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Zhifei Zhang

Publications and source records attributed to Zhifei Zhang.

At least 145 records · Page 8Linked to original sources
arXiv2022

Dynamics near Couette flow for the $β$-plane equation

In this paper, we study stationary structures near the planar Couette flow in Sobolev spaces on a channel $\mathbb{T}\times[-1,1]$, and asymptotic behavior of Couette flow in Gevrey spaces on $\mathbb{T}\times\mathbb{R}$ for the $β$-plane equation. Let $T>0$ be the horizontal period of the channel and $α={2π\over T}$ be the wave number. We obtain a sharp region $O$ in the whole $(α,β)$ half-plane such that non-parallel steadily traveling waves do not exist for $(α,β)\in O$ and such traveling waves exist for $(α,β)$ in the remaining regions, near Couette flow for $H^{\geq5}$ velocity perturbation. The borderlines between the region $O$ and its remaining are determined by two curves of the principal eigenvalues of singular Rayleigh-Kuo operators. Our results reveal that there exists $β_*>0$ such that if $|β|\leq β_*$, then non-parallel traveling waves do not exist for any $T>0$, while if $|β|>β_*$, then there exists a critical period $T_β>0$ so that such traveling waves exist for $T\in \left[T_β,\infty\right)$ and do not exist for $T\in \left(0,T_β\right)$, near Couette flow for $H^{\geq5}$ velocity perturbation. This contrasting dynamics plays an important role in studying the long time dynamics near Couette flow with Coriolis effects. Moreover, for any $β\neq0$ and $T>0$, there exist no non-parallel traveling waves with speeds converging in $(-1,1)$ near Couette flow for $H^{\geq5}$ velocity perturbation, in contrast to this, we construct non-shear stationary solutions near Couette flow for $H^{<{5\over2}}$ velocity perturbation, which is a generalization of Theorem 1 in [22] but the construction is more difficult due to the $β$'s term. Finally, we prove nonlinear inviscid damping for Couette flow in some Gevrey spaces by extending the method of [4] to the $β$-plane equation on $\mathbb{T}\times\mathbb{R}$.

math.AP↗
arXiv2022

A Multi-Implicit Neural Representation for Fonts

Fonts are ubiquitous across documents and come in a variety of styles. They are either represented in a native vector format or rasterized to produce fixed resolution images. In the first case, the non-standard representation prevents benefiting from latest network architectures for neural representations; while, in the latter case, the rasterized representation, when encoded via networks, results in loss of data fidelity, as font-specific discontinuities like edges and corners are difficult to represent using neural networks. Based on the observation that complex fonts can be represented by a superposition of a set of simpler occupancy functions, we introduce \textit{multi-implicits} to represent fonts as a permutation-invariant set of learned implict functions, without losing features (e.g., edges and corners). However, while multi-implicits locally preserve font features, obtaining supervision in the form of ground truth multi-channel signals is a problem in itself. Instead, we propose how to train such a representation with only local supervision, while the proposed neural architecture directly finds globally consistent multi-implicits for font families. We extensively evaluate the proposed representation for various tasks including reconstruction, interpolation, and synthesis to demonstrate clear advantages with existing alternatives. Additionally, the representation naturally enables glyph completion, wherein a single characteristic font is used to synthesize a whole font family in the target style.

cs.CV↗
arXiv2021

The derivation of the compressible Euler equation from quantum many-body dynamics

We study the three dimensional many-particle quantum dynamics in mean-field setting. We forge together the hierarchy method and the modulated energy method. We prove rigorously that the compressible Euler equation is the limit as the particle number tends to infinity and the Planck's constant tends to zero. We establish strong and quantitative microscopic to macroscopic convergence of mass and momentum densities up to the 1st blow up time of the limiting Euler equation. We justify that the macroscopic pressure emerges from the space-time averages of microscopic interactions, which are in fact, Strichartz-type bounds. We have hence found a physical meaning for Strichartz type bounds which were first raised by Klainerman and Machedon in this context.

math.AP↗
arXiv2021

OpenClinicalAI: enabling AI to diagnose diseases in real-world clinical settings

This paper quantitatively reveals the state-of-the-art and state-of-the-practice AI systems only achieve acceptable performance on the stringent conditions that all categories of subjects are known, which we call closed clinical settings, but fail to work in real-world clinical settings. Compared to the diagnosis task in the closed setting, real-world clinical settings pose severe challenges, and we must treat them differently. We build a clinical AI benchmark named Clinical AIBench to set up real-world clinical settings to facilitate researches. We propose an open, dynamic machine learning framework and develop an AI system named OpenClinicalAI to diagnose diseases in real-world clinical settings. The first versions of Clinical AIBench and OpenClinicalAI target Alzheimer's disease. In the real-world clinical setting, OpenClinicalAI significantly outperforms the state-of-the-art AI system. In addition, OpenClinicalAI develops personalized diagnosis strategies to avoid unnecessary testing and seamlessly collaborates with clinicians. It is promising to be embedded in the current medical systems to improve medical services.

