116 results found
Friston K, Da Costa L, Sakthivadivel DAR, et al., 2023, Path integrals, particular kinds, and strange things., Phys Life Rev, Vol: 47, Pages: 35-62
This paper describes a path integral formulation of the free energy principle. The ensuing account expresses the paths or trajectories that a particle takes as it evolves over time. The main results are a method or principle of least action that can be used to emulate the behaviour of particles in open exchange with their external milieu. Particles are defined by a particular partition, in which internal states are individuated from external states by active and sensory blanket states. The variational principle at hand allows one to interpret internal dynamics-of certain kinds of particles-as inferring external states that are hidden behind blanket states. We consider different kinds of particles, and to what extent they can be imbued with an elementary form of inference or sentience. Specifically, we consider the distinction between dissipative and conservative particles, inert and active particles and, finally, ordinary and strange particles. Strange particles can be described as inferring their own actions, endowing them with apparent autonomy or agency. In short-of the kinds of particles afforded by a particular partition-strange kinds may be apt for describing sentient behaviour.
Costa LD, Pavliotis GA, 2023, The entropy production of stationary diffusions, Journal of Physics A: Mathematical and Theoretical, Vol: 56, Pages: 1-52, ISSN: 1751-8113
The entropy production rate is a central quantity in non-equilibrium statistical physics, scoring how far a stochastic process is from being time-reversible. In this paper, we compute the entropy production of diffusion processes at non-equilibrium steady-state under the condition that the time-reversal of the diffusion remains a diffusion. We start by characterising the entropy production of both discrete and continuous-time Markov processes. We investigate the time-reversal of time-homogeneous stationary diffusions and recall the most general conditions for the reversibility of the diffusion property, which includes hypoelliptic and degenerate diffusions, and locally Lipschitz vector fields. We decompose the drift into its time-reversible and irreversible parts, or equivalently, the generator into symmetric and antisymmetric operators. We show the equivalence with a decomposition of the backward Kolmogorov equation considered in hypocoercivity theory, and a decomposition of the Fokker-Planck equation in GENERIC form. The main result shows that when the time-irreversible part of the drift is in the range of the volatility matrix (almost everywhere) the forward and time-reversed path space measures of the process are mutually equivalent, and evaluates the entropy production. When this does not hold, the measures are mutually singular and the entropy production is infinite. We verify these results using exact numerical simulations of linear diffusions. We illustrate the discrepancy between the entropy production of non-linear diffusions and their numerical simulations in several examples and illustrate how the entropy production can be used for accurate numerical simulation. Finally, we discuss the relationship between time-irreversibility and sampling efficiency, and how we can modify the definition of entropy production to score how far a process is from being generalised reversible.
Sharrock L, Kantas N, Parpas P, et al., 2023, Online parameter estimation for the McKean–Vlasov stochastic differential equation, Stochastic Processes and their Applications, Vol: 162, Pages: 481-546, ISSN: 0304-4149
We analyse the problem of online parameter estimation for a stochastic McKean–Vlasov equation, and the associated system of weakly interacting particles. We propose an online estimator for the parameters of the McKean–Vlasov SDE, or the interacting particle system, which is based on a continuous-time stochastic gradient ascent scheme with respect to the asymptotic log-likelihood of the interacting particle system. We characterise the asymptotic behaviour of this estimator in the limit as �→∞, and also in the joint limit as �→∞ and �→∞. In these two cases, we obtain almost sure or �1 convergence to the stationary points of a limiting contrast function, under suitable conditions which guarantee ergodicity and uniform-in-time propagation of chaos. We also establish, under the additional condition of global strong concavity, �2 convergence to the unique maximiser of the asymptotic log-likelihood of the McKean–Vlasov SDE, with an asymptotic convergence rate which depends on the learning rate, the number of observations, and the dimension of the non-linear process. Our theoretical results are supported by two numerical examples, a linear mean field model and a stochastic opinion dynamics model.
