Cuiyu He

Zhiqiang Cai, Cuiyu He, Shun Zhang

In \cite{CaZh:09}, we introduced and analyzed an improved Zienkiewicz-Zhu (ZZ) estimator for the conforming linear finite element approximation to elliptic interface problems. The estimator is based on the piecewise "constant" flux recovery in the $H(div;Ω)$ conforming finite element space. This paper extends the results of \cite{CaZh:09} to diffusion problems with full diffusion tensor and to the flux recovery both in piecewise constant and piecewise linear $H(div)$ space.

Zhiqiang Cai, Cuiyu He, Shun Zhang

For elliptic interface problems, this paper studies residual-based a posteriori error estimations for various finite element approximations. For the conforming and the Raviart-Thomas mixed elements in two-dimension and for the Crouzeix-Raviart nonconforming and the discontinuous Galerkin elements in both two- and three-dimensions, the global reliability bounds are established with constants independent of the jump of the diffusion coefficient. Moreover, we obtain these estimates with no assumption on the distribution of the diffusion coefficient.

Xinheng Xie, Yue Wu, Cuiyu He

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.

Xinheng Xie, Yue Wu, Hao Ni et al.

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.