Junping Wang

Junping Wang, Xiu Ye

In this paper, authors shall introduce a finite element method by using a weakly defined gradient operator over discontinuous functions with heterogeneous properties. The use of weak gradients and their approximations results in a new concept called {\em discrete weak gradients} which is expected to play important roles in numerical methods for partial differential equations. This article intends to provide a general framework for operating differential operators on functions with heterogeneous properties. As a demonstrative example, the discrete weak gradient operator is employed as a building block to approximate the solution of a model second order elliptic problem, in which the classical gradient operator is replaced by the discrete weak gradient. The resulting numerical approximation is called a weak Galerkin (WG) finite element solution. It can be seen that the weak Galerkin method allows the use of totally discontinuous functions in the finite element procedure. For the second order elliptic problem, an optimal order error estimate in both a discrete $H^1$ and $L^2$ norms are established for the corresponding weak Galerkin finite element solutions. A superconvergence is also observed for the weak Galerkin approximation.

Junping Wang, Xiu Ye

A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons/polyhedra. The WG-MFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. Allowing the use of discontinuous approximating functions on arbitrary shape of polygons/polyhedra makes the method highly flexible in practical computation. Optimal order error estimates in both discrete $H^1$ and $L^2$ norms are established for the corresponding weak Galerkin mixed finite element solutions.

Junping Wang, Xiu Ye

This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primary velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree $k\ge 1$ for the velocity and polynomials of degree $k-1$ for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree $k-1$ on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.

Lin Mu, Junping Wang, Guowei Wei et al.

Weak Galerkin methods refer to general finite element methods for PDEs in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic partial differential equations (PDEs) with discontinuous coefficients and interfaces. The paper also presents many numerical tests for validating the WG-FEM for solving second order elliptic interface problems. For such interface problems, the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design high order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order one for the solution itself in $L_\infty$ norm. It is demonstrated that the WG-FEM of lowest order is capable of delivering numerical approximations that are of order 1.75 in the usual $L_\infty$ norm for $C^1$ or Lipschitz continuous interfaces associated with a $C^1$ or $H^2$ continuous solutions. Theoretically, it is proved that high order of numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element.

Lin Mu, Junping Wang, Xiu Ye

A new weak Galerkin (WG) finite element method is introduced and analyzed in this paper for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding WG finite element solutions. Error estimates in the usual $L^2$ norm are also derived, yielding a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions.

Qiaoluan H. Li, Junping Wang

A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and discontinuous time weak Galerkin finite element schemes are developed and analyzed. Optimal order error estimates in both H^1 and L^2 norms are established. Numerical tests are performed and reported.

Lin Mu, Junping Wang, Yanqiu Wang et al.

The weak Galerkin finite element method is a novel numerical method that was first proposed and analyzed by Wang and Ye for general second order elliptic problems on triangular meshes. The goal of this paper is to conduct a computational investigation for the weak Galerkin method for various model problems with more general finite element partitions. The numerical results confirm the theory established by Wang and Ye. The results also indicate that the weak Galerkin method is efficient, robust, and reliable in scientific computing.

Lin Mu, Junping Wang, Xiu Ye

This paper introduces a new weak Galerkin (WG) finite element method for second order elliptic equations on polytopal meshes. This method, called WG-FEM, is designed by using a discrete weak gradient operator applied to discontinuous piecewise polynomials on finite element partitions of arbitrary polytopes with certain shape regularity. The paper explains how the numerical schemes are designed and why they provide reliable numerical approximations for the underlying partial differential equations. In particular, optimal order error estimates are established for the corresponding WG-FEM approximations in both a discrete $H^1$ norm and the standard $L^2$ norm. Numerical results are presented to demonstrate the robustness, reliability, and accuracy of the WG-FEM. All the results are derived for finite element partitions with polytopes. Allowing the use of discontinuous approximating functions on arbitrary polytopal elements is a highly demanded feature for numerical algorithms in scientific computing.

Lin Mu, Junping Wang, Yanqiu Wang et al.

