arXiv subjects

Xicheng Zhang

Publications and source records attributed to Xicheng Zhang.

At least 37 records · Page 2Linked to original sources

arXiv2025

LipidBERT: A Lipid Language Model Pre-trained on METiS de novo Lipid Library

In this study, we generate and maintain a database of 10 million virtual lipids through METiS's in-house de novo lipid generation algorithms and lipid virtual screening techniques. These virtual lipids serve as a corpus for pre-training, lipid representation learning, and downstream task knowledge transfer, culminating in state-of-the-art LNP property prediction performance. We propose LipidBERT, a BERT-like model pre-trained with the Masked Language Model (MLM) and various secondary tasks. Additionally, we compare the performance of embeddings generated by LipidBERT and PhatGPT, our GPT-like lipid generation model, on downstream tasks. The proposed bilingual LipidBERT model operates in two languages: the language of ionizable lipid pre-training, using in-house dry-lab lipid structures, and the language of LNP fine-tuning, utilizing in-house LNP wet-lab data. This dual capability positions LipidBERT as a key AI-based filter for future screening tasks, including new versions of METiS de novo lipid libraries and, more importantly, candidates for in vivo testing for orgran-targeting LNPs. To the best of our knowledge, this is the first successful demonstration of the capability of a pre-trained language model on virtual lipids and its effectiveness in downstream tasks using web-lab data. This work showcases the clever utilization of METiS's in-house de novo lipid library as well as the power of dry-wet lab integration.

cs.CL↗

arXiv2025

Stochastic Transport Maps in Diffusion Models and Sampling

In this work, we present a theoretical and computational framework for constructing stochastic transport maps between probability distributions using diffusion processes. We begin by proving that the time-marginal distribution of the sum of two independent diffusion processes satisfies a Fokker-Planck equation. Building on this result and applying Ambrosio-Figalli-Trevisan's superposition principle, we establish the existence and uniqueness of solutions to the associated stochastic differential equation (SDE). Leveraging these theoretical foundations, we develop a method to construct (stochastic) transport maps between arbitrary probability distributions using dynamical ordinary differential equations (ODEs) and SDEs. Furthermore, we introduce a unified framework that generalizes and extends a broad class of diffusion-based generative models and sampling techniques. Finally, we analyze the convergence properties of particle approximations for the SDEs underlying our framework, providing theoretical guarantees for their practical implementation. This work bridges theoretical insights with practical applications, offering new tools for generative modeling and sampling in high-dimensional spaces.

math.PR↗

arXiv2025

Stealthy Backdoor Attack to Real-world Models in Android Apps

Powered by their superior performance, deep neural networks (DNNs) have found widespread applications across various domains. Many deep learning (DL) models are now embedded in mobile apps, making them more accessible to end users through on-device DL. However, deploying on-device DL to users' smartphones simultaneously introduces several security threats. One primary threat is backdoor attacks. Extensive research has explored backdoor attacks for several years and has proposed numerous attack approaches. However, few studies have investigated backdoor attacks on DL models deployed in the real world, or they have shown obvious deficiencies in effectiveness and stealthiness. In this work, we explore more effective and stealthy backdoor attacks on real-world DL models extracted from mobile apps. Our main justification is that imperceptible and sample-specific backdoor triggers generated by DNN-based steganography can enhance the efficacy of backdoor attacks on real-world models. We first confirm the effectiveness of steganography-based backdoor attacks on four state-of-the-art DNN models. Subsequently, we systematically evaluate and analyze the stealthiness of the attacks to ensure they are difficult to perceive. Finally, we implement the backdoor attacks on real-world models and compare our approach with three baseline methods. We collect 38,387 mobile apps, extract 89 DL models from them, and analyze these models to obtain the prerequisite model information for the attacks. After identifying the target models, our approach achieves an average of 12.50% higher attack success rate than DeepPayload while better maintaining the normal performance of the models. Extensive experimental results demonstrate that our method enables more effective, robust, and stealthy backdoor attacks on real-world models.

cs.CR↗

arXiv2024

Distribution-flow dependent SDEs driven by (fractional) Brownian motion and Navier-Stokes equations

