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Zhenfu Wang

Publications and source records attributed to Zhenfu Wang.

36 records · Page 2Linked to original sources
arXiv2024

Gaussian fluctuations for interacting particle systems with singular kernels

We consider the asymptotic behavior of the fluctuations for the empirical measures of interacting particle systems with singular kernels. We prove that the sequence of fluctuation processes converges in distribution to a generalized Ornstein-Uhlenbeck process. Our result considerably extends classical results to singular kernels, including the Biot-Savart law. The result applies to the point vortex model approximating the 2D incompressible Navier-Stokes equation and the 2D Euler equation. We also obtain Gaussianity and optimal regularity of the limiting Ornstein-Uhlenbeck process. The method relies on the martingale approach and the Donsker-Varadhan variational formula, which transfers the uniform estimate to some exponential integrals. Estimation of those exponential integrals follows by cancellations and combinatorics techniques and is of the type of large deviation principle.

math.PR↗
arXiv2024

Propagation of Chaos for 2D Log Gas on the Whole Space

We derive the quantitative propagation of chaos in the sense of relative entropy for the first time for the 2D Log gas or the weakly interacting particle systems with 2D Coulomb interactions on the whole space. We resolve this problem by adapting the modulated free energy method in [BJW23] to the whole space setting and establishing the crucial logarithmic growth estimates for the mean-field Poisson-Nernst-Planck (PNP) equation of single component via the parabolic maximum principle.

math.AP↗
arXiv2024

Relative Entropy Method for Particle Approximation of the Landau Equation for Maxwellian Molecules

We derive the spatially homogeneous Landau equation for Maxwellian molecules from a natural stochastic interacting particle system. More precisely, we control the relative entropy between the joint law of the particle system and the tensorized law of the Landau equation. To obtain this, we establish as key tools the pointwise logarithmic gradient and Hessian estimates of the density function and also a new Law of Large Numbers result for the particle system. The logarithmic estimates are derived via the Bernstein method and the parabolic maximum principle, while the Law of Large Numbers result comes from crucial observations on the control of moments at the particle level.

math.AP↗
arXiv2024

Uniform-in-time propagation of chaos for second order interacting particle systems

We study the long time behavior of second order particle systems interacting through global Lipschitz kernels. Combining hypocoercivity method in [37] and relative entropy method in [25], we are able to overcome the degeneracy of diffusion in position direction by controlling the relative entropy and relative Fisher information together. This implies the uniform-in-time propagation of chaos through the strong convergence of all marginals. Our method works at the level of Liouville equation and relies on the log Sobolev inequality of equilibrium of Vlasov-Fokker-Planck equation.

math.AP↗
arXiv2024

Transport based particle methods for the Fokker-Planck-Landau equation

We propose a particle method for numerically solving the Landau equation, inspired by the score-based transport modeling (SBTM) method for the Fokker-Planck equation. This method can preserve some important physical properties of the Landau equation, such as the conservation of mass, momentum, and energy, and decay of estimated entropy. We prove that matching the gradient of the logarithm of the approximate solution is enough to recover the true solution to the Landau equation with Maxwellian molecules. Several numerical experiments in low and moderately high dimensions are performed, with particular emphasis on comparing the proposed method with the traditional particle or blob method.

math.NA↗
arXiv2023

Spectrum of Hatano-Nelson model with strictly ergodic potentials

We provide a precise formula for the spectrum of the Hatano-Nelson model with strictly ergodic potentials in terms of its Lyapunov exponent. As applications, one clearly observes the real-complex spectrum transition. Moreover, if the Lyapunov exponent is continuous, the spectrum of the Hatano-Nelson model in $\ell^{2}(\mathbb{Z})$ can be approximated by the spectrum of its finite-interval truncation with periodic boundary conditions. Both of these results are strikingly different from the Hatano-Nelson model with random potentials \cite{Dav01A, Dav01, Dav02}.