This article introduces and analyzes a weak Galerkin mixed finite element method for solving the biharmonic equation. The weak Galerkin method, first introduced by two of the authors (J. Wang and X. Ye) in an earlier publication for second order elliptic problems, is based on the concept of discrete weak gradients. The method allows the use of completely discrete finite element functions on partitions of arbitrary polygon or polyhedron. In this article, the idea of weak Galerkin method is applied to discretize the Ciarlet-Raviart mixed formulation for the biharmonic equation. In particular, an a priori error estimation is given for the corresponding finite element approximations. The error analysis essentially follows the framework of Babuska, Osborn, and Pitkaranta and uses specially designed mesh-dependent norms. The proof is technically tedious due to the discontinuous nature of the weak Galerkin finite element functions. Some computational results are presented to demonstrate the efficiency of the method.

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.

Yujie Liu, Junping Wang

This article presents a simplified formulation for the weak Galerkin finite element method for the Stokes equation without using the degrees of freedom associated with the unknowns in the interior of each element as formulated in the original weak Galerkin algorithm. The simplified formulation preserves the important mass conservation property locally on each element and allows the use of general polygonal partitions. A particular application of the simplified weak Galerkin on rectangular partitions yields a new class of 5- and 7-point finite difference schemes for the Stokes equation. An explicit formula is presented for the computation of the element stiffness matrices on arbitrary polygonal elements. Error estimates of optimal order are established for the simplified weak Galerkin finite element method in the H^1 and L^2 norms. Furthermore, a superconvergence of order O(h^{1.5}) is established on rectangular partitions for the velocity approximation in the H^1 norm at cell centers, and a similar superconvergence is derived for the pressure approximation in the L^2 norm at cell centers. Some numerical results are reported to confirm the convergence and superconvergence theory.

Long Chen, Junping Wang, Yanqiu Wang et al.

In this paper, the authors constructed an auxiliary space multigrid preconditioner for the weak Galerkin finite element method for second-order diffusion equations, discretized on simplicial 2D or 3D meshes. The idea of the auxiliary space multigrid preconditioner is to use an auxiliary space as a "coarse" space in the multigrid algorithm, where the discrete problem in the auxiliary space can be easily solved by an existing solver. In this construction, the authors conveniently use the $H^1$ conforming piecewise linear finite element space as an auxiliary space. The main technical difficulty is to build the connection between the weak Galerkin discrete space and the $H^1$ conforming piecewise linear finite element space. The authors successfully constructed such an auxiliary space multigrid preconditioner for the weak Galerkin method, as well as a reduced system of the weak Galerkin method involving only the degrees of freedom on edges/faces. The preconditioned systems are proved to have condition numbers independent of the mesh size. Numerical experiments are conducted to support the theoretical results.

Junping Wang, Ran Zhang

This paper derives some discrete maximum principles for $P1$-conforming finite element approximations for quasi-linear second order elliptic equations. The results are extensions of the classical maximum principles in the theory of partial differential equations to finite element methods. The mathematical tools are based on the variational approach that was commonly used in the classical PDE theory. The discrete maximum principles are established by assuming a property on the discrete variational form that is of global nature. In particular, the assumption on the variational form is verified when the finite element partition satisfies some angle conditions. For the general quasi-linear elliptic equation, these angle conditions indicate that each triangle or tetrahedron needs to be $\mathcal{O}(h^α)$-acute in the sense that each angle $α_{ij}$ (for triangle) or interior dihedral angle $α_{ij}$ (for tetrahedron) must satisfy $α_{ij}\le π/2-γh^α$ for some $α\ge 0$ and $γ>0$. For the Poisson problem where the differential operator is given by Laplacian, the angle requirement is the same as the existing ones: either all the triangles are non-obtuse or each interior edge is non-negative. It should be pointed out that the analytical tools used in this paper are based on the powerful De Giorgi's iterative method that has played important roles in the theory of partial differential equations. The mathematical analysis itself is of independent interest in the finite element analysis.