Motivated by the probabilistic representation for solutions of the Navier-Stokes equations, we introduce a novel class of stochastic differential equations that depend on the entire flow of its time marginals. We establish the existence and uniqueness of both strong and weak solutions under one-sided Lipschitz conditions and for singular drifts. These newly proposed distribution-flow dependent stochastic differential equations are closely connected to quasilinear backward Kolmogorov equations and Fokker-Planck equations. Furthermore, we investigate a stochastic version of the 2D-Navier-Stokes equation associated with fractional Brownian noise. We demonstrate the global well-posedness and smoothness of solutions when the Hurst parameter $H$ lies in the range $(0, \frac12)$ and the initial vorticity is a finite signed measure.

math.PR↗

arXiv2024

Heat kernel estimates for nonlocal kinetic operators

In this paper, we employ probabilistic techniques to derive sharp, explicit two-sided estimates for the heat kernel of the nonlocal kinetic operator $$ Δ^{α/2}_v + v \cdot \nabla_x, \quad α\in (0, 2),\ (x,v)\in {\mathbb R}^{d}\times{\mathbb R}^d,$$ where $ Δ^{α/2}_v $ represents the fractional Laplacian acting on the velocity variable $v$. Additionally, we establish logarithmic gradient estimates with respect to both the spatial variable $x$ and the velocity variable $v$. In fact, the estimates are developed for more general non-symmetric stable-like operators, demonstrating explicit dependence on the lower and upper bounds of the kernel functions. These results, in particular, provide a solution to a fundamental problem in the study of \emph{nonlocal} kinetic operators.

math.PR↗

arXiv2024

$W_{\bf d}$-convergence rate of EM schemes for invariant measures of supercritical stable SDEs

By establishing the regularity estimates for nonlocal Stein/Poisson equations under $γ$-order Hölder and dissipative conditions on the coefficients, we derive the $W_{\bf d}$-convergence rate for the Euler-Maruyama schemes applied to the invariant measure of SDEs driven by multiplicative $α$-stable noises with $α\in (\frac{1}{2}, 2)$, where $W_{\bf d}$ denotes the Wasserstein metric with ${\bf d}(x,y)=|x-y|^γ\wedge 1$ and $γ\in ((1-α)_+, 1]$.

math.PR↗

arXiv2024

Averaging principle for SDEs with singular drifts driven by $α$-stable processes

In this paper, we investigate the convergence rate of the averaging principle for stochastic differential equations (SDEs) with $β$-Hölder drift driven by $α$-stable processes. More specifically, we first derive the Schauder estimate for nonlocal partial differential equations (PDEs) associated with the aforementioned SDEs, within the framework of Besov-Hölder spaces. Then we consider the case where $(α,β)\in(0,2)\times(1-\tfracα{2},1)$. Using the Schauder estimate, we establish the strong convergence rate for the averaging principle. In particular, under suitable conditions we obtain the optimal rate of strong convergence when $(α,β)\in(\tfrac{2}{3},1]\times(2-\tfrac{3α}{2},1)\cup(1,2)\times(\tfracα{2},1)$. Furthermore, when $(α,β)\in(0,1]\times(1-α,1-\tfracα{2}]\cup(1,2)\times(\tfrac{1-α}{2},1-\tfracα{2}]$, we show the convergence of the martingale solutions of original systems to that of the averaged equation. When $α\in(1,2)$, the drift can be a distribution.

math.DS↗

arXiv2024

New algorithms for sampling and diffusion models

Drawing from the theory of stochastic differential equations, we introduce a novel sampling method for known distributions and a new algorithm for diffusion generative models with unknown distributions. Our approach is inspired by the concept of the reverse diffusion process, widely adopted in diffusion generative models. Additionally, we derive the explicit convergence rate based on the smooth ODE flow. For diffusion generative models and sampling, we establish a dimension-free particle approximation convergence result. Numerical experiments demonstrate the effectiveness of our method. Notably, unlike the traditional Langevin method, our sampling method does not require any regularity assumptions about the density function of the target distribution. Furthermore, we also apply our method to optimization problems.

math.ST↗

arXiv2024

Heat kernel estimates for kinetic SDEs with drifts being unbounded and in Kato's class

