Found 1683 results, sorted by relevancy. Show newest relevant.Update me on new preprints

On the **mean**-**field** limit for the Vlasov-Poisson-Fokker-Planck system

We rigorously justify the mean-field limit of a $N$-particle system subject to the Brownian motion and interacting through a Newtonian potential in $\mathbb{R}^3$. Expand abstract.

We rigorously justify the

**mean**-**field**limit of a $N$-particle system subject to the Brownian motion and interacting through a Newtonian potential in $\mathbb{R}^3$. Our result leads to a derivation of the Vlasov-Poisson-Fokkker-Planck (VPFP) equation from the microscopic $N$-particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and trajectories following the the**mean**-**field**is bounded by $N^{-\frac{1}{3}+\varepsilon}$ ($\frac{1}{63}\leq\varepsilon676 days ago

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arXiv

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arXiv

On a **Mean** **Field** Optimal Control Problem

We notice this model is in close connection with the theory of

**mean**-**field**games systems. Expand abstract. In this paper we consider a

**mean****field**optimal control problem with an aggregation-diffusion constraint, where agents interact through a potential, in the presence of a Gaussian noise term. Our analysis focuses on a PDE system coupling a Hamilton-Jacobi and a Fokker-Planck equation, describing the optimal control aspect of the problem and the evolution of the population of agents, respectively. The main contribution of the paper is a result on the existence of solutions for the aforementioned system. We notice this model is in close connection with the theory of**mean**-**field**games systems. However, a distinctive feature concerns the nonlocal character of the interaction; it affects the drift term in the Fokker-Planck equation as well as the Hamiltonian of the system, leading to new difficulties to be addressed.154 days ago

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arXiv

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arXiv

Improved treatment of fermion-boson vertices and Bethe-Salpeter
equations in non-local extensions of dynamical **mean** **field** theory

The method is tested on the Hubbard model and can be used in a broad range of applications of non-local extensions of dynamical

**mean**-**field**theory. Expand abstract. We reconsider the procedure of calculation of fermion-boson vertices and numerical solution of Bethe-Salpeter equations, used in non-local extensions of dynamical

**mean**-**field**theory. Because of the frequency dependence of vertices, finite frequency box for matrix inversions is typically used, which often requires some treatment of asymptotic behaviour of vertices. Recently [Phys. Rev. B 83, 085102 (2011); 97, 235140 (2018)] it was proposed to split the considered frequency box into smaller and larger one; in the smaller frequency box the numerically exact vertices are used, while beyond this box asymptotics of vertices are applied. Yet, this method requires numerical treatment of vertex asymptotics (including corresponding matrix manipulations) in the larger frequency box and/or knowing fermion-boson vertices, which may be not convenient for numerical calculations. In the present paper we derive the formulae which treat analytically contribution of vertices beyond chosen frequency box, such that only numerical operations with vertices in the chosen small frequency box are required. The method is tested on the Hubbard model and can be used in a broad range of applications of non-local extensions of dynamical**mean**-**field**theory.200 days ago

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arXiv

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arXiv

**Mean** **Field** Theory for Lyapunov Exponents and KS Entropy in Lorentz
Lattice Gases

The computer results show a distribution of values for the dynamical quantities with average values that are in good agreement with the

**mean****field**theory, and consistent with the escape-rate formalism for the coefficient of diffusion. Expand abstract. Cellular automata lattice gases are useful systems for systematically exploring the connections between non-equilibrium statisitcal mechanics and dynamical systems theory. Here the chaotic properties of a Lorentz lattice gas are studied analytically and by computer simulations. The escape-rates, Lyapunov exponents, and KS entropies are estimated for a one-dimensional example using a

**mean**-**field**theory, and the results are compared with simulations for a range of densities and scattering parameters of the lattice gas. The computer results show a distribution of values for the dynamical quantities with average values that are in good agreement with the**mean****field**theory, and consistent with the escape-rate formalism for the coefficient of diffusion.9191 days ago

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arXiv

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arXiv

Minimal-time **mean** **field** games

After a preliminary study of the corresponding minimal-time optimal control problem, we formulate the

**mean****field**game in a Lagrangian setting and prove existence of Lagrangian equilibria using a fixed point strategy. Expand abstract. This paper considers a

**mean****field**game model inspired by crowd motion where agents want to leave a given bounded domain through a part of its boundary in minimal time. Each agent is free to move in any direction, but their maximal speed is bounded in terms of the average density of agents around their position in order to take into account congestion phenomena. After a preliminary study of the corresponding minimal-time optimal control problem, we formulate the**mean****field**game in a Lagrangian setting and prove existence of Lagrangian equilibria using a fixed point strategy. We provide a further study of equilibria under the assumption that agents may leave the domain through the whole boundary, in which case equilibria are described through a system of a continuity equation on the distribution of agents coupled with a Hamilton--Jacobi equation on the value function of the optimal control problem solved by each agent. This is possible thanks to the semiconcavity of the value function, which follows from some further regularity properties of optimal trajectories obtained through Pontryagin Maximum Principle. Simulations illustrate the behavior of equilibria in some particular situations.686 days ago