Lin Mu, Junping Wang, Xiu Ye

A discrete divergence-free weak Galerkin finite element method is developed for the Stokes equations based on a weak Galerkin (WG) method introduced in the reference [15]. Discrete divergence-free bases are constructed explicitly for the lowest order weak Galerkin elements in two and three dimensional spaces. These basis functions can be derived on general meshes of arbitrary shape of polygons and polyhedrons. With the divergence-free basis derived, the discrete divergence-free WG scheme can eliminate the pressure variable from the system and reduces a saddle point problem to a symmetric and positive definite system with many fewer unknowns. Numerical results are presented to demonstrate the robustness and accuracy of this discrete divergence-free WG method.

Mu Lin, Junping Wang, Yanqiu Wang et al.

This paper focuses on interior penalty discontinuous Galerkin methods for second order elliptic equations on very general polygonal or polyhedral meshes. The mesh can be composed of any polygons or polyhedra which satisfies certain shape regularity conditions characterized in a recent paper by two of the authors in [17]. Such general meshes have important application in computational sciences. The usual $H^1$ conforming finite element methods on such meshes are either very complicated or impossible to implement in practical computation. However, the interior penalty discontinuous Galerkin method provides a simple and effective alternative approach which is efficient and robust. This article provides a mathematical foundation for the use of interior penalty discontinuous Galerkin methods in general meshes.

Lin Mu, Junping Wang, Xiu Ye et al.

A C^0-weak Galerkin (WG) method is introduced and analyzed for solving the biharmonic equation in 2D and 3D. A weak Laplacian is defined for C^0 functions in the new weak formulation. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established in both a discrete H^2 norm and the L^2 norm, for the weak Galerkin finite element solution. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott-Zhang interpolation operator is constructed to assist the corresponding error estimate. This refined interpolation preserves the volume mass of order (k+1-d) and the surface mass of order (k+2-d) for the P_{k+2} finite element functions in d-dimensional space.

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.

Lin Mu, Junping Wang, Xiu Ye

The weak Galerkin (WG) methods have been introduced in the references [11, 16] for solving the biharmonic equation. The purpose of this paper is to develop an algorithm to implement the WG methods effectively. This can be achieved by eliminating local unknowns to obtain a global system with significant reduction of size. In fact, this reduced global system is equivalent to the Schur complements of the WG methods. The unknowns of the Schur complement of the WG method are those defined on the element boundaries. The equivalence of the WG method and its Schur complement is established. The numerical results demonstrate the effectiveness of this new implementation technique.

Yujie Liu, Junping Wang, Qingsong Zou

This article presents a new finite element method for convection-diffusion equations by enhancing the continuous finite element space with a flux space for flux approximations that preserve the important mass conservation locally on each element. The numerical scheme is based on a constrained flux optimization approach where the constraint was given by local mass conservation equations and the flux error is minimized in a prescribed topology/metric. This new scheme provides numerical approximations for both the primal and the flux variables. It is shown that the numerical approximations for the primal and the flux variables are convergent with optimal order in some discrete Sobolev norms. Numerical experiments are conducted to confirm the convergence theory. Furthermore, the new scheme was employed in the computational simulation of a simplified two-phase flow problem in highly heterogeneous porous media. The numerical results illustrate an excellent performance of the method in scientific computing.

Junping Wang, Zhizhong Zhang, Yongqiang Tang et al.

Scaling individual robot capabilities is common but costly. Here we investigate a system-level design question in real-world multi-robot coordination: given matched hardware budgets, does restructuring communication among robots yield larger gains than increasing onboard model size? Using a representative transport-and-mapping task with 10 physical robots (5 runs per condition, 60 runs total), we find that switching from fully connected to modular hierarchical interactions improves normalised performance by 47 points (0--100), whereas doubling neural network hidden size yields at most 9 points. Nested mixed-effects model comparisons show a substantially larger improvement in model fit for topology than for scale. The pattern is confirmed in independent SMAC replications; heterogeneous benchmark reanalyses provide secondary supporting consistency checks rather than primary evidence. Performance saturation beyond 1024 hidden units is observed in simulation-calibrated extrapolation, not directly on hardware. These results indicate that interaction structure can play a dominant role within the tested system and task setting, while broader quantitative generalisation remains to be established.