math.SP↗
arXiv2023

Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEs

We extend the concept of self-consistency for the Fokker-Planck equation (FPE) to the more general McKean-Vlasov equation (MVE). While FPE describes the macroscopic behavior of particles under drift and diffusion, MVE accounts for the additional inter-particle interactions, which are often highly singular in physical systems. Two important examples considered in this paper are the MVE with Coulomb interactions and the vorticity formulation of the 2D Navier-Stokes equation. We show that a generalized self-consistency potential controls the KL-divergence between a hypothesis solution to the ground truth, through entropy dissipation. Built on this result, we propose to solve the MVEs by minimizing this potential function, while utilizing the neural networks for function approximation. We validate the empirical performance of our approach by comparing with state-of-the-art NN-based PDE solvers on several example problems.

math.NA↗
arXiv2023

Non-Self-Adjoint Quasi-periodic Operators with complex spectrum

We give a precise and complete description on the spectrum for a class of non-self-adjoint quasi-periodic operators acting on $\ell^2(\mathbb{Z}^d)$ which contains the Sarnak's model as a special case. As a consequence, one can see various interesting spectral phenomena including $\mathscr{P}\mathscr{T}$ symmetric breaking, the non-simply-connected two-dimensional spectrum in this class of operators. Particularly, we provide new examples of non-self-adjoint operator in $\mathscr{l}^{2}(\mathbb{Z})$ whose spectra (actually a two-dimensional subset of $\mathbb{C}$) can not be approximated by the spectra of its finite-interval truncations.

math.SP↗
arXiv2023

Winding number, density of states and acceleration

Winding number and density of states are two fundamental physical quantities for non-self-adjoint quasi-periodic Schrödinger operators, which reflect the asymptotic distribution of zeros of the characteristic determinants of the truncated operators under Dirichlet boundary condition, with respect to complexified phase and the energy respectively. We will prove that the winding number is in fact Avila's acceleration and it is also closely related to the density of states by a generalized Thouless formula for non-self-adjoint Schrödinger operators and Avila's global theory.

math-ph↗
arXiv2022

Self-Consistency of the Fokker-Planck Equation

The Fokker-Planck equation (FPE) is the partial differential equation that governs the density evolution of the Itô process and is of great importance to the literature of statistical physics and machine learning. The FPE can be regarded as a continuity equation where the change of the density is completely determined by a time varying velocity field. Importantly, this velocity field also depends on the current density function. As a result, the ground-truth velocity field can be shown to be the solution of a fixed-point equation, a property that we call self-consistency. In this paper, we exploit this concept to design a potential function of the hypothesis velocity fields, and prove that, if such a function diminishes to zero during the training procedure, the trajectory of the densities generated by the hypothesis velocity fields converges to the solution of the FPE in the Wasserstein-2 sense. The proposed potential function is amenable to neural-network based parameterization as the stochastic gradient with respect to the parameter can be efficiently computed. Once a parameterized model, such as Neural Ordinary Differential Equation is trained, we can generate the entire trajectory to the FPE.

cs.LG↗
arXiv2020

Mean-field limit and quantitative estimates with singular attractive kernels

This paper proves the mean field limit and quantitative estimates for many-particle systems with singular attractive interactions between particles. As an important example, a full rigorous derivation (with quantitative estimates) of the Patlak-Keller-Segel model in optimal subcritical regimes is obtained for the first time. To give an answer to this longstanding problem, we take advantage of a new modulated free energy and we prove some precise large deviation estimates encoding the competition between diffusion and attraction. Combined with the range of repulsive kernels, already treated in the s{é}minaire Laurent Schwartz proceeding [https://slsedp.centre-mersenne.org/journals/SLSEDP/ ], we provide the full proof of results announced by the authors in [C. R. Acad. Sciences Section Maths (2019)].

math.AP↗
arXiv2020

Sinkhorn Natural Gradient for Generative Models

We consider the problem of minimizing a functional over a parametric family of probability measures, where the parameterization is characterized via a push-forward structure. An important application of this problem is in training generative adversarial networks. In this regard, we propose a novel Sinkhorn Natural Gradient (SiNG) algorithm which acts as a steepest descent method on the probability space endowed with the Sinkhorn divergence. We show that the Sinkhorn information matrix (SIM), a key component of SiNG, has an explicit expression and can be evaluated accurately in complexity that scales logarithmically with respect to the desired accuracy. This is in sharp contrast to existing natural gradient methods that can only be carried out approximately. Moreover, in practical applications when only Monte-Carlo type integration is available, we design an empirical estimator for SIM and provide the stability analysis. In our experiments, we quantitatively compare SiNG with state-of-the-art SGD-type solvers on generative tasks to demonstrate its efficiency and efficacy of our method.

