arXiv subjects

Cuiyu He

Publications and source records attributed to Cuiyu He.

At least 19 recordsLinked to original sources

arXiv2026

A Unified Equilibrated Flux Recovery Framework with Robust A Posteriori Error Estimation

We introduce the Equilibrated Averaging Residual Method (EARM), a unified equilibrated flux-recovery framework for elliptic interface problems that applies to a broad class of finite element discretizations. The method is applicable in both two and three dimensions and for arbitrary polynomial orders, and it enables the construction of computationally efficient recovered fluxes. We develop EARM for both discontinuous Galerkin (DG) and conforming finite element discretizations. For DG methods, EARM can be applied directly and yields an explicit recovered flux that coincides with state-of-the-art conservative flux reconstructions. For conforming discretizations, we further propose the Orthogonal Null-space--Eliminated EARM (ON-EARM), which ensures uniqueness by restricting the correction flux to the orthogonal complement of the divergence-free null space. We prove local conservation and establish a robust a~posteriori error estimator for the recovered flux in two dimensions, with robustness measured with respect to jumps in the diffusion coefficient. Numerical results in two and three dimensions confirm the theoretical findings.

math.NA↗

arXiv2025

Elliptic interface problem approximated by CutFEM: I. Conservative flux recovery and numerical validation of adaptive mesh refinement

We study an elliptic interface problem with discontinuous diffusion coefficients on unfitted meshes using the CutFEM method. Our main contribution is the reconstruction of conservative fluxes from the CutFEM solution and their use in a posteriori error estimation. We introduce a hybrid mixed formulation with locally computable Lagrange multipliers and reconstruct the flux in the immersed Raviart-Thomas space. Based on this, we propose a new a posteriori error estimator that includes both volume and interface terms. We state its robust reliability and local efficiency, and validate the approach through numerical experiments.

math.NA↗

arXiv2025

Equilibrated Averaging Residual Method: A General Approach to Conservative Flux Recovery

Many equilibrated flux recovery methods for finite element solutions rely on ad hoc or method-specific techniques, limiting their generalizability and efficiency. In this work, we introduce the Equilibrated Averaging Residual Method (EARM), a unified framework for flux recovery that not only reproduces state-of-the-art locally conservative fluxes but also enables the derivation of new equilibrated fluxes with improved properties. In this paper, EARM is applied to conforming, nonconforming, and discontinuous Galerkin methods, ensuring local conservation and robust a posteriori error estimation. Despite the unified nature of the variational problem, the framework retains the flexibility to fully leverage the inherent properties of finite element spaces. Moreover, EARM offers explicit and computationally efficient flux reconstructions for all methods in two dimensions. In three dimensions, only simple local problems need to be solved for the conforming finite element methods.

math.NA↗

arXiv2024

NODE-AdvGAN: Improving the transferability and perceptual similarity of adversarial examples by dynamic-system-driven adversarial generative model

Understanding adversarial examples is crucial for improving model robustness, as they introduce imperceptible perturbations to deceive models. Effective adversarial examples, therefore, offer the potential to train more robust models by eliminating model singularities. We propose NODE-AdvGAN, a novel approach that treats adversarial generation as a continuous process and employs a Neural Ordinary Differential Equation (NODE) to simulate generator dynamics. By mimicking the iterative nature of traditional gradient-based methods, NODE-AdvGAN generates smoother and more precise perturbations that preserve high perceptual similarity when added to benign images. We also propose a new training strategy, NODE-AdvGAN-T, which enhances transferability in black-box attacks by tuning the noise parameters during training. Experiments demonstrate that NODE-AdvGAN and NODE-AdvGAN-T generate more effective adversarial examples that achieve higher attack success rates while preserving better perceptual quality than baseline models.

cs.LG↗

arXiv2024

Best approximation results and essential boundary conditions for novel types of weak adversarial network discretizations for PDEs

In this paper, we provide a theoretical analysis of the recently introduced weakly adversarial networks (WAN) method, used to approximate partial differential equations in high dimensions. We address the existence and stability of the solution, as well as approximation bounds. More precisely, we prove the existence of discrete solutions, intended in a suitable weak sense, for which we prove a quasi-best approximation estimate similar to Cea's lemma, a result commonly found in finite element methods. We also propose two new stabilized WAN-based formulas that avoid the need for direct normalization. Furthermore, we analyze the method's effectiveness for the Dirichlet boundary problem that employs the implicit representation of the geometry. The key requirement for achieving the best approximation outcome is to ensure that the space for the test network satisfies a specific condition, known as the inf-sup condition, essentially requiring that the test network set is sufficiently large when compared to the trial space. The method's accuracy, however, is only determined by the space of the trial network. We also devise a pseudo-time XNODE neural network class for static PDE problems, yielding significantly faster convergence results than the classical DNN network.