Yujie Liu, Junping Wang

In this article a simplified weak Galerkin finite element method is developed for the Dirichlet boundary value problem of convection-diffusion-reaction equations. The simplified weak Galerkin method utilizes only the degrees of freedom on the boundary of each element and, hence, has significantly reduced computational complexity over the regular weak Galerkin finite element method. A stability and some optimal order error estimates in the $H^1$ and $L^2$ norms are established for the corresponding numerical solutions. Numerical results are presented to verify the theory error estimates and a superconvergence phenomena on rectangular partitions.

Lin Mu, Junping Wang, Xiu Ye et al.

Weak Galerkin (WG) refers to general finite element methods for partial differential equations in which differential operators are approximated by weak forms through the usual integration by parts. In particular, WG methods allow the use of discontinuous finite element functions in the algorithm design. One of such examples was recently introduced by Wang and Ye for solving second order elliptic problems. The goal of this paper is to apply the WG method of Wang and Ye to the Helmholtz equation with high wave numbers. Several test scenarios are designed for a numerical investigation on the accuracy, convergence, and robustness of the WG method in both inhomogeneous and homogeneous media over convex and non-convex domains. Our numerical experiments indicate that weak Galerkin is a finite element technique that is easy to implement, and provides very accurate and robust numerical solutions for the Helmholtz problem with high wave numbers.

Jianxin Lin, Yongqiang Tang, Junping Wang et al.

As a recent noticeable topic, domain generalization aims to learn a generalizable model on multiple source domains, which is expected to perform well on unseen test domains. Great efforts have been made to learn domain-invariant features by aligning distributions across domains. However, existing works are often designed based on some relaxed conditions which are generally hard to satisfy and fail to realize the desired joint distribution alignment. In this paper, we propose a novel domain generalization method, which originates from an intuitive idea that a domain-invariant classifier can be learned by minimizing the KL-divergence between posterior distributions from different domains. To enhance the generalizability of the learned classifier, we formalize the optimization objective as an expectation computed on the ground-truth marginal distribution. Nevertheless, it also presents two obvious deficiencies, one of which is the side-effect of entropy increase in KL-divergence and the other is the unavailability of ground-truth marginal distributions. For the former, we introduce a term named maximum in-domain likelihood to maintain the discrimination of the learned domain-invariant representation space. For the latter, we approximate the ground-truth marginal distribution with source domains under a reasonable convex hull assumption. Finally, a Constrained Maximum Cross-domain Likelihood (CMCL) optimization problem is deduced, by solving which the joint distributions are naturally aligned. An alternating optimization strategy is carefully designed to approximately solve this optimization problem. Extensive experiments on four standard benchmark datasets, i.e., Digits-DG, PACS, Office-Home and miniDomainNet, highlight the superior performance of our method.

Zhizhong Zhang, Jiangming Wang, Xin Tan et al.

Unsupervised visible infrared person re-identification (USVI-ReID) is a challenging retrieval task that aims to retrieve cross-modality pedestrian images without using any label information. In this task, the large cross-modality variance makes it difficult to generate reliable cross-modality labels, and the lack of annotations also provides additional difficulties for learning modality-invariant features. In this paper, we first deduce an optimization objective for unsupervised VI-ReID based on the mutual information between the model's cross-modality input and output. With equivalent derivation, three learning principles, i.e., "Sharpness" (entropy minimization), "Fairness" (uniform label distribution), and "Fitness" (reliable cross-modality matching) are obtained. Under their guidance, we design a loop iterative training strategy alternating between model training and cross-modality matching. In the matching stage, a uniform prior guided optimal transport assignment ("Fitness", "Fairness") is proposed to select matched visible and infrared prototypes. In the training stage, we utilize this matching information to introduce prototype-based contrastive learning for minimizing the intra- and cross-modality entropy ("Sharpness"). Extensive experimental results on benchmarks demonstrate the effectiveness of our method, e.g., 60.6% and 90.3% of Rank-1 accuracy on SYSU-MM01 and RegDB without any annotations.

