Mean-field Systems Research Articles

Maximum principle for infinite horizon optimal control of mean-field backward stochastic systems with delay and noisy memory

In this paper, we consider a problem of optimal control of an infinite horizon mean-field backward stochastic differential equation with delay and noisy memory under partial information. We derive necessary and sufficient maximum principles using Malliavin calculus technique for such a system. A class of mean-field time-advanced stochastic differential equations is introduced as the adjoint process which involves partial derivatives of the Hamiltonian functions and their Malliavin derivatives. To illustrate our theoretical results, we give an example for a linear-quadratic mean-field backward delay stochastic system with noisy memory on infinite horizon to obtain the optimal control. Also, we apply our results to pension fund problems with delay and noisy memory which are arising from the financial market.

A Hamiltonian Interacting Particle System for Compressible Flow

The decomposition of the energy of a compressible fluid parcel into slow (deterministic) and fast (stochastic) components is interpreted as a stochastic Hamiltonian interacting particle system (HIPS). It is shown that the McKean–Vlasov equation associated to the mean field limit yields the barotropic Navier–Stokes equation with density-dependent viscosity. Capillary forces can also be treated by this approach. Due to the Hamiltonian structure, the mean field system satisfies a Kelvin circulation theorem along stochastic Lagrangian paths.

On a mean field optimal control problem

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.

Non-Markovian Mpemba effect in mean-field systems.

Under certain conditions, a counterintuitive behavior-an initially hotter sample freezes faster when quenched to a cold bath than an identical system initialled at a lower temperature-is known as the Mpemba effect (ME). Here we identify the existence of the ME in mean-field systems (MFS). Specifically, the thermal contact between MFS and a large thermal reservoir is built up using the microcanonical Monte Carlo algorithm. The simulation results unambiguously demonstrate that an initial hotter system undergoes the paramagnetic-ferromagnetic phase transition faster than the initial cooler one. The ME here originates from the back-reaction of the MFS system on the reservoir, which is thus an embodiment of non-Markovianness in relaxation. In addition, we confirm that the ME survives in the thermodynamic limit. And the significance of reservoir size is also explored: A smaller heat reservoir facilitates the overall relaxation process. In general, this work establishes a theoretical non-Markovian ME framework, which may shed light on widening the understanding of the mechanism behind the ME in other substances, including water.

Solvability and optimal stabilization controls of discrete-time mean-field stochastic system with infinite horizon

The paper addresses the optimal control and stabilization problems for the indefinite discrete-time mean-field system over infinite horizon. Firstly, we show the convergence of the generalized algebraic Riccati equations (GAREs) and establish their compact form GARE. By dealing with the GARE, we derive the existence of the maximal solution to the original GAREs along with the fact that the maximal solution is the stabilizing solution. Then, the maximal solution is employed to design the linear-quadratic (LQ) optimal controller and the optimal value of the control problem. Specifically, we deduce that under the assumption of exact observability, the mean-field system is L^{2}-stabilizable if and only if the GAREs have a solution, which is also the maximal solution. By semi-definite programming (SDP) method, the solvability of the GAREs is discussed. Our results generalize and improve previous results. Finally, some numerical examples are exploited to illustrate the validity of the obtained results.

Metamagnetism and kinetic arrest in a long-range ferromagnetically ordered multicaloric double perovskite Y2CoMnO6

A systematic magnetic study of the double perovskite oxide Y2CoMnO6 (YCMO) has been performed. A monoclinic P21/n phase of YCMO was synthesized using a sol-gel method. Neutron diffraction (ND) measurements evidence the onset of long-range ferromagnetic (FM) ordering at TC ~ 76 K, which persists down to 5 K. The presence of 25% antisite disorder, estimated from the ND data, leads to the appearance of short-range antiferromagnetic (AFM) interactions. The existence of thermal hysteresis due to competing interactions between FM and AFM phases is observed in the M vs. T measurements. The pinning of magnetic domain walls at the Co/Mn antiphase boundaries results in a metamagnetic-like behavior. The field dependence of thermomagnetic irreversibility and the nature of virgin curves indicate the occurrence of a kinetic arrest phenomenon, which is further verified via cooling and heating of the system in an unequal fields (CHUF) protocol. In full agreement with the ND data, a detailed analysis of critical exponents near the paramagnetic (PM)-FM phase transition also establishes YCMO as a long-range interacting mean-field system. A careful examination of the temperature- and field-dependent magnetic entropy change yields an in-depth understanding of coexisting magnetically ordered and disordered phases in YCMO, leading to a comprehensive magnetic phase diagram of this multifunctional double perovskite system.