math.NA↗

arXiv2023

NODE-ImgNet: a PDE-informed effective and robust model for image denoising

Inspired by the traditional partial differential equation (PDE) approach for image denoising, we propose a novel neural network architecture, referred as NODE-ImgNet, that combines neural ordinary differential equations (NODEs) with convolutional neural network (CNN) blocks. NODE-ImgNet is intrinsically a PDE model, where the dynamic system is learned implicitly without the explicit specification of the PDE. This naturally circumvents the typical issues associated with introducing artifacts during the learning process. By invoking such a NODE structure, which can also be viewed as a continuous variant of a residual network (ResNet) and inherits its advantage in image denoising, our model achieves enhanced accuracy and parameter efficiency. In particular, our model exhibits consistent effectiveness in different scenarios, including denoising gray and color images perturbed by Gaussian noise, as well as real-noisy images, and demonstrates superiority in learning from small image datasets.

eess.IV↗

arXiv2023

p-Adic Statistical Field Theory and Convolutional Deep Boltzmann Machines

Understanding how deep learning architectures work is a central scientific problem. Recently, a correspondence between neural networks (NNs) and Euclidean quantum field theories (QFTs) has been proposed. This work investigates this correspondence in the framework of p-adic statistical field theories (SFTs) and neural networks (NNs). In this case, the fields are real-valued functions defined on an infinite regular rooted tree with valence p, a fixed prime number. This infinite tree provides the topology for a continuous deep Boltzmann machine (DBM), which is identified with a statistical field theory (SFT) on this infinite tree. In the p-adic framework, there is a natural method to discretize SFTs. Each discrete SFT corresponds to a Boltzmann machine (BM) with a tree-like topology. This method allows us to recover the standard DBMs and gives new convolutional DBMs. The new networks use O(N) parameters while the classical ones use O(N^{2}) parameters.

hep-th↗

arXiv2021

An a posteriori error estimate of the outer normal derivative using dual weights

We derive a residual based a-posteriori error estimate for the outer normal flux of approximations to {the diffusion problem with variable coefficient}. By analyzing the solution of the adjoint problem, we show that error indicators in the bulk may be defined to be of higher order than those close to the boundary, which lead to more economic meshes. The theory is illustrated with some numerical examples.

math.NA↗

arXiv2021

Towards fast weak adversarial training to solve high dimensional parabolic partial differential equations using XNODE-WAN

Due to the curse of dimensionality, solving high dimensional parabolic partial differential equations (PDEs) has been a challenging problem for decades. Recently, a weak adversarial network (WAN) proposed in (Y.Zang et al., 2020) offered a flexible and computationally efficient approach to tackle this problem defined on arbitrary domains by leveraging the weak solution. WAN reformulates the PDE problem as a generative adversarial network, where the weak solution (primal network) and the test function (adversarial network) are parameterized by the multi-layer deep neural networks (DNNs). However, it is not yet clear whether DNNs are the most effective model for the parabolic PDE solutions as they do not take into account the fundamentally different roles played by time and spatial variables in the solution. To reinforce the difference, we design a novel so-called XNODE model for the primal network, which is built on the neural ODE (NODE) model with additional spatial dependency to incorporate the a priori information of the PDEs and serve as a universal and effective approximation to the solution. The proposed hybrid method (XNODE-WAN), by integrating the XNODE model within the WAN framework, leads to significant improvement in the performance and efficiency of training. Numerical results show that our method can reduce the training time to a fraction of that of the WAN model.

math.NA↗

arXiv2021

Flux recovery for Cut finite element method and its application in a posteriori error estimation

In this article, we aim to recover locally conservative and $H(div)$ conforming fluxes for the linear Cut Finite Element Solution with Nitsche's method for Poisson problems with Dirichlet boundary condition. The computation of the conservative flux in the Raviart-Thomas space is completely local and does not require to solve any mixed problem. The $L^2$-norm of the difference between the numerical flux and the recovered flux can then be used as a posteriori error estimator in the adaptive mesh refinement procedure. Theoretically we are able to prove the global reliability and local efficiency. The theoretical results are verified in the numerical results. Moreover, in the numerical results we also observe optimal convergence rate for the flux error.

math.NA↗

arXiv2021

Optimal Control of Convection-Cooling and Numerical Implementation

This paper is concerned with the problem of enhancing convection-cooling via active control of the incompressible velocity field, described by a stationary diffusion-convection model. This essentially leads to a bilinear optimal control problem. A rigorous proof of the existence of an optimal control is presented and the first order optimality conditions are derived for solving the control using a variational inequality. Moreover, the second order sufficient conditions are established to characterize the local minimizer. Finally, numerical experiments are conducted utilizing finite elements methods together with nonlinear iterative schemes, to demonstrate and validate the effectiveness of our control design.

math.OC↗

arXiv2020

Comparison of Shape Derivatives using CutFEM for Ill-posed Bernoulli Free Boundary Problem