In this paper we investigate the existence and uniqueness of weak solutions for kinetic stochastic differential equations with Hölder diffusion and unbounded singular drifts in Kato's class. Moreover, we also establish sharp two-sided estimates for the density of the solution. In particular, the drift $b$ can be in the mixed $L^q_tL^{p_1}_{x_1}L^{p_2}_{x_2}$ space with $\frac2q+\frac{d}{p_1}+\frac{3d}{p_2}<1$. As an application, we show the existence and uniqueness of weak solution to a second order singular interacting particle system in ${\mathbb R}^{d N}$.

math.PR↗

arXiv2023

Compound Poisson particle approximation for McKean-Vlasov SDEs

We present a comprehensive discretization scheme for linear and nonlinear stochastic differential equations (SDEs) driven by either Brownian motions or $α$-stable processes. Our approach utilizes compound Poisson particle approximations, allowing for simultaneous discretization of both the time and space variables in McKean-Vlasov SDEs. Notably, the approximation processes can be represented as a Markov chain with values on a lattice. Importantly, we demonstrate the propagation of chaos under relatively mild assumptions on the coefficients, including those with polynomial growth. This result establishes the convergence of the particle approximations towards the true solutions of the McKean-Vlasov SDEs. By only imposing moment conditions on the intensity measure of compound Poisson processes, our approximation exhibits universality. In the case of ordinary differential equations (ODEs), we investigate scenarios where the drift term satisfies the one-sided Lipschitz assumption. We prove the optimal convergence rate for Filippov solutions in this setting. Additionally, we establish a functional central limit theorem (CLT) for the approximation of ODEs and show the convergence of invariant measures for linear SDEs. As a practical application, we construct a compound Poisson approximation for 2D-Navier Stokes equations on the torus and demonstrate the optimal convergence rate.

math.PR↗

arXiv2023

Second order fractional mean-field SDEs with singular kernels and measure initial data

In this paper we establish the local and global well-posedness of weak and strong solutions to second order fractional mean-field SDEs with singular/distribution interaction kernels and measure initial value, where the kernel can be Newton or Coulomb potential, Riesz potential, Biot-Savart law, etc. Moreover, we also show the stability, smoothness and the short time singularity and large time decay estimates of the distribution density. Our results reveal a phenomenon that for {\it nonlinear} mean-field equations, the regularity of the initial distribution could balance the singularity of the kernel. The precise relationship between the singularity of kernels and the regularity of initial values are calculated, which belongs to the subcritical regime in the scaling sense. In particular, our results provide a microscopic probabilistic explanation and establish a unified treatment for many physical models such as the fractional Vlasov-Poisson-Fokker-Planck system, the vorticity formulation of 2D-fractal Navier-Stokes equations, surface quasi-geostrophic models, fractional porous medium equation with viscosity, etc.

math.AP↗

arXiv2022

Strong convergence of propagation of chaos for McKean-Vlasov SDEs with singular interactions

In this work we show the strong convergence of propagation of chaos for the particle approximation of McKean-Vlasov SDEs with singular $L^p$-interactions as well as for the moderate interaction particle systems on the level of particle trajectories. One of the main obstacles is to establish the strong well-posedness of the SDEs for particle systems with singular interaction. To this end, we extend the results on strong well-posedness of Krylov and Röckner \cite{Kr-Ro} to the case of mixed $L^p$-drifts, where the heat kernel estimates play a crucial role. Moreover, when the interaction kernel is bounded measurable, we also obtain the optimal rate of strong convergence, which is partially based on Jabin and Wang's entropy method \cite{JW16} and Zvonkin's transformation.

math.PR↗

arXiv2022

Second order McKean-Vlasov SDEs and kinetic Fokker-Planck-Kolmogorov equations

In this paper we study second order stochastic differential equations with measurable and density-distribution dependent coefficients. Through establishing a maximum principle for kinetic Fokker-Planck-Kolmogorov equations with distribution-valued inhomogeneous term, we show the existence of weak solutions under mild assumptions. Moreover, by using the Hölder regularity estimate obtained recently in \cite{GIMV19}, we also show the well-posedness of generalized martingale problems when diffusion coefficients only depend on the position variable (not necessarily continuous). Even in the non density-distribution dependent case, it seems that this is the first result about the well-posedness of SDEs with measurable diffusion coefficients.

