Interacting Particle Systems Research Articles

Ergodicity and long-time behavior of the Random Batch Method for interacting particle systems

We study the geometric ergodicity and the long-time behavior of the Random Batch Method for interacting particle systems, which exhibits superior numerical performance in recent large-scale scientific computing experiments. We show that for both the interacting particle system (IPS) and the random batch interacting particle system (RB–IPS), the distribution laws converge to their respective invariant distributions exponentially, and the convergence rate does not depend on the number of particles [Formula: see text], the time step [Formula: see text] for batch divisions or the batch size [Formula: see text]. Moreover, the Wasserstein-1 distance between the invariant distributions of the IPS and the RB–IPS is bounded by [Formula: see text], showing that the RB–IPS can be used to sample the invariant distribution of the IPS accurately with greatly reduced computational cost.

Stein variational gradient descent: Many-particle and long-time asymptotics

Stein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well as a stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statistics:variational inferenceandMarkov chain Monte Carlo. As it turns out, these are tightly linked through a correspondence between gradient flow structures and large-deviation principles rooted in statistical physics. To expose this relationship, we develop the cotangent space construction for the Stein geometry, prove its basic properties, and determine the large-deviation functional governing the many-particle limit for the empirical measure. Moreover, we identify theStein-Fisher information(orkernelised Stein discrepancy) as its leading order contribution in the long-time and many-particle regime in the sense of $ \Gamma $-convergence, shedding some light on the finite-particle properties of SVGD. Finally, we establish a comparison principle between the Stein-Fisher information and RKHS-norms that might be of independent interest.

Fluctuations for Some Nonstationary Interacting Particle Systems via Boltzmann–Gibbs Principle

Abstract Conjecture II.3.6 of Spohn in [47] and Lecture 7 of Jensen–Yau in [35] ask for a general derivation of universal fluctuations of hydrodynamic limits in large-scale stochastic interacting particle systems. However, the past few decades have witnessed only minimal progress according to [26]. In this paper, we develop a general method for deriving the so-called Boltzmann–Gibbs principle for a general family of nonintegrable and nonstationary interacting particle systems, thereby responding to Spohn and Jensen–Yau. Most importantly, our method depends mostly on local and dynamical, and thus more general/universal, features of the model. This contrasts with previous work [6, 8, 24, 34], all of which rely on global and nonuniversal assumptions on invariant measures or initial measures of the model. As a concrete application of the method, we derive the KPZ equation as a large-scale limit of the height functions for a family of nonstationary and nonintegrable exclusion processes with an environment-dependent asymmetry. This establishes a first result to Big Picture Question 1.6 in [54] for nonstationary and nonintegrable ‘speed-change’ models that have also been of interest beyond KPZ [18, 22, 23, 38].

Path dependent McKean-Vlasov SDEs with Hölder continuous diffusion

In this paper, by using the Yamada-Watanabe approximation, the well-posedness for one-dimensional path dependent SDEs with $ \alpha $($ \alpha\geq \frac{1}{2} $)-Hölder continuous diffusion is investigated, which together with the Banach fixed point theorem derives the well-posedness for the corresponding path dependent McKean-Vlasov SDEs. Moreover, the well-posedness for the interacting particle system is also obtained. Finally, the associated quantitative propagation of chaos in the sense of Wasserstein distance is studied, which combined with the Girsanov's transform yields the quantitative propagation of chaos in total variation distance as well as relative entropy.

Reproducing kernel Hilbert spaces in the mean field limit

Kernel methods, being supported by a well-developed theory and coming with efficient algorithms, are among the most popular and successful machine learning techniques. From a mathematical point of view, these methods rest on the concept of kernels and function spaces generated by kernels, so–called reproducing kernel Hilbert spaces. Motivated by recent developments of learning approaches in the context of interacting particle systems, we investigate kernel methods acting on data with many measurement variables. We show the rigorous mean field limit of kernels and provide a detailed analysis of the limiting reproducing kernel Hilbert space. Furthermore, several examples of kernels, that allow a rigorous mean field limit, are presented.