Delgadino MG, Gvalani RS, Pavliotis GA, et al., 2023, Phase transitions, logarithmic sobolev inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions, Communications in Mathematical Physics, Vol: 401, Pages: 275-323, ISSN: 0010-3616
In this article, we study the mean field limit of weakly interacting diffusions for confining and interaction potentials that are not necessarily convex. We explore the relationship between the large N limit of the constant in the logarithmic Sobolev inequality (LSI) for the N-particle system and the presence or absence of phase transitions for the mean field limit. We show that the non-degeneracy of the LSI constant implies uniform-in-time propagation of chaos and Gaussianity of the fluctuations at equilibrium. As byproducts of our analysis, we provide concise and, to our knowledge, new proofs of a generalised form of Talagrand’s inequality and of quantitative propagation of chaos by employing techniques from the theory of gradient flows, specifically the Riemannian calculus on the space of probability measures.
Pavliotis GA, Stoltz G, Vaes U, 2023, Mobility estimation for langevin dynamics using control variates, SIAM: Multiscale Modeling and Simulation, Vol: 21, Pages: 680-715, ISSN: 1540-3459
The scaling of the mobility of two-dimensional Langevin dynamics in a periodicpotential as the friction vanishes is not well understood for nonseparable potentials. Theoreticalresults are lacking, and numerical calculation of the mobility in the underdamped regime is chal-lenging because the computational cost of standard Monte Carlo methods is inversely proportionalto the friction coefficient, while deterministic methods are ill-conditioned. In this work, we proposea new variance-reduction method based on control variates for efficiently estimating the mobility ofLangevin-type dynamics. We provide bounds on the bias and variance of the proposed estimatorand illustrate its efficacy through numerical experiments, first in simple one-dimensional settingsand then for two-dimensional Langevin dynamics. Our results corroborate prior numerical evidencethat the mobility scales as\gamma - \sigma , with 0<\sigma \leqslant 1, in the low friction regime for a simple nonseparablepotential.
Diamantakis T, Holm DD, Pavliotis GA, 2023, Variational principles on geometric rough paths and the Lévy area correction, SIAM Journal on Applied Dynamical Systems, Vol: 22, Pages: 1182-1218, ISSN: 1536-0040
In this paper, we describe two effects of the Lévy area correction on the invariant measure of stochastic rigid body dynamics on geometric rough paths. From the viewpoint of dynamics, the Lévy area correction introduces an additional deterministic torque into the rigid body motion equation on geometric rough paths. When the rigid body dynamics is driven by colored noise, and damped by double-bracket dissipation, our theoretical and numerical results show that the additional deterministic torque due to the the Lévy area correction shifts the center of the probability distribution function by shifting the Hamiltonian function in the exponent of the Gibbsian invariant measure.
Friston K, Da Costa L, Sajid N, et al., 2023, The free energy principle made simpler but not too simple, Physics Reports, Vol: 1024, Pages: 1-29, ISSN: 0370-1573
This paper provides a concise description of the free energy principle, starting from a formulation of random dynamical systems in terms of a Langevin equation and ending with a Bayesian mechanics that can be read as a physics of sentience. It rehearses the key steps using standard results from statistical physics. These steps entail (i) establishing a particular partition of states based upon conditional independencies that inherit from sparsely coupled dynamics, (ii) unpacking the implications of this partition in terms of Bayesian inference and (iii) describing the paths of particular states with a variational principle of least action. Teleologically, the free energy principle offers a normative account of self-organisation in terms of optimal Bayesian design and decision-making, in the sense of maximising marginal likelihood or Bayesian model evidence. In summary, starting from a description of the world in terms of random dynamical systems, we end up with a description of self-organisation as sentient behaviour that can be interpreted as self-evidencing; namely, self-assembly, autopoiesis or active inference.
Duncan AB, Duong MH, Pavliotis GA, 2023, Brownian motion in an N-scale periodic potential, Journal of Statistical Physics, Vol: 190, Pages: 1-34, ISSN: 0022-4715
We study the problem of Brownian motion in a multiscale potential. The potential is assumed to have N+1 scales (i.e. N small scales and one macroscale) and to depend periodically on all the small scales. We show that for nonseparable potentials, i.e. potentials in which the microscales and the macroscale are fully coupled, the homogenized equation is an overdamped Langevin equation with multiplicative noise driven by the free energy, for which the detailed balance condition still holds. This means, in particular, that homogenized dynamics is reversible and that the coarse-grained Fokker–Planck equation is still a Wasserstein gradient flow with respect to the coarse-grained free energy. The calculation of the effective diffusion tensor requires the solution of a system of N coupled Poisson equations.
