arXiv subjects

Zhifei Zhang

Publications and source records attributed to Zhifei Zhang.

At least 109 records · Page 6Linked to original sources

arXiv2024

Landau-de Gennes model with sextic potentials: asymptotic behavior of minimizers

We study a class of Landau-de Gennes energy functionals with a sextic bulk energy density in a three-dimensional domain. We examine the asymptotic behavior of uniformly bounded minimizers in two distinct scenarios: one where their energy remains uniformly bounded, and another where it logarithmically diverges as a function of the elastic constant. In the first case, we show that up to a subsequence, the minimizers converge to a locally minimizing harmonic map in both the $ H_{\loc}^1 $ and $ C_{\loc}^j, j\in \Z_+ $ norms within compact subsets that are distant from the singularities of the limit. For the second case, we establish the existence of a closed set denoted as $ \cS_{\op{line}} $. This set has finite length and consists of finite segments of lines locally such that the energy of minimizers are locally uniformly bounded away from it. This work solves an open question raised by Canevari (ARMA, 223 (2017), 591-676), specifically concerning point and line defects in the Landau-de Gennes model with sextic potentials.

math.AP↗

arXiv2024

Evaluatology: The Science and Engineering of Evaluation

Evaluation is a crucial aspect of human existence and plays a vital role in various fields. However, it is often approached in an empirical and ad-hoc manner, lacking consensus on universal concepts, terminologies, theories, and methodologies. This lack of agreement has significant repercussions. This article aims to formally introduce the discipline of evaluatology, which encompasses the science and engineering of evaluation. We propose a universal framework for evaluation, encompassing concepts, terminologies, theories, and methodologies that can be applied across various disciplines. Our research reveals that the essence of evaluation lies in conducting experiments that intentionally apply a well-defined evaluation condition to diverse subjects and infer the impact of different subjects by measuring and/or testing. Derived from the essence of evaluation, we propose five axioms focusing on key aspects of evaluation outcomes as the foundational evaluation theory. These axioms serve as the bedrock upon which we build universal evaluation theories and methodologies. When evaluating a single subject, it is crucial to create evaluation conditions with different levels of equivalency. By applying these conditions to diverse subjects, we can establish reference evaluation models. These models allow us to alter a single independent variable at a time while keeping all other variables as controls. When evaluating complex scenarios, the key lies in establishing a series of evaluation models that maintain transitivity. Building upon the science of evaluation, we propose a formal definition of a benchmark as a simplified and sampled evaluation condition that guarantees different levels of equivalency. This concept serves as the cornerstone for a universal benchmark-based engineering approach to evaluation across various disciplines, which we refer to as benchmarkology.

cs.HC↗

arXiv2024

Self-similar imploding solutions of the relativistic Euler equations

Motivated by recent breakthrough on smooth imploding solutions of compressible Euler, we construct self-similar smooth imploding solutions of isentropic relativistic Euler equations with isothermal equation of state $p=\frac1\ell\varrho$ for \textit{all} $\ell>1$ in physical space dimension $d=2,3$ and for $\ell>1$ close to 1 in higher dimensions. This work is a crucial step toward solving the long-standing problem: finite time blow-up of the supercritical defocusing nonlinear wave equation.

math.AP↗

arXiv2024

IMPRINT: Generative Object Compositing by Learning Identity-Preserving Representation

Generative object compositing emerges as a promising new avenue for compositional image editing. However, the requirement of object identity preservation poses a significant challenge, limiting practical usage of most existing methods. In response, this paper introduces IMPRINT, a novel diffusion-based generative model trained with a two-stage learning framework that decouples learning of identity preservation from that of compositing. The first stage is targeted for context-agnostic, identity-preserving pretraining of the object encoder, enabling the encoder to learn an embedding that is both view-invariant and conducive to enhanced detail preservation. The subsequent stage leverages this representation to learn seamless harmonization of the object composited to the background. In addition, IMPRINT incorporates a shape-guidance mechanism offering user-directed control over the compositing process. Extensive experiments demonstrate that IMPRINT significantly outperforms existing methods and various baselines on identity preservation and composition quality.

