the system is symmetric, well-posed, and is satisfied by the exact solution. A certain stability and error estimates were derived in discrete Sobolev norms, including one in a weak $L^2$ topology. Some numerical results are reported to illustrate and validate the theory developed in the paper.

Dan Li, Chunmei Wang, Junping Wang

This article presents a superconvergence for the gradient approximation of the second order elliptic equation discretized by the weak Galerkin finite element methods on nonuniform rectangular partitions. The result shows a convergence of ${\cal O}(h^r)$, $1.5\leq r \leq 2$, for the numerical gradient obtained from the lowest order weak Galerkin element consisting of piecewise linear and constant functions. For this numerical scheme, the optimal order of error estimate is ${\cal O}(h)$ for the gradient approximation. The superconvergence reveals a superior performance of the weak Galerkin finite element methods. Some computational results are included to numerically validate the superconvergence theory.

Chunmei Wang

This paper introduces new discretization schemes for time-harmonic Maxwell equations in a connected domain by using the weak Galerkin (WG) finite element method. The corresponding WG algorithms are analyzed for their stability and convergence. Error estimates of optimal order in various discrete Sobolev norms are established for the resulting finite element approximations.

Ke Chen, Chunmei Wang, Haizhao Yang

Deep neural networks (DNNs) have achieved remarkable success in numerous domains, and their application to PDE-related problems has been rapidly advancing. This paper provides an estimate for the generalization error of learning Lipschitz operators over Banach spaces using DNNs with applications to various PDE solution operators. The goal is to specify DNN width, depth, and the number of training samples needed to guarantee a certain testing error. Under mild assumptions on data distributions or operator structures, our analysis shows that deep operator learning can have a relaxed dependence on the discretization resolution of PDEs and, hence, lessen the curse of dimensionality in many PDE-related problems including elliptic equations, parabolic equations, and Burgers equations. Our results are also applied to give insights about discretization-invariance in operator learning.

Ming-Jun Lai, Chunmei Wang

A bivariate spline method is developed to numerically solve second order elliptic partial differential equations (PDE) in non-divergence form. The existence, uniqueness, stability as well as approximation properties of the discretized solution will be established by using the well-known Ladyzhenskaya-Babuska-Brezzi (LBB) condition. Bivariate splines, discontinuous splines with smoothness constraints are used to implement the method. A plenty of computational results based on splines of various degrees are presented to demonstrate the effectiveness and efficiency of our method.

Jianda Du, Senwei Liang, Chunmei Wang

Modeling and forecasting the spread of infectious diseases is essential for effective public health decision-making. Traditional epidemiological models rely on expert-defined frameworks to describe complex dynamics, while neural networks, despite their predictive power, often lack interpretability due to their ``black-box" nature. This paper introduces the Finite Expression Method, a symbolic learning framework that leverages reinforcement learning to derive explicit mathematical expressions for epidemiological dynamics. Through numerical experiments on both synthetic and real-world datasets, FEX demonstrates high accuracy in modeling and predicting disease spread, while uncovering explicit relationships among epidemiological variables. These results highlight FEX as a powerful tool for infectious disease modeling, combining interpretability with strong predictive performance to support practical applications in public health.

Senwei Liang, Chunmei Wang, Xingjian Xu

Modeling stochastic differential equations (SDEs) is crucial for understanding complex dynamical systems in various scientific fields. Recent methods often employ neural network-based models, which typically represent SDEs through a combination of deterministic and stochastic terms. However, these models usually lack interpretability and have difficulty generalizing beyond their training domain. This paper introduces the Finite Expression Method (FEX), a symbolic learning approach designed to derive interpretable mathematical representations of the deterministic component of SDEs. For the stochastic component, we integrate FEX with advanced generative modeling techniques to provide a comprehensive representation of SDEs. The numerical experiments on linear, nonlinear, and multidimensional SDEs demonstrate that FEX generalizes well beyond the training domain and delivers more accurate long-term predictions compared to neural network-based methods. The symbolic expressions identified by FEX not only improve prediction accuracy but also offer valuable scientific insights into the underlying dynamics of the systems, paving the way for new scientific discoveries.

Zhongyi Jiang, Chunmei Wang, Haizhao Yang

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.

Fan Chen, Jianguo Huang, Chunmei Wang et al.

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.