Chak M, Kantas N, Pavliotis GA, 2023, On the generalized langevin equation for simulated annealing, SIAM/ASA Journal on Uncertainty Quantification, Vol: 11, Pages: 139-167, ISSN: 2166-2525
In this paper, we consider the generalized (higher order) Langevin equation for the purpose of simulated annealing and optimization of nonconvex functions. Our approach modifies the underdamped Langevin equation by replacing the Brownian noise with an appropriate Ornstein–Uhlenbeck process to account for memory in the system. Under reasonable conditions on the loss function and the annealing schedule, we establish convergence of the continuous time dynamics to a global minimum. In addition, we investigate the performance numerically and show better performance and higher exploration of the state space compared to the underdamped Langevin dynamics with the same annealing schedule.
Gaskin T, Pavliotis GA, Girolami M, 2023, Neural parameter calibration for large-scale multiagent models, Proceedings of the National Academy of Sciences, Vol: 120, ISSN: 0027-8424
<jats:p>Computational models have become a powerful tool in the quantitative sciences to understand the behavior of complex systems that evolve in time. However, they often contain a potentially large number of free parameters whose values cannot be obtained from theory but need to be inferred from data. This is especially the case for models in the social sciences, economics, or computational epidemiology. Yet, many current parameter estimation methods are mathematically involved and computationally slow to run. In this paper, we present a computationally simple and fast method to retrieve accurate probability densities for model parameters using neural differential equations. We present a pipeline comprising multiagent models acting as forward solvers for systems of ordinary or stochastic differential equations and a neural network to then extract parameters from the data generated by the model. The two combined create a powerful tool that can quickly estimate densities on model parameters, even for very large systems. We demonstrate the method on synthetic time series data of the SIR model of the spread of infection and perform an in-depth analysis of the Harris–Wilson model of economic activity on a network, representing a nonconvex problem. For the latter, we apply our method both to synthetic data and to data of economic activity across Greater London. We find that our method calibrates the model orders of magnitude more accurately than a previous study of the same dataset using classical techniques, while running between 195 and 390 times faster.</jats:p>
Abdulle A, Garegnani G, Pavliotis GA, et al., 2023, Drift estimation of multiscale diffusions based on filtered data, Foundations of Computational Mathematics, Vol: 23, Pages: 33-84, ISSN: 1615-3375
We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure.
Pavliotis GA, Zanoni A, 2022, Eigenfunction martingale estimators for interacting particle systems and their mean field limit, SIAM Journal on Applied Dynamical Systems, Vol: 21, Pages: 2338-2370, ISSN: 1536-0040
We study the problem of parameter estimation for large exchangeable interacting particle systems when a sample of discrete observations from a single particle is known. We propose a novel method based on martingale estimating functions constructed by employing the eigenvalues and eigenfunctions of the generator of the mean field limit, where the law of the process is replaced by the (unique) invariant measure of the mean field dynamics. We then prove that our estimator is asymptotically unbiased and asymptotically normal when the number of observations and the number of particles tend to infinity, and we provide a rate of convergence toward the exact value of the parameters. Finally, we present several numerical experiments which show the accuracy of our estimator and corroborate our theoretical findings, even in the case that the mean field dynamics exhibit more than one steady state.