cs.AI↗
arXiv2021

Asymptotic Behavior of the Steady Prandtl Equation

We study the asymptotic behavior of the Oleinik's solution to the steady Prandtl equation when the outer flow $U(x)=1$. Serrin proved that the Oleinik's solution converges to the famous Blasius solution $\bar u$ in $L^\infty_y$ sense as $x\rightarrow+\infty$. The explicit decay estimates of $u-\bar u$ and its derivatives were proved by Iyer[ARMA 237(2020)] when the initial data is a small localized perturbation of the Blasius profile. In this paper, we prove the explicit decay estimate of $\|u(x,y)-\bar{u}(x,y)\|_{L^\infty_y}$ for general initial data with exponential decay. We also prove the decay estimates of its derivatives when the data has an additional concave assumption. Our proof is based on the maximum principle technique. The key ingredient is to find a series of barrier functions.

math.AP↗
arXiv2021

Font Completion and Manipulation by Cycling Between Multi-Modality Representations

Generating font glyphs of consistent style from one or a few reference glyphs, i.e., font completion, is an important task in topographical design. As the problem is more well-defined than general image style transfer tasks, thus it has received interest from both vision and machine learning communities. Existing approaches address this problem as a direct image-to-image translation task. In this work, we innovate to explore the generation of font glyphs as 2D graphic objects with the graph as an intermediate representation, so that more intrinsic graphic properties of font styles can be captured. Specifically, we formulate a cross-modality cycled image-to-image model structure with a graph constructor between an image encoder and an image renderer. The novel graph constructor maps a glyph's latent code to its graph representation that matches expert knowledge, which is trained to help the translation task. Our model generates improved results than both image-to-image baseline and previous state-of-the-art methods for glyph completion. Furthermore, the graph representation output by our model also provides an intuitive interface for users to do local editing and manipulation. Our proposed cross-modality cycled representation learning has the potential to be applied to other domains with prior knowledge from different data modalities. Our code is available at https://github.com/VITA-Group/Font_Completion_Graph.

cs.CV↗
arXiv2021

Matrix-valued Allen-Cahn equation and the Keller-Rubinstein-Sternberg problem

In this paper, we consider the sharp interface limit of a matrix-valued Allen-Cahn equation, which takes the form: $$\partial_t A=ΔA-\varepsilon^{-2}( A A^{\mathrm{T}}A- A)~~\text{with}~~A:Ω\subset\mathbb{R}^m\to\mathbb{R}^{n\times n}.$$ We show that the sharp interface system is a two-phases flow system: the interface evolves according to the motion by mean curvature; in the two bulk phase regions, the solution obeys the heat flow of harmonic maps with values in $O^+(n)$ and $O^-(n)$ (represent the sets of $n\times n$ orthogonal matrices with determinant $+1$ and $-1$ respectively); on the interface, the phase matrices in two sides satisfy a novel mixed boundary condition. The above result provides a solution to the Keller-Rubinstein-Sternberg's (conjecture) problem in this $O(n)$ setting. Our proof relies on two key ingredients. First, in order to construct the approximate solution by matched asymptotic expansions as the standard approach does not seem to work, we introduce the notion of quasi-minimal connecting orbits. They satisfy the usual leading order equations up to some small higher order terms. In addition, the linearized systems around these quasi-minimal orbits needs to be solvable up to some good remainders. These flexibilities are needed for the possible "degenerations" or higher dimensional kernels for the linearized operators due to intriguing boundary conditions at the sharp interface. The second key point is to establish a spectral uniform lower bound estimate for the linearized operator around approximate solutions. To this end, we introduce additional decompositions to reduce the problem into the coercive estimates of several linearized operators for scalar functions and some singular product estimates which is accomplished by exploring special cancellation structures between eigenfunctions of these linearized operators.

math.AP↗
arXiv2021

Linear stability of the Couette flow in the 3D isentropic compressible Navier-Stokes equations

Consider the linear stability of the three dimensional isentropic compressible Navier-Stokes equations on $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. We prove the enhanced dissipation phenomenon for the linearized isentropic compressible Navier-Stokes equations around the Couette flow $(y, 0, 0)^\top$. Moreover, the lift-up phenomenon is also shown in this paper. Compared with the 3D incompressible Navier-Stokes equations [Ann. of Math.,185(2017), 541--608], the lift-up effect here is stronger due to the loss of the incompressible condition.