Jianxin Lin, Yongqiang Tang, Junping Wang et al.

Domain generalization (DG) aims to learn a model on several source domains, hoping that the model can generalize well to unseen target domains. The distribution shift between domains contains the covariate shift and conditional shift, both of which the model must be able to handle for better generalizability. In this paper, a novel DG method is proposed to deal with the distribution shift via Visual Alignment and Uncertainty-guided belief Ensemble (VAUE). Specifically, for the covariate shift, a visual alignment module is designed to align the distribution of image style to a common empirical Gaussian distribution so that the covariate shift can be eliminated in the visual space. For the conditional shift, we adopt an uncertainty-guided belief ensemble strategy based on the subjective logic and Dempster-Shafer theory. The conditional distribution given a test sample is estimated by the dynamic combination of that of source domains. Comprehensive experiments are conducted to demonstrate the superior performance of the proposed method on four widely used datasets, i.e., Office-Home, VLCS, TerraIncognita, and PACS.

Wenhua Wu, Qi Wang, Guangming Wang et al.

Road surface reconstruction plays a vital role in autonomous driving systems, enabling road lane perception and high-precision mapping. Recently, neural implicit encoding has achieved remarkable results in scene representation, particularly in the realistic rendering of scene textures. However, it faces challenges in directly representing geometric information for large-scale scenes. To address this, we propose EMIE-MAP, a novel method for large-scale road surface reconstruction based on explicit mesh and implicit encoding. The road geometry is represented using explicit mesh, where each vertex stores implicit encoding representing the color and semantic information. To overcome the difficulty in optimizing road elevation, we introduce a trajectory-based elevation initialization and an elevation residual learning method based on Multi-Layer Perceptron (MLP). Additionally, by employing implicit encoding and multi-camera color MLPs decoding, we achieve separate modeling of scene physical properties and camera characteristics, allowing surround-view reconstruction compatible with different camera models. Our method achieves remarkable road surface reconstruction performance in a variety of real-world challenging scenarios.

Bicheng Wang, Junping Wang, Yibo Xue

Learning complex network dynamics is fundamental to understanding, modelling and controlling real-world complex systems. There are two main problems in the task of predicting the dynamic evolution of complex networks: on the one hand, existing methods usually use simple graphs to describe the relationships in complex networks; however, this approach can only capture pairwise relationships, while there may be rich non-pairwise structured relationships in the network. First-order GNNs have difficulty in capturing dynamic non-pairwise relationships. On the other hand, theoretical prediction models lack accuracy and data-driven prediction models lack interpretability. To address the above problems, this paper proposes a higher-order network dynamics identification method for long-term dynamic prediction of complex networks. Firstly, to address the problem that traditional graph machine learning can only deal with pairwise relations, dynamic hypergraph learning is introduced to capture the higher-order non-pairwise relations among complex networks and improve the accuracy of complex network modelling. Then, a dual-driven dynamic prediction module for physical data is proposed. The Koopman operator theory is introduced to transform the nonlinear dynamical differential equations for the dynamic evolution of complex networks into linear systems for solving. Meanwhile, the physical information neural differential equation method is utilised to ensure that the dynamic evolution conforms to the physical laws. The dual-drive dynamic prediction module ensures both accuracy and interpretability of the prediction. Validated on public datasets and self-built industrial chain network datasets, the experimental results show that the method in this paper has good prediction accuracy and long-term prediction performance.