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arXiv

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arXiv

Entropic curvature and convergence to equilibrium for **mean**-field
dynamics on discrete spaces

We establish explicit curvature bounds for several examples of

**mean**-**field**limits of various classical models from statistical mechanics. Expand abstract. We consider non-linear evolution equations arising from

**mean**-**field**limits of particle systems on discrete spaces. We investigate a notion of curvature bounds for these dynamics based on convexity of the free energy along interpolations in a discrete transportation distance related to the gradient flow structure of the dynamics. This notion extends the one for linear Markov chain dynamics studied by Erbar and Maas. We show that positive curvature bounds entail several functional inequalities controlling the convergence to equilibrium of the dynamics. We establish explicit curvature bounds for several examples of**mean**-**field**limits of various classical models from statistical mechanics.199 days ago

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arXiv

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arXiv

**Mean** **field** theory for deep dropout networks: digging up gradient
backpropagation deeply

The

**mean****field**theory shows that the existence of depth scales that limit the maximum depth of signal propagation and gradient backpropagation. Expand abstract. In recent years, the

**mean****field**theory has been applied to the study of neural networks and has achieved a great deal of success. The theory has been applied to various neural network structures, including CNNs, RNNs, Residual networks, and Batch normalization. Inevitably, recent work has also covered the use of dropout. The**mean****field**theory shows that the existence of depth scales that limit the maximum depth of signal propagation and gradient backpropagation. However, the gradient backpropagation is derived under the gradient independence assumption that weights used during feed forward are drawn independently from the ones used in backpropagation. This is not how neural networks are trained in a real setting. Instead, the same weights used in a feed-forward step needs to be carried over to its corresponding backpropagation. Using this realistic condition, we perform theoretical computation on linear dropout networks and a series of experiments on dropout networks. Our empirical results show an interesting phenomenon that the length gradients can backpropagate for a single input and a pair of inputs are governed by the same depth scale. Besides, we study the relationship between variance and**mean**of statistical metrics of the gradient and shown an emergence of universality. Finally, we investigate the maximum trainable length for deep dropout networks through a series of experiments using MNIST and CIFAR10 and provide a more precise empirical formula that describes the trainable length than original work.67 days ago

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arXiv

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arXiv

**Mean** **Field** Limit for Coulomb-Type Flows

In the appendix, it is also adapted to prove the

**mean**-**field**convergence of the solutions to Newton's law with Coulomb or Riesz interaction in the monokinetic case to solutions of an Euler-Poisson type system. Expand abstract. We establish the

**mean**-**field**convergence for systems of points evolving along the gradient flow of their interaction energy when the interaction is the Coulomb potential or a super-coulombic Riesz potential, for the first time in arbitrary dimension. The proof is based on a modulated energy method using a Coulomb or Riesz distance, assumes that the solutions of the limiting equation are regular enough and exploits a weak-strong stability property for them. The method can handle the addition of a regular interaction kernel, and applies also to conservative and mixed flows. In the appendix, it is also adapted to prove the**mean**-**field**convergence of the solutions to Newton's law with Coulomb or Riesz interaction in the monokinetic case to solutions of an Euler-Poisson type system.704 days ago

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arXiv

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arXiv

**Mean** **field** games with branching

For this reason, we introduce a branching mechanism in the population of agents and obtain a variation on the

**mean****field**game problem. Expand abstract.**Mean**

**field**games are concerned with the limit of large-population stochastic differential games where the agents interact through their empirical distribution. In the classical setting, the number of players is large but fixed throughout the game. However, in various applications, such as population dynamics or economic growth, the number of players can vary across time which may lead to different Nash equilibria. For this reason, we introduce a branching mechanism in the population of agents and obtain a variation on the

**mean**

**field**game problem. As a first step, we study a simple model using a PDE approach to illustrate the main differences with the classical setting. We prove existence of a solution and show that it provides an approximate Nash-equilibrium for large population games. We also present a numerical example for a linear--quadratic model. Then we study the problem in a general setting by a probabilistic approach. It is based upon the relaxed formulation of stochastic control problems which allows us to obtain a general existence result.

60 days ago

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arXiv

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arXiv

**Mean** **Field** Game for Linear Quadratic Stochastic Recursive Systems

This paper focuses on linear-quadratic (LQ for short) mean-field games described by forward-backward stochastic differential equations (FBSDEs for short), in which the individual control region is postulated to be convex. Expand abstract.

This paper focuses on linear-quadratic (LQ for short)

**mean**-**field**games described by forward-backward stochastic differential equations (FBSDEs for short), in which the individual control region is postulated to be convex. The decentralized strategies and consistency condition are represented by a kind of coupled**mean**-**field**FBSDEs with projection operators. The well-posedness of consistency condition system is obtained using the monotonicity condition method. The $\epsilon$-Nash equilibrium property is discussed as well.194 days ago

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arXiv

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arXiv