cs.CV↗

arXiv2024

Linear inviscid damping in the presence of an embedding eigenvalue

In this paper, we investigate the long-time dynamics of the linearized 2-D Euler equations around a hyperbolic tangent flow $(\tanh y,0)$. A key difference compared to previous results is that the linearized operator has an embedding eigenvalue, which has a significant impact on the dynamics of the linearized system. For the first mode, the dynamics consists of there parts: non-decay part related to the eigenspace associated with the embedding eigenvalue, slow decay part due to the resolvent singularity, and fast decay part related to the inviscid damping. For higher modes, the dynamics is similar to the inviscid damping phenomena in the case without embedding eigenvalues.

math.AP↗

arXiv2024

Nonlinear stability for 3-D plane Poiseuille flow in a finite channel

In this paper, we study the nonlinear stability for the 3-D plane Poiseuille flow $(1-y^2,0,0)$ at high Reynolds number $Re$ in a finite channel $\mathbb{T}\times [-1,1 ]\times \mathbb{T}$ with non-slip boundary condition. We prove that if the initial velocity $v_0$ satisfies $\|v_0-(1-y^2,0,0)\|_{H^{4}}\leq c_0 Re^{-\frac{7}{4}}$ for some $c_0>0$ independent of $Re$, then the solution of 3-D Naiver-Stokes equations is global in time and does not transit away from the plane Poiseuille flow. To our knowledge, this is the first nonlinear stability result for the 3-D plane Poiseuille flow and the transition threshold is accordant with the numerical result by Lundbladh et al. \cite{LHR}.

math.AP↗

arXiv2024

PatSTEG: Modeling Formation Dynamics of Patent Citation Networks via The Semantic-Topological Evolutionary Graph

Patent documents in the patent database (PatDB) are crucial for research, development, and innovation as they contain valuable technical information. However, PatDB presents a multifaceted challenge compared to publicly available preprocessed databases due to the intricate nature of the patent text and the inherent sparsity within the patent citation network. Although patent text analysis and citation analysis bring new opportunities to explore patent data mining, no existing work exploits the complementation of them. To this end, we propose a joint semantic-topological evolutionary graph learning approach (PatSTEG) to model the formation dynamics of patent citation networks. More specifically, we first create a real-world dataset of Chinese patents named CNPat and leverage its patent texts and citations to construct a patent citation network. Then, PatSTEG is modeled to study the evolutionary dynamics of patent citation formation by considering the semantic and topological information jointly. Extensive experiments are conducted on CNPat and public datasets to prove the superiority of PatSTEG over other state-of-the-art methods. All the results provide valuable references for patent literature research and technical exploration.

cs.IR↗

arXiv2023

Nonlinear stability of entropy waves for the Euler equations

In this article, we consider a class of the contact discontinuity for the full compressible Euler equations, namely the entropy wave, where the velocity is continuous across the interface while the density and the entropy can have jumps. The nonlinear stability of entropy waves is a longstanding open problem in multi-dimensional hyperbolic conservation laws. The rigorous treatments are challenging due to the characteristic discontinuity nature of the problem (G.-Q. Chen and Y.-G. Wang in \textit{Nonlinear partial differential equations}, Volume 7 of \textit{Abel Symp.}(2012)). In this article, we discover that the Taylor sign condition plays an essential role in the nonlinear stability of entropy waves. By deriving the evolution equation of the interface in the Eulerian coordinates, we relate the Taylor sign condition to the hyperbolicity of this evolution equation, which reveals a stability condition of the entropy wave. With the optimal regularity estimates of the interface, we can derive the a priori estimates without loss of regularity.

math.AP↗

arXiv2023

Sharp Global Well-posedness and Scattering of the Boltzmann Equation

We consider the 3D Boltzmann equation for the Maxwellian particle and soft potential with an angular cutoff. We prove sharp global well-posedness with initial data small in the scaling-critical space. The solution also remains in $L^{1}$ if the initial datum is in $L^{1}$, even at such low regularity. The key to existence, uniqueness and regularity criteria is the new bilinear spacetime estimates for the gain term, the proof of which is based on novel techniques from nonlinear dispersive PDEs including the atomic $U$-$V$ spaces, multi-linear frequency analysis, dispersive estimates, etc. To our knowledge, this is the first 3D sharp global result for the Boltzmann equation.