Emergence of two-magnon modes below spin-reorientation transition and phonon-magnon coupling in bulk BiFeO3: An infrared spectroscopic study

Abstract We have studied the low-temperature magnetic and infrared data of the BiFeO3 ceramic sample to investigate the behavior of phonons and magnon-phonon coupling below 300 K. From the magnetic study, we noticed two magnetic transitions near 150 and 257 K due to the spin-reorientation. Besides, this sample is found to be a mean-field system rather than a glassy system. In our assignment of the infrared modes, all the 13 allowed phonon modes are accounted for rhombohedral BiFeO3 at room temperature itself. Almost all the modes show anomalous behavior in their frequency dependence, close to the observed magnetic transitions. In addition, the emergence of two-magnon modes at 106 and 111 cm−1 below 160 K is observed due to a modification in the noncollinear spin structure. These results indicate the presence of magnon-phonon coupling in multiferroic BiFeO3 below room temperature.

The dynamics of polarized beliefs in networks governed by viral diffusion and media influence

This paper studies the evolution of polarized beliefs governed by the intertwined dynamics of viral diffusion and media influence in influence networks. It addresses the question of how different forms of influence interact with each other. First, we propose a Markov chain to model the dynamics of individuals as they transition between three belief states (neutral, positive and negative) based on the states of their neighbors. This stochastic system assumes that individuals are influenced via the links of the network or through the global effect of mass media. For exponential and scale-free networks, we approximate this stochastic system by deterministic differential equations and define the homogeneous mean-field system and heterogeneous mean-field system, respectively. Studying stability conditions for these deterministic dynamical systems, we analyze the fraction of neutral, positive and negative individuals in the population. Also, we determine the conditions under which desired dynamical transitions happen for the targeted population. These conditions allow us to predict macroscopic measures of dynamics of adoption in influence networks. Finally, the derived analytical results are validated using simulations of four synthetic networks: Watts–Strogatz, random regular, Barabasi–Albert and small-world forest-fire, as well as five real-world networks: ego-Facebook, Deezer, Livemocha, a Facebook interaction network and Douban. Also, we demonstrate how the proposed model can be leveraged by marketing campaigns for optimal resource allocations between viral marketing and media marketing to minimize the number of final negative individuals in different network settings.

Conditional propagation of chaos for mean field systems of interacting neurons

We study the stochastic system of interacting neurons introduced in [5] and in [10] in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order 1∕ N. In between successive spikes, each neuron’s potential follows a deterministic flow. We prove the convergence of the system, as N→∞, to a limit nonlinear jumping stochastic differential equation driven by Poisson random measure and an additional Brownian motion W which is created by the central limit theorem. This Brownian motion is underlying each particle’s motion and induces a common noise factor for all neurons in the limit system. Conditionally on W, the different neurons are independent in the limit system. This is the conditional propagation of chaos property. We prove the well-posedness of the limit equation by adapting the ideas of [12] to our frame. To prove the convergence in distribution of the finite system to the limit system, we introduce a new martingale problem that is well suited for our framework. The uniqueness of the limit is deduced from the exchangeability of the underlying system.

Pareto optimal control of the mean-field stochastic systems by adaptive dynamic programming algorithm

The Pareto game for the model-free continuous-time stochastic system is studied through approximate/adaptive dynamic programming (ADP) in this paper. Firstly, the model-based online iterative algorithm is proposed, and it is proved that the control iterative sequence converges to the Pareto efficient solution, but the algorithm requires complete system parameters. Then, we derive the model-free iterative equation and develop the ADP algorithm to calculate the equation by collecting updated states and input information online. From the derivation of the ADP algorithm, the model-free iterative equation and the model-based iterative equation have the same solution, which means that the ADP algorithm can approximate the Pareto optimal solution. Next, the convergence analysis shows that the Pareto optimal strategy is uniquely determined by the ADP algorithm. Finally, two simulation examples confirm the feasibility of the ADP algorithm.

Social optima in linear quadratic mean field control with unmodeled dynamics and multiplicative noise

AbstractThis paper investigates the linear‐quadratic social control problem for mean field systems with unmodeled dynamics and multiplicative noise. The objective of each agent is to optimize the social cost in the worst‐case disturbance. We first analyze the centralized strategies by the person‐by‐person optimality and then construct an auxiliary zero‐sum game according to mean field approximations. By solving the auxiliary problem subject to consistency conditions, we design a set of decentralized strategies, which is further shown to have asymptotic robust social optimality.

A global maximum principle for optimal control of general mean-field forward-backward stochastic systems with jumps

In this paper we prove a maximum principle of optimal control problem for a class of general mean-field forward-backward stochastic systems with jumps in the case where the diffusion coefficients depend on control, the control set does not need to be convex, the coefficients of jump terms are independent of control as well as the coefficients of mean-field backward stochastic differential equations depend on the joint law of (X(t), Y (t)). Since the coefficients depend on measure, higher mean-field terms could be involved. In order to analyse them, two new adjoint equations are brought in and several new generic estimates of their solutions are investigated. Utilizing these subtle estimates, the second-order expansion of the cost functional, which is the key point to analyse the necessary condition, is obtained, and where after the stochastic maximum principle. An illustrative application to mean-field game is considered.