Barp A, Da Costa L, França G, et al., 2022, Geometric methods for sampling, optimization, inference, and adaptive agents, Handbook of Statistics, Vol: 46, Pages: 21-78, ISSN: 0169-7161
In this chapter, we identify fundamental geometric structures that underlie the problems of sampling, optimization, inference, and adaptive decision-making. Based on this identification, we derive algorithms that exploit these geometric structures to solve these problems efficiently. We show that a wide range of geometric theories emerge naturally in these fields, ranging from measure-preserving processes, information divergences, Poisson geometry, and geometric integration. Specifically, we explain how (i) leveraging the symplectic geometry of Hamiltonian systems enables us to construct (accelerated) sampling and optimization methods, (ii) the theory of Hilbertian subspaces and Stein operators provides a general methodology to obtain robust estimators, and (iii) preserving the information geometry of decision-making yields adaptive agents that perform active inference. Throughout, we emphasize the rich connections between these fields; e.g., inference draws on sampling and optimization, and adaptive decision-making assesses decisions by inferring their counterfactual consequences. Our exposition provides a conceptual overview of underlying ideas, rather than a technical discussion, which can be found in the references herein.
Abdulle A, Pavliotis GA, Zanoni A, 2022, Eigenfunction martingale estimating functions and filtered data for drift estimation of discretely observed multiscale diffusions, STATISTICS AND COMPUTING, Vol: 32, ISSN: 0960-3174
Pavliotis GA, Stuart AM, Vaes U, 2022, Derivative-free Bayesian inversion using multiscale dynamics, SIAM Journal on Applied Dynamical Systems, Vol: 21, Pages: 284-326, ISSN: 1536-0040
Inverse problems are ubiquitous because they formalize the integration of data with mathematical models. In many scientific applications the forward model is expensive to evaluate, and adjoint computations are difficult to employ; in this setting derivative-free methods which involve a small number of forward model evaluations are an attractive proposition. Ensemble Kalman-based interacting particle systems (and variants such as consensus-based and unscented Kalman approaches) have proven empirically successful in this context, but suffer from the fact that they cannot be systematically refined to return the true solution, except in the setting of linear forward models [A. Garbuno-Inigo et al., SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 412--441]. In this paper, we propose a new derivative-free approach to Bayesian inversion, which may be employed for posterior sampling or for maximum a posteriori estimation, and may be systematically refined. The method relies on a fast/slow system of stochastic differential equations for the local approximation of the gradient of the log-likelihood appearing in a Langevin diffusion. Furthermore the method may be preconditioned by use of information from ensemble Kalman--based methods (and variants), providing a methodology which leverages the documented advantages of those methods, while also being provably refinable. We define the methodology, highlighting its flexibility and many variants, provide a theoretical analysis of the proposed approach, and demonstrate its efficacy by means of numerical experiments.
Da Costa L, Friston K, Heins C, et al., 2021, Bayesian mechanics for stationary processes, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 477, Pages: 1-26, ISSN: 1364-5021
This paper develops a Bayesian mechanics for adaptive systems. Firstly, we model the interface between a system and its environment with a Markov blanket. This affords conditions under which states internal to the blanket encode information about external states. Second, we introduce dynamics and represent adaptive systems as Markov blankets at steady state. This allows us to identify a wide class of systems whose internal states appear to infer external states, consistent with variational inference in Bayesian statistics and theoretical neuroscience. Finally, we partition the blanket into sensory and active states. It follows that active states can be seen as performing active inference and well-known forms of stochastic control (such as PID control), which are prominent formulations of adaptive behaviour in theoretical biology and engineering.
Goddard BD, Gooding B, Short H, et al., 2021, Noisy bounded confidence models for opinion dynamics: the effect of boundary conditions on phase transitions, IMA Journal of Applied Mathematics, Vol: 87, Pages: 80-110, ISSN: 0272-4960
We study SDE and PDE models for opinion dynamics under bounded confidence, for a range of different boundary conditions, with and without the inclusion of a radical population. We perform exhaustive numerical studies with pseudo-spectral methods to determine the effects of the boundary conditions, suggesting that the no-flux case most faithfully reproduces the underlying mechanisms in the associated deterministic models of Hegselmann and Krause. We also compare the SDE and PDE models, and use tools from analysis to study phase transitions, including a systematic description of an appropriate order parameter.
Delgadino MG, Gvalani RS, Pavliotis G, 2021, On the diffusive-mean field limit for weakly interacting diffusionsexhibiting phase transitions, Archive for Rational Mechanics and Analysis, Vol: 241, Pages: 91-148, ISSN: 0003-9527
The objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive (homogenisation) limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition, that is to say, if it admits more than one steady state. A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature.