math.PR↗

arXiv2022

Ergodicity of supercritical SDEs driven by $α$-stable processes and heavy-tailed sampling

Let $α\in(0,2)$ and $d\in\mathbb{N}$. Consider the following stochastic differential equation (SDE) driven by $α$-stable process in $\mathbb{R}^d$: $$ dX_t=b(X_t)dt+σ(X_{t-})d L^α_t, \quad X_0=x\in\mathbb{R}^d, $$ where $b:\mathbb{R}^d\to\mathbb{R}^d$ and $σ:\mathbb{R}^d\to\mathbb{R}^d\otimes\mathbb{R}^d$ are locally $γ$-Hölder continuous with $γ\in(0\vee(1-α)^+,1]$, $L^α_t$ is a $d$-dimensional rotationally invariant $α$-stable process. Under some dissipative and non-degenerate assumptions on $b,σ$, we show the $V$-uniformly exponential ergodicity for the semigroup $P_t$ associated with $(X_t(x),t\geq 0)$. Our proofs are mainly based on the heat kernel estimates recently established in \cite{MZ20} through showing the strong Feller property and the irreducibility of $P_t$. It is interesting that when $α$ goes to zero, the diffusion coefficient $σ$ can grow faster than drift $b$. As applications, we put forward a new heavy-tailed sampling scheme.

math.PR↗

arXiv2021

Singular kinetic equations and applications

In this paper we study singular kinetic equations on $\mathbb{R}^{2d}$ by the paracontrolled distribution method introduced in \cite{GIP15}. We first develop paracontrolled calculus in the kinetic setting, and use it to establish the global well-posedness for the linear singular kinetic equations under the assumptions that the products of singular terms are well-defined. We also demonstrate how the required products can be defined in the case that singular term is a Gaussian random field by probabilistic calculation. Interestingly, although the terms in the zeroth Wiener chaos of regularization approximation are not zero, they converge in suitable weighted Besov spaces and no renormalization is required. As applications the global well-posedness for a nonlinear kinetic equation with singular coefficients is obtained by the entropy method. Moreover, we also solve the martingale problem for nonlinear kinetic distribution dependent stochastic differential equations with singular drifts.

math.PR↗

arXiv2021

Maximum principle for non-uniformly parabolic equations and applications

In this paper we study the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi's iteration. As applications, we show the existence of weak solutions for possibly degenerate stochastic differential equations with singular diffusion and drift coefficients. Moreover, by the Markov selection theorem of Krylov [8], we also establish the existence of the associated strong Markov family.

math.AP↗

arXiv2021

Cauchy Problem of Stochastic Kinetic Equations

In this paper we establish the optimal regularity estimates for the Cauchy problem of stochastic kinetic equations with random coefficients in anisotropic Besov spaces. As applications, we study the nonlinear filtering problem for a degenerate diffusion process, and obtain the existence and regularity of conditional probability densities under few assumptions. Moreover, we also show the well-posedness for a class of super-linear growth stochastic kinetic equations driven by velocity-time white noises, as well as a kinetic version of Parabolic Anderson Model with measure as initial values.

math.PR↗

arXiv2020

Weak solutions of McKean-Vlasov SDEs with supercritical drifts

Consider the following McKean-Vlasov SDE: $$ d X_t=\sqrt{2}d W_t+\int_{{\mathbb R}^d}K(t,X_t-y)μ_{X_t}(dy)d t,\ \ X_0=x, $$ where $μ_{X_t}$ stands for the distribution of $X_t$ and $K(t,x): {\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ is a time-dependent divergence free vector field. Under the assumption $K\in L^q_t(\widetilde L_x^p)$ with $\frac dp+\frac2q<2$, where $\widetilde L^p_x$ stands for the localized $L^p$-space, we show the existence of weak solutions to the above SDE. As an application, we provide a new proof for the existence of weak solutions to 2D-Navier-Stokes equations with measure as initial vorticity.

math.PR↗