math.AP↗

arXiv2023

Well/Ill-posedness of the Boltzmann Equation with Soft Potential

We consider the Boltzmann equation with the soft potential and angular cutoff. Inspired by the methods from dispersive PDEs, we establish its sharp local well-posedness and ill-posedness in $H^{s}$ Sobolev space. We find the well/ill-posedness separation at regularity $s=\frac{d-1}{2}$, strictly $\frac{1}{2}$-derivative higher than the scaling-invariant index $s=\frac{d-2}{2}$, the usually expected separation point.

math.AP↗

arXiv2023

Localization and Regularity of the Integrated Density of States for Schrödinger Operators on $\mathbb{Z}^d$ with $C^2$-cosine Like Quasi-periodic Potential

In this paper, we study the multidimensional lattice Schrödinger operators with $C^2$-cosine like quasi-periodic (QP) potential. We establish quantitative Green's function estimates, the arithmetic version of Anderson (and dynamical) localization, and the finite volume version of $(\frac 12-)$-Hölder continuity of the integrated density of states (IDS) for such QP Schrödinger operators. Our proof is based on an extension of the fundamental multi-scale analysis (MSA) type method of Fröhlich-Spencer-Wittwer [\textit{Comm. Math. Phys.} 132 (1990): 5--25] to the higher lattice dimensions. We resolve the level crossing issue on eigenvalues parameterizations in the case of both higher lattice dimension and $C^2$ regular potential.

math-ph↗

arXiv2023

OpenAPMax: Abnormal Patterns-based Model for Real-World Alzheimer's Disease Diagnosis

Alzheimer's disease (AD) cannot be reversed, but early diagnosis will significantly benefit patients' medical treatment and care. In recent works, AD diagnosis has the primary assumption that all categories are known a prior -- a closed-set classification problem, which contrasts with the open-set recognition problem. This assumption hinders the application of the model in natural clinical settings. Although many open-set recognition technologies have been proposed in other fields, they are challenging to use for AD diagnosis directly since 1) AD is a degenerative disease of the nervous system with similar symptoms at each stage, and it is difficult to distinguish from its pre-state, and 2) diversified strategies for AD diagnosis are challenging to model uniformly. In this work, inspired by the concerns of clinicians during diagnosis, we propose an open-set recognition model, OpenAPMax, based on the anomaly pattern to address AD diagnosis in real-world settings. OpenAPMax first obtains the abnormal pattern of each patient relative to each known category through statistics or a literature search, clusters the patients' abnormal pattern, and finally, uses extreme value theory (EVT) to model the distance between each patient's abnormal pattern and the center of their category and modify the classification probability. We evaluate the performance of the proposed method with recent open-set recognition, where we obtain state-of-the-art results.

cs.LG↗

arXiv2023

OpenClinicalAI: An Open and Dynamic Model for Alzheimer's Disease Diagnosis

Although Alzheimer's disease (AD) cannot be reversed or cured, timely diagnosis can significantly reduce the burden of treatment and care. Current research on AD diagnosis models usually regards the diagnosis task as a typical classification task with two primary assumptions: 1) All target categories are known a priori; 2) The diagnostic strategy for each patient is consistent, that is, the number and type of model input data for each patient are the same. However, real-world clinical settings are open, with complexity and uncertainty in terms of both subjects and the resources of the medical institutions. This means that diagnostic models may encounter unseen disease categories and need to dynamically develop diagnostic strategies based on the subject's specific circumstances and available medical resources. Thus, the AD diagnosis task is tangled and coupled with the diagnosis strategy formulation. To promote the application of diagnostic systems in real-world clinical settings, we propose OpenClinicalAI for direct AD diagnosis in complex and uncertain clinical settings. This is the first powerful end-to-end model to dynamically formulate diagnostic strategies and provide diagnostic results based on the subject's conditions and available medical resources. OpenClinicalAI combines reciprocally coupled deep multiaction reinforcement learning (DMARL) for diagnostic strategy formulation and multicenter meta-learning (MCML) for open-set recognition. The experimental results show that OpenClinicalAI achieves better performance and fewer clinical examinations than the state-of-the-art model. Our method provides an opportunity to embed the AD diagnostic system into the current health care system to cooperate with clinicians to improve current health care.