Applying monoid duality to a double contact process

In this paper we use duality techniques to study a coupling of the well-known contact process (CP) and the annihilating branching process. As the latter can be seen as a cancellative version of the contact process, we rebrand it as the cancellative contact process (cCP). Our process of interest will consist of two components, the first being a CP and the second being a cCP. We call this process the double contact process (2CP) and prove that it has (depending on the model parameters) at most one invariant law under which ones are present in both processes. In particular, we can choose the model parameters in such a way that CP and cCP are monotonely coupled. In this case also the above mentioned invariant law will have the property that, under it, ones (modeling “infected individuals”) can only be present in the cCP at sites where there are also ones in the CP. Along the way we extend the dualities for Markov processes discovered in our paper “Commutative monoid duality” to processes on infinite state spaces so that they, in particular, can be used for interacting particle systems.

Hydrodynamic limits of interacting particle systems on crystal lattices in periodic realizations

Hydrodynamic limits of interacting particle systems on crystal lattices in periodic realizations

Singular limit of an Allen–Cahn equation with nonlinear diffusion

We consider an Allen-Cahn equation with nonlinear diffusion, motivated by the study of the scaling limit of certain interacting particle systems. We investigate its singular limit and show the generation and propagation of an interface in the limit. The evolution of this limit interface is governed by mean curvature flow with a novel, homogenized speed in terms of a surface tension-mobility parameter emerging from the nonlinearity in our equation.

Controlled interacting particle algorithms for simulation-based reinforcement learning

This paper is concerned with optimal control problems for control systems in continuous time, and interacting particle system methods designed to construct approximate control solutions. Particular attention is given to the linear quadratic (LQ) control problem. There is a growing interest in re-visiting this classical problem, in part due to the successes of reinforcement learning (RL). The main question of this body of research (and also of our paper) is to approximate the optimal control law without explicitly solving the Riccati equation. A novel simulation-based algorithm, namely a dual ensemble Kalman filter (EnKF), is introduced. The algorithm is used to obtain formulae for optimal control, expressed entirely in terms of the EnKF particles. An extension to the nonlinear case is also presented. The theoretical results and algorithms are illustrated with numerical experiments. • First controlled interacting particle system to solve the optimal control problem. • Suitably coupling the simulations solves the policy optimization problem. • Numerical demonstrations and comparison with state-of-the-art approaches for LQR problem. • For LQR, proposed algorithm is two orders of magnitude more computationally efficient.

A consistence-stability approach to hydrodynamic limit of interacting particle systems on lattices

A consistence-stability approach to hydrodynamic limit of interacting particle systems on lattices

On a family of coupled diffusions that can never change their initial order

We introduce a real-valued family of interacting diffusions where their paths can meet but cannot cross each other in a way that would alter their initial order. Any given interacting pair is a solution to coupled stochastic differential equations with time-dependent coefficients satisfying certain regularity conditions with respect to each other. These coefficients explicitly determine whether these processes bounce away from each other or stick to one another if/when their paths collide. When all interacting diffusions in the system follow a martingale behaviour, and if all these paths ultimately come into collision, we show that the system reaches a random steady-state with zero fluctuation thereafter. We prove that in a special case when certain paths abide to a deterministic trend, the system reduces down to the topology of captive diffusions. We also show that square-root diffusions form a subclass of the proposed family of processes. Applications include order-driven interacting particle systems in physics, adhesive microbial dynamics in biology and risk-bounded quadratic optimization solutions in control theory.

Interacting Particle System based estimation of reach probability of General Stochastic Hybrid Systems

For diffusions, a well-developed approach in rare event estimation is to introduce a suitable factorization of the reach probability and then to estimate these factors through simulation of an Interacting Particle System (IPS). This paper studies IPS based reach probability estimation for General Stochastic Hybrid Systems (GSHS). The continuous-time executions of a GSHS evolve in a hybrid state space under influence of combinations of diffusions, spontaneous jumps and forced jumps. In applying IPS to a GSHS, simulation of the GSHS execution plays a central role. From literature, two basic approaches in simulating GSHS execution are known. One approach is direct simulation of a GSHS execution. An alternative is to first transform the spontaneous jumps of a GSHS to forced transitions, and then to simulate executions of this transformed version. This paper will show that the latter transformation yields an extra Markov state component that should be treated as being unobservable for the IPS process. To formally make this state component unobservable for IPS, this paper also develops an enriched GSHS transformation prior to transforming spontaneous jumps to forced jumps. The expected improvements in IPS reach probability estimation are also illustrated through simulation results for a simple GSHS example.