Zagli N, Lucarini V, Pavliotis GA, 2021, Spectroscopy of phase transitions for multiagent systems, Chaos: an interdisciplinary journal of nonlinear science, Vol: 31, Pages: 1-8, ISSN: 1054-1500
In this paper, we study phase transitions for weakly interacting multiagent systems. By investigating the linear response of a system composed of a finite number of agents, we are able to probe the emergence in the thermodynamic limit of a singular behavior of the susceptibility. We find clear evidence of the loss of analyticity due to a pole crossing the real axis of frequencies. Such behavior has a degree of universality, as it does not depend on either the applied forcing or on the considered observable. We present results relevant for both equilibrium and nonequilibrium phase transitions by studying the Desai–Zwanzig and Bonilla–Casado–Morillo models.Multiagent models feature in a very vast range of applications in natural sciences, social sciences, and engineering. We study here the Desai–Zwanzig (DZ) and Bonilla–Casado–Morillo (BCM) models, which are paradigmatic for equilibrium and nonequilibrium conditions, respectively. Phase transitions result from the coordination between the individual agents and are associated with the divergence of the linear response of the system. The occurrence of phase transitions is universal: it does not depend on the acting forcing and can be detected by looking at virtually any observable of the system. We showcase here how response theory is capable of providing a useful angle for understanding the universal properties of phase transitions in complex systems.
Borovykh A, Kantas N, Parpas P, et al., 2021, On stochastic mirror descent with interacting particles: Convergence properties and variance reduction, Physica D: Nonlinear Phenomena, Vol: 418, Pages: 1-21, ISSN: 0167-2789
An open problem in optimization with noisy information is the computation of an exact minimizer that is independent of the amount of noise. A standard practice in stochastic approximation algorithms is to use a decreasing step-size. This however leads to a slower convergence. A second alternative is to use a fixed step-size and run independent replicas of the algorithm and average these. A third option is to run replicas of the algorithm and allow them to interact. It is unclear which of these options works best. To address this question, we reduce the problem of the computation of an exact minimizer with noisy gradient information to the study of stochastic mirror descent with interacting particles. We study the convergence of stochastic mirror descent and make explicit the tradeoffs between communication and variance reduction. We provide theoretical and numerical evidence to suggest that interaction helps to improve convergence and reduce the variance of the estimate.
Pavliotis GA, Stoltz G, Vaes U, 2021, Scaling limits for the generalized langevin equation, Journal of Nonlinear Science, Vol: 31, Pages: 1-58, ISSN: 0938-8974
In this paper, we study the diffusive limit of solutions to the generalized Langevin equation (GLE) in a periodic potential. Under the assumption of quasi-Markovianity, we obtain sharp longtime equilibration estimates for the GLE using techniques from the theory of hypocoercivity. We then show asymptotic results for the effective diffusion coefficient in the small correlation time regime, as well as in the overdamped and underdamped limits. Finally, we employ a recently developed numerical method (Roussel and Stoltz in ESAIM Math Model Numer Anal 52(3):1051–1083, 2018) to calculate the effective diffusion coefficient for a wide range of (effective) friction coefficients, confirming our asymptotic results.
Lucarini V, Pavliotis GA, Zagli N, 2020, Response theory and phase transitions for the thermodynamic limit of interacting identical systems: Phase Transitions in Interacting Systems, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 476, Pages: 1-27, ISSN: 1364-5021
We study the response to perturbations in the thermodynamic limit of a network of coupled identical agents undergoing a stochastic evolution which, in general, describes non-equilibrium conditions. All systems are nudged towards the common centre of mass. We derive Kramers-Kronig relations and sum rules for the linear susceptibilities obtained through mean field Fokker-Planck equations and then propose corrections relevant for the macroscopic case, which incorporates in a self-consistent way the effect of the mutual interaction between the systems. Such an interaction creates a memory effect. We are able to derive conditions determining the occurrence of phase transitions specifically due to system-to-system interactions. Such phase transitions exist in the thermodynamic limit and are associated with the divergence of the linear response but are not accompanied by the divergence in the integrated autocorrelation time for a suitably defined observable. We clarify that such endogenous phase transitions are fundamentally different from other pathologies in the linear response that can be framed in the context of critical transitions. Finally, we show how our results can elucidate the properties of the Desai-Zwanzig model and of the Bonilla-Casado-Morillo model, which feature paradigmatic equilibrium and non-equilibrium phase transitions, respectively.