Dan Li, Yufeng Nie, Chunmei Wang

A superconvergence error estimate for the gradient approximation of the second order elliptic problem in three dimensions is analyzed by using weak Galerkin finite element scheme on the uniform and non-uniform cubic partitions. Due to the loss of the symmetric property from two dimensions to three dimensions, this superconvergence result in three dimensions is not a trivial extension of the recent superconvergence result in two dimensions \cite{sup_LWW2018} from rectangular partitions to cubic partitions. The error estimate for the numerical gradient in the $L^{2}$-norm arrives at a superconvergence order of ${\cal O}(h^r) (1.5 \leq r\leq 2)$ when the lowest order weak Galerkin finite elements consisting of piecewise linear polynomials in the interior of the elements and piecewise constants on the faces of the elements are employed. A series of numerical experiments are illustrated to confirm the established superconvergence theory in three dimensions.

Chunmei Wang

A new numerical method is devised and analyzed for a type of ill-posed elliptic Cauchy problems by using the primal-dual weak Galerkin finite element method. This new primal-dual weak Galerkin algorithm is robust and efficient in the sense that the system arising from the scheme is symmetric, well-posed, and is satisfied by the exact solution (if it exists). An error estimate of optimal order is established for the corresponding numerical solutions in a scaled residual norm. In addition, a mathematical convergence is established in a weak $L^2$ topology for the new numerical method. Numerical results are reported to demonstrate the efficiency of the primal-dual weak Galerkin method as well as the accuracy of the numerical approximations.

Chunmei Wang

In this paper, the author derives an $O(h^4)$-superconvergence for the piecewise linear Ritz-Galerkin finite element approximations for the second order elliptic equation $-\nabla \cdot(A\nabla u)= f$ equipped with Dirichlet boundary conditions. This superconvergence error estimate is established between the finite element solution and the usual Lagrange nodal point interpolation of the exact solution, and thus the superconvergence at the nodal points of each element. The result is based on a condition for the finite element partition characterized by the coefficient tensor $A$ and the usual shape functions on each element, called $A$-equilateral assumption in this paper. Several examples are presented for the coefficient tensor $A$ and finite element triangulations which satisfy the conditions necessary for superconvergence. Some numerical experiments are conducted to confirm this new theory of superconvergence.

Chunmei Wang, Haomin Zhou

In this paper, a new and efficient numerical algorithm by using weak Galerkin (WG) finite element methods is proposed for a type of fourth order problem arising from fluorescence tomography(FT). Fluorescence tomography is an emerging, in vivo non-invasive 3-D imaging technique which reconstructs images that characterize the distribution of molecules that are tagged by fluorophores. Weak second order elliptic operator and its discrete version are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. An error estimate of optimal order is derived in an $H^2$-equivalent norm for the WG finite element solutions. Error estimates in the usual $L^2$ norm are established, yielding optimal order of convergence for all the WG finite element algorithms except the one corresponding to the lowest order (i.e., piecewise quadratic elements). Some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme.

Chunmei Wang, Junping Wang

This article proposes a new numerical algorithm for second order elliptic equations in non-divergence form. The new method is based on a discrete weak Hessian operator locally constructed by following the weak Galerkin strategy. The numerical solution is characterized as a minimization of a non-negative quadratic functional with constraints that mimic the second order elliptic equation by using the discrete weak Hessian. The resulting Euler-Lagrange equation offers a symmetric finite element scheme involving both the primal and a dual variable known as the Lagrange multiplier, and thus the name of primal-dual weak Galerkin finite element method. Optimal order error estimates are derived for the finite element approximations in a discrete $H^2$-norm, as well as the usual $H^1$- and $L^2$-norms. Some numerical results are presented for smooth and non-smooth coefficients on convex and non-convex domains.

Chunmei Wang, Junping Wang, Ruishu Wang et al.

This paper presents an arbitrary order locking-free numerical scheme for linear elasticity on general polygonal/polyhedral partitions by using weak Galerkin (WG) finite element methods. Like other WG methods, the key idea for the linear elasticity is to introduce discrete weak strain and stress tensors which are defined and computed by solving inexpensive local problems on each element. Such local problems are derived from weak formulations of the corresponding differential operators through integration by parts. Locking-free error estimates of optimal order are derived in a discrete $H^1$-norm and the usual $L^2$-norm for the approximate displacement when the exact solution is smooth. Numerical results are presented to demonstrate the efficiency, accuracy, and the locking-free property of the weak Galerkin finite element method.