Learning physics-consistent particle interactions.

Interacting particle systems play a key role in science and engineering. Access to the governing particle interaction law is fundamental for a complete understanding of such systems. However, the inherent system complexity keeps the particle interaction hidden in many cases. Machine learning methods have the potential to learn the behavior of interacting particle systems by combining experiments with data analysis methods. However, most existing algorithms focus on learning the kinetics at the particle level. Learning pairwise interaction, e.g., pairwise force or pairwise potential energy, remains an open challenge. Here, we propose an algorithm that adapts the Graph Networks framework, which contains an edge part to learn the pairwise interaction and a node part to model the dynamics at particle level. Different from existing approaches that use neural networks in both parts, we design a deterministic operator in the node part that allows to precisely infer the pairwise interactions that are consistent with underlying physical laws by only being trained to predict the particle acceleration. We test the proposed methodology on multiple datasets and demonstrate that it achieves superior performance in inferring correctly the pairwise interactions while also being consistent with the underlying physics on all the datasets. While the previously proposed approaches are able to be applied as simulators, they fail to infer physically consistent particle interactions that satisfy Newton's laws. Moreover, the proposed physics-induced graph network for particle interaction also outperforms the other baseline models in terms of generalization ability to larger systems and robustness to significant levels of noise. The developed methodology can support a better understanding and discovery of the underlying particle interaction laws, and hence, guide the design of materials with targeted properties.

A Conversation with David J. Aldous

David John Aldous was born in Exeter U.K. on July 13, 1952. He received a B.A. and Ph.D. in Mathematics in 1973 and 1977, respectively from Cambridge. After spending two years as a research fellow at St. John’s College, Cambridge, he joined the Department of Statistics at the University of California, Berkeley in 1979 where he spent the rest of his academic career until retiring in 2018. He is known for seminal contributions on many topics within probability including weak convergence and tightness, exchangeability, Markov chain mixing times, Poisson clumping heuristic and limit theory for large discrete random structures including random trees, stochastic coagulation and fragmentation systems, models of complex networks and interacting particle systems on such structures. For his contributions to the field, he has received numerous honors and awards including the Rollo Davidson prize in 1980, the inaugural Loeve prize in Probability in 1993, and the Brouwer medal in 2021, and being elected as an IMS fellow in 1985, Fellow of the Royal Society in 1994, Fellow of the American Academy of Arts and Sciences in 2004, elected to the National Academy of Sciences (foreign associate) in 2010, ICM plenary speaker in 2010 and AMS fellow in 2012.

Asymptotic expansions for a class of Fredholm Pfaffians and interacting particle systems

Motivated by the phenomenon of duality for interacting particle systems, we introduce two classes of Pfaffian kernels describing a number of Pfaffian point processes in the “bulk” and at the “edge.” Using the probabilistic method due to Mark Kac, we prove two Szegő-type asymptotic expansion theorems for the corresponding Fredholm Pfaffians. The idea of the proof is to introduce an effective random walk with transition density determined by the Pfaffian kernel, express the logarithm of the Fredholm Pfaffian through expectations with respect to the random walk, and analyse the expectations using general results on random walks. We demonstrate the utility of the theorems by calculating asymptotics for the empty interval and noncrossing probabilities for a number of examples of Pfaffian point processes: coalescing/annihilating Brownian motions, massive coalescing Brownian motions, real zeros of Gaussian power series and Kac polynomials, and real eigenvalues for the real Ginibre ensemble.

