arXiv subjects
Publications and source records attributed to Yuzhe Zhu.
We prove interior Hölder regularity for solutions of the stationary kinetic Fokker-Planck equation in non-divergence form in space dimension one. The estimate is independent of any small-oscillation condition on the diffusion coefficient.
We consider the Fisher information for spatially homogeneous multi-species Landau system. We show that the mass-weighted Fisher information is monotone decreasing in time along the solutions of the Landau system with a general class of interaction potentials.
Propelled by the breakthrough in deep generative models, audio-to-image generation has emerged as a pivotal cross-modal task that converts complex auditory signals into rich visual representations. However, previous works only focus on single-source audio inputs for image generation, ignoring the multi-source characteristic in natural auditory scenes, thus limiting the performance in generating comprehensive visual content. To bridge this gap, we propose a method called MACS to conduct multi-source audio-to-image generation. To our best knowledge, this is the first work that explicitly separates multi-source audio to capture the rich audio components before image generation. MACS is a two-stage method. In the first stage, multi-source audio inputs are separated by a weakly supervised method, where the audio and text labels are semantically aligned by casting into a common space using the large pre-trained CLAP model. We introduce a ranking loss to consider the contextual significance of the separated audio signals. In the second stage, effective image generation is achieved by mapping the separated audio signals to the generation condition using only a trainable adapter and a MLP layer. We preprocess the LLP dataset as the first full multi-source audio-to-image generation benchmark. The experiments are conducted on multi-source, mixed-source, and single-source audio-to-image generation tasks. The proposed MACS outperforms the current state-of-the-art methods in 17 out of the 21 evaluation indexes on all tasks and delivers superior visual quality.
We establish sharp boundary regularity results for solutions to kinetic Fokker-Planck equations under prescribed inflow boundary conditions, providing precise quantification of the boundary hypoelliptic regularization effect. For equations with rough coefficients, we characterize the behaviours for solutions on grazing and incoming boundaries. In particular, in the absence of influxes and sources, an explicit exponential infinite-order vanishing estimate is derived near incoming boundaries. When the coefficients are regular, we obtained the optimal Hölder regularity on grazing boundaries and general Schauder-type estimates away from them.
We consider kinetic models for Fermi-Dirac-like particles obeying the exclusion principle. A generalized notion of Fisher information, tailored to kinetic equations of Fermi-Dirac-Fokker-Planck type, is introduced via the associated entropy dissipation identity. We show that, subject to a suitable upper bound on the initial data, this quantity decreases along solutions of the Fermi-Dirac-Fokker-Planck equation, while monotonicity can fail in the absence of such a bound. We also discuss the time evolution of this Fermi-Dirac-type Fisher information for the heat equation and the linear-type Landau-Fermi-Dirac equation with Maxwell molecules.
Neural Radiance Field (NeRF) models are implicit neural scene representation methods that offer unprecedented capabilities in novel view synthesis. Semantically-aware NeRFs not only capture the shape and radiance of a scene, but also encode semantic information of the scene. The training of semantically-aware NeRFs typically requires pixel-level class labels, which can be prohibitively expensive to collect. In this work, we explore active learning as a potential solution to alleviate the annotation burden. We investigate various design choices for active learning of semantically-aware NeRF, including selection granularity and selection strategies. We further propose a novel active learning strategy that takes into account 3D geometric constraints in sample selection. Our experiments demonstrate that active learning can effectively reduce the annotation cost of training semantically-aware NeRF, achieving more than 2X reduction in annotation cost compared to random sampling.
We use the methods of commutator and fundamental solutions to establish averaging lemmas and hypoelliptic estimates for purely kinetic transport equations. Assuming certain amount of velocity regularity for solutions, we extend our analysis using the commutator method to derive the averaging and hypoelliptic regularity properties for kinetic equations in the presence of general inhomogeneous fluxes. These results find applications in the study of hypoelliptic advection-diffusion equations and kinetic formulations of hyperbolic conservation laws including Burgers' equation with transport and isentropic gas dynamics.
We establish $L^2$-exponential decay properties for linear dissipative kinetic equations, including the time-relaxation and Fokker-Planck models, in bounded spatial domains with general boundary conditions that may not conserve mass. Their diffusion asymptotics in $L^2$ is also derived under general Maxwell boundary conditions. The proofs are simply based on energy estimates together with previous ideas from $L^2$-hypocoercivity and relative entropy methods.
We obtain the existence, uniqueness and regularity results for solutions to kinetic Fokker-Planck equations with bounded measurable coefficients in the presence of boundary conditions, including the inflow, diffuse reflection and specular reflection cases.
We explore quantitative propagation of smallness for solutions of two-dimensional elliptic equations from sets of positive $δ$-dimensional Hausdorff content for any $δ>0$. In particular, the gradients of solutions to divergence form equations with Hölder continuous coefficients, as well as those of nondivergence form equations with measurable coefficients, can be quantitatively estimated from the small sets.
We investigate the Cauchy problem and the diffusion asymptotics for a spatially inhomogeneous kinetic model associated to a nonlinear Fokker-Planck operator. We derive the global well-posedness result with instantaneous smoothness effect, when the initial data lies below a Maxwellian. The proof relies on the hypoelliptic analog of classical parabolic theory, as well as a positivity-spreading result based on the Harnack inequality and barrier function methods. Moreover, the scaled equation leads to the fast diffusion flow under the low field limit. The relative phi-entropy method enables us to see the connection between the overdamped dynamics of the nonlinearly coupled kinetic model and the correlated fast diffusion. The global in time quantitative diffusion asymptotics is then derived by combining entropic hypocoercivity, relative phi-entropy and barrier function methods.
This article addresses the local boundedness and Hölder continuity of weak solutions to kinetic Fokker-Planck equations with general transport operators and rough coefficients. These results are due to the mixing effect of diffusion and transport. Although the equation is parabolic only in the velocity variable, it has a hypoelliptic structure provided that the transport part $\partial_t+b(v)\cdot\nabla_x$ is nondegenerate in some sense. We achieve the results by revisiting the method, proposed by Golse, Imbert, Mouhot and Vasseur in the case $b(v)= v$, that combines the elliptic De Giorgi-Nash-Moser theory with velocity averaging lemmas.