In this paper we discuss a level set approach for the identification of an unknown boundary in a computational domain. The problem takes the form of a Bernoulli problem where only the Dirichlet datum is known on the boundary that is to be identified, but additional information on the Neumann condition is available on the known part of the boundary. The approach uses a classical constrained optimization problem, where a cost functional is minimized with respect to the unknown boundary, the position of which is defined implicitly by a level set function. To solve the optimization problem a steepest descent algorithm using shape derivatives is applied. In each iteration the cut finite element method is used to obtain high accuracy approximations of the pde-model constraint for a given level set configuration without re-meshing. We consider three different shape derivatives. First the classical one, derived using the continuous optimization problem (optimize then discretize). Then the functional is first discretized using the CutFEM method and the shape derivative is evaluated on the finite element functional (discretize then optimize). Finally we consider a third approach, also using a discretized functional. In this case we do not perturb the domain, but consider a so-called boundary value correction method, where a small correction to the boundary position may be included in the weak boundary condition. Using this correction the shape derivative may be obtained by perturbing a distance parameter in the discrete variational formulation. The theoretical discussion is illustrated with a series of numerical examples showing that all three approaches produce similar result on the proposed Bernoulli problem.

math.NA↗

arXiv2020

A Mesh-free Method Using Piecewise Deep Neural Network for Elliptic Interface Problems

In this paper, we propose a novel mesh-free numerical method for solving the elliptic interface problems based on deep learning. We approximate the solution by the neural networks and, since the solution may change dramatically across the interface, we employ different neural networks in different sub-domains. By reformulating the interface problem as a least-squares problem, we discretize the objective function using mean squared error via sampling and solve the proposed deep least-squares method by standard training algorithms such as stochastic gradient descent. The discretized objective function utilizes only the point-wise information on the sampling points and thus no underlying mesh is required. Doing this circumvents the challenging meshing procedure as well as the numerical integration on the complex interface. To improve the computational efficiency for more challenging problems, we further design an adaptive sampling strategy based on the residual of the least-squares function and propose an adaptive algorithm. Finally, we present several numerical experiments in both 2D and 3D to show the flexibility, effectiveness, and accuracy of the proposed deep least-square method for solving interface problems.

math.NA↗

arXiv2020

Generalized Prager-Synge Inequality and Equilibrated Error Estimators for Discontinuous Elements

The well-known Prager-Synge identity is valid in $H^1(Ω)$ and serves as a foundation for developing equilibrated a posteriori error estimators for continuous elements. In this paper, we introduce a new inequality, that may be regarded as a generalization of the Prager-Synge identity, to be valid for piecewise $H^1(Ω)$ functions for diffusion problems. The inequality is proved to be identity in two dimensions. For nonconforming finite element approximation of arbitrary odd order, we propose a fully explicit approach that recovers an equilibrated flux in $H(div; Ω)$ through a local element-wise scheme and that recovers a gradient in $H(curl;Ω)$ through a simple averaging technique over edges. The resulting error estimator is then proved to be globally reliable and locally efficient. Moreover, the reliability and efficiency constants are independent of the jump of the diffusion coefficient regardless of its distribution.

math.NA↗

arXiv2019

Residual-based a posteriori error estimation for immersed finite element methods

In this paper we introduce and analyze the residual-based a posteriori error estimation of the partially penalized immersed finite element method for solving elliptic interface problems. The immersed finite element method can be naturally utilized on interface-unfitted meshes. Our a posteriori error estimate is proved to be both reliable and efficient with reliability constant independent of the location of the interface. Numerical results indicate that the efficiency constant is independent of the interface location and that the error estimation is robust with respect to the coefficient contrast.

math.NA↗

arXiv2019

A Posteriori Error Estimates with Boundary Correction for a Cut Finite Element Method

In this work we study a residual based a posteriori error estimation for the CutFEM method applied to an elliptic model problem. We consider the problem with non-polygonal boundary and the analysis takes into account the geometry and data approximation on the boundary. The reliability and efficiency are theoretically proved. Moreover, constants are robust with respect to how the domain boundary cuts the mesh.

math.NA↗

arXiv2018

Primal dual mixed finite element methods for indefinite advection--diffusion equations

We consider primal-dual mixed finite element methods for the advection--diffusion equation. For the primal variable we use standard continuous finite element space and for the flux we use the Raviart-Thomas space. We prove optimal a priori error estimates in the energy- and the $L^2$-norms for the primal variable in the low Peclet regime. In the high Peclet regime we also prove optimal error estimates for the primal variable in the $H(div)$ norm for smooth solutions. Numerically we observe that the method eliminates the spurious oscillations close to interior layers that pollute the solution of the standard Galerkin method when the local Peclet number is high. This method, however, does produce spurious solutions when outflow boundary layer presents. In the last section we propose two simple strategies to remove such numerical artefacts caused by the outflow boundary layer and validate them numerically.

math.NA↗

arXiv2016

Residual-based a Posteriori Error Estimate for Interface Problems: Nonconforming Linear Elements

In this paper, we study a modified residual-based a posteriori error estimator for the nonconforming linear finite element approximation to the interface problem. The reliability of the estimator is analyzed by a new and direct approach without using the Helmholtz decomposition. It is proved that the estimator is reliable with constant independent of the jump of diffusion coefficients across the interfaces, without the assumption that the diffusion coefficient is quasi-monotone. Numerical results for one test problem with intersecting interfaces are also presented.

math.NA↗