Bicheng Wang, Junping Wang, Yibo Xue

Industrial chain plays an increasingly important role in the sustainable development of national economy. However, as a typical complex network, data-driven deep learning is still in its infancy in describing and analyzing the resilience of complex networks, and its core is the lack of a theoretical framework to describe the system dynamics. In this paper, we propose a physically informative neural symbolic approach to describe the evolutionary dynamics of complex networks for resilient prediction. The core idea is to learn the dynamics of the activity state of physical entities and integrate it into the multi-layer spatiotemporal co-evolution network, and use the physical information method to realize the joint learning of physical symbol dynamics and spatiotemporal co-evolution topology, so as to predict the industrial chain resilience. The experimental results show that the model can obtain better results and predict the elasticity of the industry chain more accurately and effectively, which has certain practical significance for the development of the industry.

Junping Wang, Bicheng Wang, Yibo Xuea et al.

Resilience non-equilibrium measurement, the ability to maintain fundamental functionality amidst failures and errors, is crucial for scientific management and engineering applications of industrial chain. The problem is particularly challenging when the number or types of multiple co-evolution of resilience (for example, randomly placed) are extremely chaos. Existing end-to-end deep learning ordinarily do not generalize well to unseen full-feld reconstruction of spatiotemporal co-evolution structure, and predict resilience of network topology, especially in multiple chaos data regimes typically seen in real-world applications. To address this challenge, here we propose industrial brain, a human-like autonomous cognitive decision-making and planning framework integrating higher-order activity-driven neuro network and CT-OODA symbolic reasoning to autonomous plan resilience directly from observational data of global variable. The industrial brain not only understands and model structure of node activity dynamics and network co-evolution topology without simplifying assumptions, and reveal the underlying laws hidden behind complex networks, but also enabling accurate resilience prediction, inference, and planning. Experimental results show that industrial brain significantly outperforms resilience prediction and planning methods, with an accurate improvement of up to 10.8\% over GoT and OlaGPT framework and 11.03\% over spectral dimension reduction. It also generalizes to unseen topologies and dynamics and maintains robust performance despite observational disturbances. Our findings suggest that industrial brain addresses an important gap in resilience prediction and planning for industrial chain.

JunPing Wang, WenSheng Zhang, Ian Thomas et al.

Generating sequential decision process from huge amounts of measured process data is a future research direction for collaborative factory automation, making full use of those online or offline process data to directly design flexible make decisions policy, and evaluate performance. The key challenges for the sequential decision process is to online generate sequential decision-making policy directly, and transferring knowledge across tasks domain. Most multi-task policy generating algorithms often suffer from insufficient generating cross-task sharing structure at discrete-time nonlinear systems with applications. This paper proposes the multi-task generative adversarial nets with shared memory for cross-domain coordination control, which can generate sequential decision policy directly from raw sensory input of all of tasks, and online evaluate performance of system actions in discrete-time nonlinear systems. Experiments have been undertaken using a professional flexible manufacturing testbed deployed within a smart factory of Weichai Power in China. Results on three groups of discrete-time nonlinear control tasks show that our proposed model can availably improve the performance of task with the help of other related tasks.

Yujie Liu, Junping Wang

This article establishes a discrete maximum principle (DMP) for the approximate solution of convection-diffusion-reaction problems obtained from the weak Galerkin finite element method on nonuniform rectangular partitions. The DMP analysis is based on a simplified formulation of the weak Galerkin involving only the approximating functions defined on the boundary of each element. The simplified weak Galerkin method has a reduced computational complexity over the usual weak Galerkin, and indeed provides a discretization scheme different from the weak Galerkin when the reaction term presents. An application of the simplified weak Galerkin on uniform rectangular partitions yields some $5$- and $7$-point finite difference schemes for the second order elliptic equation. Numerical experiments are presented to verify the discrete maximum principle and the accuracy of the scheme, particularly the finite difference 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.

Lin Mu, Junping Wang, Xiu Ye

This paper presents a hybridized formulation for the weak Galerkin mixed finite element method (WG-MFEM) which was introduced and analyzed for second order elliptic equations. The WG-MFEM method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element. The hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising from the Lagrange multiplier in hybridization. Optimal-order error estimates are derived for the hybridized WG-MFEM approximations. Some numerical results are reported to confirm the theory and a superconvergence for the Lagrange multiplier.

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.