Exit-time of mean-field particles system

The current article is devoted to the study of a mean-field system of particles. The question that we are interested in is the behaviour of the exit-time of the first particle (and the one of any particle) from a domain D on ℝd as the diffusion coefficient goes to 0. We establish a Kramers’ type law. In other words, we show that the exit-time is exponentially equivalent to [see formula in PDF], HN being the exit-cost. We also show that this exit-cost converges to some quantity H.

Propagation of chaos and moderate interaction for a piecewise deterministic system of geometrically enriched particles

In this article we study a system of $N$ particles, each of them being defined by the couple of a position (in $\mathbb{R}^d$) and a so-called orientation which is an element of a compact Riemannian manifold. This orientation can be seen as a generalisation of the velocity in Vicsek-type models such as [Degond et al. 2017; Degond, Motsch 2008]. We will assume that the orientation of each particle follows a jump process whereas its position evolves deterministically between two jumps. The law of the jump depends on the position of the particle and the orientations of its neighbours. In the limit $N\to\infty$, we first prove a propagation of chaos result which can be seen as an adaptation of the classical result on McKean-Vlasov systems [Sznitman 1991] to Piecewise Deterministic Markov Processes (PDMP). As in [Jourdain, M\'el\'eard 1998], we then prove that under a proper rescaling with respect to $N$ of the interaction radius between the agents (moderate interaction), the law of the limiting mean-field system satisfies a BGK equation with localised interactions which has been studied as a model of collective behaviour in [Degond et al. 2018]. Finally, in the spatially homogeneous case, we give an alternative approach based on martingale arguments.

Finite-time guaranteed cost control for uncertain mean-field stochastic systems

This paper deals with the finite-time guaranteed cost control problem for uncertain mean-field stochastic systems. Some sufficient conditions for the existence of state and output feedback finite-time guaranteed cost laws are given using finite-time stability theory. A guaranteed cost control gain is obtained through a convex optimization problem. The effectiveness of the proposed results is illustrated by a numerical example.

Comparing dynamics: deep neural networks versus glassy systems**This article is an updated version of a paper presented at ICML 2018, and is present in the following informal collection of proceedings: 2018 Proceedings Of Machine Learning Research 80 324–333.

We analyze numerically the training dynamics of deep neural networks (DNN) by using methods developed in statistical physics of glassy systems. The two main issues we address are (1) the complexity of the loss landscape and of the dynamics within it, and (2) to what extent DNNs share similarities with glassy systems. Our findings, obtained for different architectures and datasets, suggest that during the training process the dynamics slows down because of an increasingly large number of flat directions. At large times, when the loss is approaching zero, the system diffuses at the bottom of the landscape. Despite some similarities with the dynamics of mean-field glassy systems, in particular, the absence of barrier crossing, we find distinctive dynamical behaviors in the two cases, showing that the statistical properties of the corresponding loss and energy landscapes are different. In contrast, when the network is under-parametrized we observe a typical glassy behavior, thus suggesting the existence of different phases depending on whether the network is under-parametrized or over-parametrized.

On mean field systems with multi-classes

This work focuses on stochastic systems of weakly interacting particles containing different populations represented by multi-classes. The dynamics of each particle depends not only on the empirical measure of the whole population but also on those of different populations. The limits of such systems as the number of particles tends to infinity are investigated. We establish the existence, uniqueness, and basic properties of solutions to the limiting McKean-Vlasov equations of these systems and then obtain the rate of convergence of the sequences of empirical measures associated with the systems to their limits in terms of the \begin{document}$ p^{\text{th}} $\end{document} Monge-Wasserstein distance.

Critical exponents in mean-field classical spin systems.

For mean-field classical spin systems exhibiting a second-order phase transition in the stationary state, we obtain within the corresponding phase-space evolution according to the Vlasov equation the values of the critical exponents describing power-law behavior of response to a small external field. The exponent values so obtained significantly differ from the ones obtained on the basis of an analysis of the static phase-space distribution, with no reference to dynamics. This work serves as an illustration that cautions against relying on a static approach, with no reference to the dynamical evolution, to extract critical exponent values for mean-field systems.

Open Problem—Weakly Interacting Particle Systems on Dense Random Graphs

We first briefly introduce the following classic mean-field particle system given as a collection of [Formula: see text]-valued stochastic processes {[Formula: see text](t) : t ∈ [0, T][Formula: see text].[Formula: see text]Here, {Wi} are independent and identically distributed standard d-dimensional Brownian motions, and [Formula: see text] × [Formula: see text] → [Formula: see text] is assumed to be bounded and Lipschitz, but one could also allow for more general initial states, drift coefficient b, and a state-dependent diffusion coefficient.

Maximum principle for delayed stochastic mean-field control problem with state constraint

In this paper, we consider the optimal control problem for the mean-field stochastic differential equations with delay and state constraint. By virtue of the classical Ekeland’s variational principle, the duality method and a new type of mean-field anticipated backward stochastic differential equation, we obtain the maximum principle of the optimal control for this problem. Our result can be applied to a harvest model from a mean-field system with delay.