cs.LG↗

arXiv2023

Photoswap: Personalized Subject Swapping in Images

In an era where images and visual content dominate our digital landscape, the ability to manipulate and personalize these images has become a necessity. Envision seamlessly substituting a tabby cat lounging on a sunlit window sill in a photograph with your own playful puppy, all while preserving the original charm and composition of the image. We present Photoswap, a novel approach that enables this immersive image editing experience through personalized subject swapping in existing images. Photoswap first learns the visual concept of the subject from reference images and then swaps it into the target image using pre-trained diffusion models in a training-free manner. We establish that a well-conceptualized visual subject can be seamlessly transferred to any image with appropriate self-attention and cross-attention manipulation, maintaining the pose of the swapped subject and the overall coherence of the image. Comprehensive experiments underscore the efficacy and controllability of Photoswap in personalized subject swapping. Furthermore, Photoswap significantly outperforms baseline methods in human ratings across subject swapping, background preservation, and overall quality, revealing its vast application potential, from entertainment to professional editing.

cs.CV↗

arXiv2023

Modulating Pretrained Diffusion Models for Multimodal Image Synthesis

We present multimodal conditioning modules (MCM) for enabling conditional image synthesis using pretrained diffusion models. Previous multimodal synthesis works rely on training networks from scratch or fine-tuning pretrained networks, both of which are computationally expensive for large, state-of-the-art diffusion models. Our method uses pretrained networks but \textit{does not require any updates to the diffusion network's parameters}. MCM is a small module trained to modulate the diffusion network's predictions during sampling using 2D modalities (e.g., semantic segmentation maps, sketches) that were unseen during the original training of the diffusion model. We show that MCM enables user control over the spatial layout of the image and leads to increased control over the image generation process. Training MCM is cheap as it does not require gradients from the original diffusion net, consists of only $\sim$1$\%$ of the number of parameters of the base diffusion model, and is trained using only a limited number of training examples. We evaluate our method on unconditional and text-conditional models to demonstrate the improved control over the generated images and their alignment with respect to the conditioning inputs.

cs.CV↗

arXiv2023

DualVector: Unsupervised Vector Font Synthesis with Dual-Part Representation

Automatic generation of fonts can be an important aid to typeface design. Many current approaches regard glyphs as pixelated images, which present artifacts when scaling and inevitable quality losses after vectorization. On the other hand, existing vector font synthesis methods either fail to represent the shape concisely or require vector supervision during training. To push the quality of vector font synthesis to the next level, we propose a novel dual-part representation for vector glyphs, where each glyph is modeled as a collection of closed "positive" and "negative" path pairs. The glyph contour is then obtained by boolean operations on these paths. We first learn such a representation only from glyph images and devise a subsequent contour refinement step to align the contour with an image representation to further enhance details. Our method, named DualVector, outperforms state-of-the-art methods in vector font synthesis both quantitatively and qualitatively. Our synthesized vector fonts can be easily converted to common digital font formats like TrueType Font for practical use. The code is released at https://github.com/thuliu-yt16/dualvector.

cs.CV↗

arXiv2023

Zero-viscosity limit of the compressible Naiver-Stokes equations in the analytic setting

In this paper, we study the zero-viscosity limit of the compressible Navier-Stokes equations in a half-space with non-slip boundary condition. We justify the Prandtl boundary layer expansion for the analytic data: the compressible Navier-Stokes equations can be approximated by the compressible Euler equations away from the boundary, and by the compressible Prandtl equation near the boundary.

math.AP↗

arXiv2023

Tollmien-Schlichting waves in the subsonic regime

The Tollmien-Schlichting (T-S) waves play a key role in the early stages of boundary layer transition. In a breakthrough work, Grenier, Guo and Nguyen gave the first rigorous construction of the T-S waves of temporal mode for the incompressible fluid. Yang and Zhang recently made an important contribution by constructing the compressible T-S waves of temporal mode for certain boundary layer profiles with Mach number $m<\frac{1}{\sqrt 3}$. In this paper, we construct the T-S waves of both temporal mode and spatial mode to the linearized compressible Navier-Stokes system around the boundary layer flow in the whole subsonic regime $m<1$, including the Blasius profile. Our approach is based on a novel iteration scheme between the quasi-incompressible and quasi-compressible systems, with a key ingredient being the solution of an Orr-Sommerfeld type equation using a new Airy-Airy-Rayleigh iteration instead of Rayleigh-Airy iteration introduced by Grenier, Guo and Nguyen. We believe the method developed in this work can be applied in solving other related problems for subsonic flows.

math.AP↗