math.AP↗
arXiv2021

On the $L^\infty$ stability of Prandtl expansions in Gevrey class

In this paper, we prove the $L^\infty\cap L^2$ stability of Prandtl expansions of shear flow type as $\big(U(y/\sqrtν),0\big)$ for the initial perturbation in the Gevrey class, where $U(y)$ is a monotone and concave function and $ν$ is the viscosity coefficient. To this end, we develop the direct resolvent estimate method for the linearized Orr-Sommerfeld operator instead of the Rayleigh-Airy iteration method introduced by Grenier, Guo and Nguyen.

math.AP↗
arXiv2020

Rethinking Text Segmentation: A Novel Dataset and A Text-Specific Refinement Approach

Text segmentation is a prerequisite in many real-world text-related tasks, e.g., text style transfer, and scene text removal. However, facing the lack of high-quality datasets and dedicated investigations, this critical prerequisite has been left as an assumption in many works, and has been largely overlooked by current research. To bridge this gap, we proposed TextSeg, a large-scale fine-annotated text dataset with six types of annotations: word- and character-wise bounding polygons, masks and transcriptions. We also introduce Text Refinement Network (TexRNet), a novel text segmentation approach that adapts to the unique properties of text, e.g. non-convex boundary, diverse texture, etc., which often impose burdens on traditional segmentation models. In our TexRNet, we propose text specific network designs to address such challenges, including key features pooling and attention-based similarity checking. We also introduce trimap and discriminator losses that show significant improvement on text segmentation. Extensive experiments are carried out on both our TextSeg dataset and other existing datasets. We demonstrate that TexRNet consistently improves text segmentation performance by nearly 2% compared to other state-of-the-art segmentation methods. Our dataset and code will be made available at https://github.com/SHI-Labs/Rethinking-Text-Segmentation.

cs.CV↗
arXiv2020

Texture Hallucination for Large-Factor Painting Super-Resolution

We aim to super-resolve digital paintings, synthesizing realistic details from high-resolution reference painting materials for very large scaling factors (e.g., 8X, 16X). However, previous single image super-resolution (SISR) methods would either lose textural details or introduce unpleasing artifacts. On the other hand, reference-based SR (Ref-SR) methods can transfer textures to some extent, but is still impractical to handle very large factors and keep fidelity with original input. To solve these problems, we propose an efficient high-resolution hallucination network for very large scaling factors with an efficient network structure and feature transferring. To transfer more detailed textures, we design a wavelet texture loss, which helps to enhance more high-frequency components. At the same time, to reduce the smoothing effect brought by the image reconstruction loss, we further relax the reconstruction constraint with a degradation loss which ensures the consistency between downscaled super-resolution results and low-resolution inputs. We also collected a high-resolution (e.g., 4K resolution) painting dataset PaintHD by considering both physical size and image resolution. We demonstrate the effectiveness of our method with extensive experiments on PaintHD by comparing with SISR and Ref-SR state-of-the-art methods.

eess.IV↗
arXiv2020

Scaling invariant Serrin criterion via one velocity component for the Navier-Stokes equations

In this paper, we prove that the Leray weak solution $u : \mathbb{R}^3\times (0, T)\rightarrow\mathbb{R}^3 $ of the Navier-Stokes equations is regular in $\mathbb{R}^3\times (0,T)$ under the scaling invariant Serrin condition imposed on one component of the velocity $u_3\in L^{q,1}(0, T;L^p(\mathbb{R}^3))$ with \[ \frac{2}{q}+\frac{3}{p}\leq 1,\quad 3<p<+\infty. \] This result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.

math.AP↗
arXiv2020

Transition threshold for the 3D Couette flow in a finite channel

In this paper, we study nonlinear stability of the 3D plane Couette flow $(y,0,0)$ at high Reynolds number ${Re}$ in a finite channel $\mathbb{T}\times [-1,1]\times \mathbb{T}$. It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1}$ for some $c_0>0$ independent of $Re$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow in the $L^\infty$ sense, and rapidly converges to a streak solution for $t\gg Re^{\frac 13}$ due to the mixing-enhanced dissipation effect. This result confirms the transition threshold conjecture proposed by Trefethen et al.(Science, 261(1993), 578-584). To this end, we develop the resolvent estimate method to establish the space-time estimates for the full linearized Navier-Stokes system around the flow $(V(t,y,z), 0,0)$, where $V(t,y,z)$ is a small perturbation(but independent of $Re$) of the Couette flow $y$.

math.AP↗
arXiv2020

Blow-up criterion for the 2-D Prandtl equation

In this paper, we consider the 2-D Prandtl equation with constant outer flow and monotonic data. We prove that if the curvature of the velocity distribution(i.e., $\partial_y^2u$) is bounded near the boundary, then the solution can not develop the singularity.

math.AP↗