Lucarini V, Pavliotis G, Zagli N, 2020, Response theory and phase transitions for the thermodynamic limit of interacting identical systems, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 476, ISSN: 1364-5021
We study the response to perturbations in the thermodynamic limit of a network of coupled identical agents undergoing a stochastic evolution which, in general, describes non-equilibrium conditions. All systems are nudged towards the common centre of mass. We derive Kramers–Kronig relations and sum rules for the linear susceptibilities obtained through mean field Fokker–Planck equations and then propose corrections relevant for the macroscopic case, which incorporates in a self-consistent way the effect of the mutual interaction between the systems. Such an interaction creates a memory effect. We are able to derive conditions determining the occurrence of phase transitions specifically due to system-to-system interactions. Such phase transitions exist in the thermodynamic limit and are associated with the divergence of the linear response but are not accompanied by the divergence in the integrated autocorrelation time for a suitably defined observable. We clarify that such endogenous phase transitions are fundamentally different from other pathologies in the linear response that can be framed in the context of critical transitions. Finally, we show how our results can elucidate the properties of the Desai–Zwanzig model and of the Bonilla–Casado–Morillo model, which feature paradigmatic equilibrium and non-equilibrium phase transitions, respectively.
Zelati MC, Pavliotis GA, 2020, Homogenization and hypocoercivity for Fokker-Planck equations driven by weakly compressible shear flows, IMA Journal of Applied Mathematics, Vol: 85, Pages: 951-979, ISSN: 0272-4960
We study the long-time dynamics of 2D linear Fokker–Planck equations driven by a drift that can be decomposed in the sum of a large shear component and the gradient of a regular potential depending on one spatial variable. The problem can be interpreted as that of a passive scalar advected by a slightly compressible shear flow, and undergoing small diffusion. For the corresponding stochastic differential equation, we give explicit homogenization rates in terms of a family of time-scales depending on the parameter measuring the strength of the incompressible perturbation. This is achieved by exploiting an auxiliary Poisson problem, and by computing the related effective diffusion coefficients. Regarding the long-time behavior of the solution of the Fokker–Planck equation, we provide explicit decay rates to the unique invariant measure by employing a quantitative version of the classical hypocoercivity scheme. From a fluid mechanics perspective, this turns out to be equivalent to quantifying the phenomenon of enhanced diffusion for slightly compressible shear flows.
Pozharskiy D, Wichrowski NJ, Duncan AB, et al., 2020, Manifold learning for accelerating coarse-grained optimization, Journal of Computational Dynamics, Vol: 7, Pages: 511-536, ISSN: 2158-2505
Algorithms proposed for solving high-dimensional optimization problems with no derivative information frequently encounter the "curse of dimensionality, " becoming ineffective as the dimension of the parameter space grows. One feature of a subclass of such problems that are effectively low-dimensional is that only a few parameters (or combinations thereof) are important for the optimization and must be explored in detail. Knowing these parameters/combinations in advance would greatly simplify the problem and its solution. We propose the data-driven construction of an effective (coarse-grained, "trend") optimizer, based on data obtained from ensembles of brief simulation bursts with an "inner" optimization algorithm, that has the potential to accelerate the exploration of the parameter space. The trajectories of this "effective optimizer" quickly become attracted onto a slow manifold parameterized by the few relevant parameter combinations. We obtain the parameterization of this low-dimensional, effective optimization manifold on the fly using data mining/manifold learning techniques on the results of simulation (inner optimizer iteration) burst ensembles and exploit it locally to "jump" forward along this manifold. As a result, we can bias the exploration of the parameter space towards the few, important directions and, through this "wrapper algorithm, " speed up the convergence of traditional optimization algorithms.
