Multiplicity of positive solutions for a fourth-order quasilinear singular differential equation

This paper is concerned with the multiplicity of positive solutions of boundary value problem for the fourth-order quasilinear singular differential equation $$(|u''|^{p-2}u'')''=\lambda g(t)f(u),\quad 0<t<1,$$ where $p>1$, $\lambda>0$. We apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of positive solutions. We have found a threshold $\lambda^*<+\infty$, such that if $0<\lambda\leq\lambda^*$, then the problem admits at least one positive solution; while if $\lambda \lambda^*$, then the problem has no positive solution. In particular, there exist at least two positive solutions for $0<\lambda<\lambda^*$

Evaluating Similitude and Robustness of Deep Image Denoising Models via Adversarial Attack

Deep neural networks (DNNs) have a wide range of applications in the field of image denoising, and they are superior to traditional image denoising. However, DNNs inevitably show vulnerability, which is the weak robustness in the face of adversarial attacks. In this paper, we find some similitudes between existing deep image denoising methods, as they are consistently fooled by adversarial attacks. First, denoising-PGD is proposed which is a denoising model full adversarial method. The current mainstream non-blind denoising models (DnCNN, FFDNet, ECNDNet, BRDNet), blind denoising models (DnCNN-B, Noise2Noise, RDDCNN-B, FAN), and plug-and-play (DPIR, CurvPnP) and unfolding denoising models (DeamNet) applied to grayscale and color images can be attacked by the same set of methods. Second, since the transferability of denoising-PGD is prominent in the image denoising task, we design experiments to explore the characteristic of the latent under the transferability. We correlate transferability with similitude and conclude that the deep image denoising models have high similitude. Third, we investigate the characteristic of the adversarial space and use adversarial training to complement the vulnerability of deep image denoising to adversarial attacks on image denoising. Finally, we constrain this adversarial attack method and propose the L2-denoising-PGD image denoising adversarial attack method that maintains the Gaussian distribution. Moreover, the model-driven image denoising BM3D shows some resistance in the face of adversarial attacks.Comment: 12 pages, 15 figure

A Review of Adversarial Attacks in Computer Vision

Deep neural networks have been widely used in various downstream tasks, especially those safety-critical scenario such as autonomous driving, but deep networks are often threatened by adversarial samples. Such adversarial attacks can be invisible to human eyes, but can lead to DNN misclassification, and often exhibits transferability between deep learning and machine learning models and real-world achievability. Adversarial attacks can be divided into white-box attacks, for which the attacker knows the parameters and gradient of the model, and black-box attacks, for the latter, the attacker can only obtain the input and output of the model. In terms of the attacker's purpose, it can be divided into targeted attacks and non-targeted attacks, which means that the attacker wants the model to misclassify the original sample into the specified class, which is more practical, while the non-targeted attack just needs to make the model misclassify the sample. The black box setting is a scenario we will encounter in practice

Adversarial Training for Physics-Informed Neural Networks

Physics-informed neural networks have shown great promise in solving partial differential equations. However, due to insufficient robustness, vanilla PINNs often face challenges when solving complex PDEs, especially those involving multi-scale behaviors or solutions with sharp or oscillatory characteristics. To address these issues, based on the projected gradient descent adversarial attack, we proposed an adversarial training strategy for PINNs termed by AT-PINNs. AT-PINNs enhance the robustness of PINNs by fine-tuning the model with adversarial samples, which can accurately identify model failure locations and drive the model to focus on those regions during training. AT-PINNs can also perform inference with temporal causality by selecting the initial collocation points around temporal initial values. We implement AT-PINNs to the elliptic equation with multi-scale coefficients, Poisson equation with multi-peak solutions, Burgers equation with sharp solutions and the Allen-Cahn equation. The results demonstrate that AT-PINNs can effectively locate and reduce failure regions. Moreover, AT-PINNs are suitable for solving complex PDEs, since locating failure regions through adversarial attacks is independent of the size of failure regions or the complexity of the distribution