stat.ML↗
arXiv2020

Sinkhorn Barycenter via Functional Gradient Descent

In this paper, we consider the problem of computing the barycenter of a set of probability distributions under the Sinkhorn divergence. This problem has recently found applications across various domains, including graphics, learning, and vision, as it provides a meaningful mechanism to aggregate knowledge. Unlike previous approaches which directly operate in the space of probability measures, we recast the Sinkhorn barycenter problem as an instance of unconstrained functional optimization and develop a novel functional gradient descent method named Sinkhorn Descent (SD). We prove that SD converges to a stationary point at a sublinear rate, and under reasonable assumptions, we further show that it asymptotically finds a global minimizer of the Sinkhorn barycenter problem. Moreover, by providing a mean-field analysis, we show that SD preserves the weak convergence of empirical measures. Importantly, the computational complexity of SD scales linearly in the dimension $d$ and we demonstrate its scalability by solving a $100$-dimensional Sinkhorn barycenter problem.

cs.LG↗
arXiv2019

Modulated Free Energy and Mean Field Limit

This is the document corresponding to the talk the first author gave at IH{É}S for the Laurent Schwartz seminar on November 19, 2019. It concerns our recent introduction of a modulated free energy in mean-field theory in BrJaWa [4]. This physical object may be seen as a combination of the modulated potential energy introduced by S. Serfaty [See Proc. Int. Cong. Math. (2018)] and of the relative entropy introduced in mean field limit theory by P.-E. Jabin, Z. Wang [See Inventiones 2018]. It allows to obtain, for the first time, a convergence rate in the mean field limit for Riesz and Coulomb repulsive kernels in the presence of viscosity using the estimates in Du [8] and Se1 [20]. The main objective in this paper is to explain how it is possible to cover more general repulsive kernels through a Fourier transform approach as announced in BrJaWa [4] first in the case $σ$N $\rightarrow$ 0 when N $\rightarrow$ +$\infty$ and then if $σ$ > 0 is fixed. Then we end the paper with comments on the particle approximation of the Patlak-Keller-Segel system which is associated to an attractive kernel and refer to [C.R.

math.AP↗
arXiv2019

On Mean Field Limit and Quantitative Estimates with a Large Class of Singular Kernels: Application to the Patlak-Keller-Segel Model

In this note, we propose a new relative entropy combination of the methods developed by P.--E. Jabin and Z.~Wang [Inventiones (2018)] and by S. Serfaty [Proc. Int. Cong. of Math, (2018) and references therein] to treat more general kernels in mean field limit theory. This new relative entropy may be understood as introducing appropriate weights in the relative entropy developed by P.-E. Jabin and Z. Wang (in the spirit of what has been recently developed by D.~Bresch and P.--E. Jabin [Annals of Maths (2018)]) to cancel the more singular terms involving the divergence of the flow. As an example, a full rigorous derivation (with quantitative estimates) of the Patlak-Keller-Segel model in some subcritical regimes is obtained. Our new relative entropy allows to treat singular potentials which combine large smooth part, small attractive singular part and large repulsive singular part.

math.AP↗
arXiv2018

Quantitative estimate of propagation of chaos for stochastic systems with $W^{-1, \infty}$ kernels

We derive quantitative estimates proving the propagation of chaos for large stochastic systems of interacting particles. We obtain explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit. We have to develop for this new laws of large numbers at the exponential scale. But our result only requires very weak regularity on the interaction kernel in the negative Sobolev space $\dot W^{-1,\infty}$, thus including the Biot-Savart law and the point vortices dynamics for the 2d incompressible Navier-Stokes.

math.AP↗
arXiv2015

Mean Field Limit and Propagation of Chaos for Vlasov Systems with Bounded Forces

We consider large systems of particles interacting through rough but bounded interaction kernels. We are able to control the relative entropy between the $N$-particle distribution and the expected limit which solves the corresponding Vlasov system. This implies the Mean Field limit to the Vlasov system together with Propagation of Chaos through the strong convergence of all the marginals. The method works at the level of the Liouville equation and relies on precise combinatorics results.

math.AP↗