Carrillo JA, Delgadino MG, Pavliotis GA, 2020, A λ-convexity based proof for the propagation of chaos for weakly interacting stochastic particles, Journal of Functional Analysis, Vol: 279, Pages: 1-30, ISSN: 0022-1236
In this work we give a proof of the mean-field limit for λ-convex potentials using a purely variational viewpoint. Our approach is based on the observation that all evolution equations that we study can be written as gradient flows of functionals at different levels: in the set of probability measures, in the set of symmetric probability measures on N variables, and in the set of probability measures on probability measures. This basic fact allows us to rely on Γ-convergence tools for gradient flows to complete the proof by identifying the limits of the different terms in the Evolutionary Variational Inequalities (EVIs) associated to each gradient flow. The λ-convexity of the confining and interaction potentials is crucial for the unique identification of the limits and for deriving the EVIs at each description level of the interacting particle system.
Gomes SN, Pavliotis GA, Vaes U, 2020, Mean field limits for interacting diffusions with colored noise: phase transitions and spectral numerical methods, SIAM: Multiscale Modeling and Simulation, Vol: 18, Pages: 1343-1370, ISSN: 1540-3459
In this paper we consider systems of weakly interacting particles driven by colored noise in a bistable potential, and we study the effect of the correlation time of the noise on the bifurcation diagram for the equilibrium states. We accomplish this by solving the corresponding McKean--Vlasov equation using a Hermite spectral method, and we verify our findings using Monte Carlo simulations of the particle system. We consider both Gaussian and non-Gaussian noise processes, and for each model of the noise we also study the behavior of the system in the small correlation time regime using perturbation theory. The spectral method that we develop in this paper can be used for solving linear and nonlinear, local and nonlocal (mean field) Fokker--Planck equations, without requiring that they have a gradient structure.
Carrillo de la Plata JA, Gvalani R, Pavliotis G, et al., 2020, Long-time behaviour and phase transitions for the McKean—Vlasov equation on the torus, Archive for Rational Mechanics and Analysis, Vol: 235, Pages: 635-690, ISSN: 0003-9527
We study the McKean-Vlasov equation∂t% = β−1∆% + κ ∇·(%∇(W ? %)) ,with periodic boundary conditions on the torus. We first study the global asymptotic stability of thehomogeneous steady state. We then focus our attention on the stationary system, and prove the existenceof nontrivial solutions branching from the homogeneous steady state, through possibly infinitely manybifurcations, under appropriate assumptions on the interaction potential. We also provide sufficientconditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase theseresults by applying them to several examples of interaction potentials such as the noisy Kuramoto modelfor synchronisation, the Keller–Segel model for bacterial chemotaxis, and the noisy Hegselmann–Kraussemodel for opinion dynamics.
Abdulle A, Pavliotis GA, Vilmart G, 2019, Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations, Comptes Rendus Mathematique (Academie des Sciences), Vol: 357, Pages: 349-354, ISSN: 0764-4442
In this paper, we propose a new approach for sampling from probability measures in, possibly, high-dimensional spaces. By perturbing the standard overdamped Langevin dynamics by a suitable Stratonovich perturbation that preserves the invariant measure of the original system, we show that accelerated convergence to equilibrium and reduced asymptotic variance can be achieved, leading, thus, to a computationally advantageous sampling algorithm. The new perturbed Langevin dynamics is reversible with respect to the target probability measure and, consequently, does not suffer from the drawbacks of the nonreversible Langevin samplers that were introduced in C.-R. Hwang et al. (1993)and studied in, e.g., T. Lelièvre et al. (2013)and A.B. Duncan et al. (2016), while retaining all of their advantages in terms of accelerated convergence and reduced asymptotic variance. In particular, the reversibility of the dynamics ensures that there is no oscillatory transient behaviour. The improved performance of the proposed methodology, in comparison to the standard overdamped Langevin dynamics and its nonreversible perturbation, is illustrated on an example of sampling from a two-dimensional warped Gaussian target distribution.
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