Image Denoising via Nonlinear Hybrid Diffusion

A nonlinear anisotropic hybrid diffusion equation is discussed for image denoising, which is a combination of mean curvature smoothing and Gaussian heat diffusion. First, we propose a new edge detection indicator, that is, the diffusivity function. Based on this diffusivity function, the new diffusion is nonlinear anisotropic and forward-backward. Unlike the Perona-Malik (PM) diffusion, the new forward-backward diffusion is adjustable and under control. Then, the existence, uniqueness, and long-time behavior of the new regularization equation of the model are established. Finally, using the explicit difference scheme (PM scheme) and implicit difference scheme (AOS scheme), we do numerical experiments for different images, respectively. Experimental results illustrate the effectiveness of the new model with respect to other known models

SaaFormer: Spectral-spatial Axial Aggregation Transformer for Hyperspectral Image Classification

Hyperspectral images (HSI) captured from earth observing satellites and aircraft is becoming increasingly important for applications in agriculture, environmental monitoring, mining, etc. Due to the limited available hyperspectral datasets, the pixel-wise random sampling is the most commonly used training-test dataset partition approach, which has significant overlap between samples in training and test datasets. Furthermore, our experimental observations indicates that regions with larger overlap often exhibit higher classification accuracy. Consequently, the pixel-wise random sampling approach poses a risk of data leakage. Thus, we propose a block-wise sampling method to minimize the potential for data leakage. Our experimental findings also confirm the presence of data leakage in models such as 2DCNN. Further, We propose a spectral-spatial axial aggregation transformer model, namely SaaFormer, to address the challenges associated with hyperspectral image classifier that considers HSI as long sequential three-dimensional images. The model comprises two primary components: axial aggregation attention and multi-level spectral-spatial extraction. The axial aggregation attention mechanism effectively exploits the continuity and correlation among spectral bands at each pixel position in hyperspectral images, while aggregating spatial dimension features. This enables SaaFormer to maintain high precision even under block-wise sampling. The multi-level spectral-spatial extraction structure is designed to capture the sensitivity of different material components to specific spectral bands, allowing the model to focus on a broader range of spectral details. The results on six publicly available datasets demonstrate that our model exhibits comparable performance when using random sampling, while significantly outperforming other methods when employing block-wise sampling partition.Comment: arXiv admin note: text overlap with arXiv:2107.02988 by other author

Modeled Antarctic Precipitation. Part I: Spatial and Temporal Variability*

Surface snow accumulation is the primary mass input to the Antarctic ice sheets. As the dominant term among various components of surface snow accumulation (precipitation, sublimation/deposition, and snow drift), pre-cipitation is of particular importance in helping to assess the mass balance of the Antarctic ice sheets and their contribution to global sea level change. The Polar MM5, a mesoscale atmospheric model based on the fifth-generation Pennsylvania State University– NCAR Mesoscale Model, has been run for the period of July 1996 through June 1999 to evaluate the spatial and temporal variability of Antarctic precipitation. Drift snow effects on the redistribution of surface snow over Antarctica are also assessed with surface wind fields from Polar MM5 in this study. It is found that areas with large drift snow transport convergence and divergence are located around escarpment areas where there is considerable katabatic wind acceleration. It is also found that the drift snow transport generally diverges over most areas of East and West Antarctica with relatively small values. The use of the dynamic retrieval method (DRM) to calculate precipitation has been developed and verified for the Greenland ice sheet. The DRM is also applied to retrieve the precipitation over Antarctica from 1979 to 1999 in this study. Most major features in the spatial distribution of Antarctic accumulation are well capture

An Alternative Variational Framework for Image Denoising

We propose an alternative framework for total variation based image denoising models. The model is based on the minimization of the total variation with a functional coefficient, where, in this case, the functional coefficient is a function of the magnitude of image gradient. We determine the considerations to bear on the choice of the functional coefficient. With the use of an example functional, we demonstrate the effectiveness of a model chosen based on the proposed consideration. In addition, for the illustrative model, we prove the existence and uniqueness of the minimizer of the variational problem. The existence and uniqueness of the solution associated evolution equation are also established. Experimental results are included to demonstrate the effectiveness of the selected model in image restoration over the traditional methods of Perona-Malik (PM), total variation (TV), and the D-α-PM method