Efficient derivative-free Bayesian inference for large-scale inverse problems

We consider Bayesian inference for large-scale inverse problems, where computational challenges arise from the need for repeated evaluations of an expensive forward model. This renders most Markov chain Monte Carlo approaches infeasible, since they typically require model runs, or more. Moreover, the forward model is often given as a black box or is impractical to differentiate. Therefore derivative-free algorithms are highly desirable. We propose a framework, which is built on Kalman methodology, to efficiently perform Bayesian inference in such inverse problems. The basic method is based on an approximation of the filtering distribution of a novel mean-field dynamical system, into which the inverse problem is embedded as an observation operator. Theoretical properties are established for linear inverse problems, demonstrating that the desired Bayesian posterior is given by the steady state of the law of the filtering distribution of the mean-field dynamical system, and proving exponential convergence to it. This suggests that, for nonlinear problems which are close to Gaussian, sequentially computing this law provides the basis for efficient iterative methods to approximate the Bayesian posterior. Ensemble methods are applied to obtain interacting particle system approximations of the filtering distribution of the mean-field model; and practical strategies to further reduce the computational and memory cost of the methodology are presented, including low-rank approximation and a bi-fidelity approach. The effectiveness of the framework is demonstrated in several numerical experiments, including proof-of-concept linear/nonlinear examples and two large-scale applications: learning of permeability parameters in subsurface flow; and learning subgrid-scale parameters in a global climate model. Moreover, the stochastic ensemble Kalman filter and various ensemble square-root Kalman filters are all employed and are compared numerically. The results demonstrate that the proposed method, based on exponential convergence to the filtering distribution of a mean-field dynamical system, is competitive with pre-existing Kalman-based methods for inverse problems.

Derivation of the stochastic Burgers equation from totally asymmetric interacting particle systems

We consider equilibrium fluctuations for a class of totally asymmetric zero-range type interacting particle systems, which is particularly related to q-totally asymmetric simple exclusion processes (q-TASEPs). As a main result, we show that density fluctuation of our process converges to the stationary energy solution of the stochastic Burgers equation. Our proof is based on the second-order Boltzmann–Gibbs principle and a Taylor expansion argument.

Sharp threshold for the FA-2f kinetically constrained model

The Fredrickson-Andersen 2-spin facilitated model on $${{\mathbb {Z}}} ^d$$ (FA-2f) is a paradigmatic interacting particle system with kinetic constraints (KCM) featuring dynamical facilitation, an important mechanism in condensed matter physics. In FA-2f a site may change its state only if at least two of its nearest neighbours are empty. Although the process is reversible w.r.t. a product Bernoulli measure, it is not attractive and features degenerate jump rates and anomalous divergence of characteristic time scales as the density q of empty sites tends to 0. A natural random variable encoding the above features is $$\tau _0$$ , the first time at which the origin becomes empty for the stationary process. Our main result is the sharp threshold $$\begin{aligned} \tau _0=\exp \Big (\frac{d\cdot \lambda (d,2)+o(1)}{q^{1/(d-1)}}\Big )\quad \text {w.h.p.} \end{aligned}$$ with $$\lambda (d,2)$$ the sharp threshold constant for 2-neighbour bootstrap percolation on $${{\mathbb {Z}}} ^d$$ , the monotone deterministic automaton counterpart of FA-2f. This is the first sharp result for a critical KCM and it compares with Holroyd’s 2003 result on bootstrap percolation and its subsequent improvements. It also settles various controversies accumulated in the physics literature over the last four decades. Furthermore, our novel techniques enable completing the recent ambitious program on the universality phenomenon for critical KCM and establishing sharp thresholds for other two-dimensional KCM.

A random walk through Canadian contributions on empirical processes and their applications in probability and statistics

AbstractIn this article, we present a review of important results and statistical applications obtained or generalized by Canadian pioneers and their collaborators, for empirical processes of independent and identically distributed observations, pseudo‐observations, and time series. In particular, we consider weak convergence and strong approximations results, as well as tests for model adequacy such as tests of independence, tests of goodness‐of‐fit, tests of change point, and tests of serial dependence for time series. We also consider applications of empirical processes of interacting particle systems for the approximation of measure‐valued processes.

Learning anisotropic interaction rules from individual trajectories in a heterogeneous cellular population

Interacting particle system (IPS) models have proven to be highly successful for describing the spatial movement of organisms. However, it is challenging to infer the interaction rules directly from data. In the field of equation discovery, the weak-form sparse identification of nonlinear dynamics (WSINDy) methodology has been shown to be computationally efficient for identifying the governing equations of complex systems from noisy data. Motivated by the success of IPS models to describe the spatial movement of organisms, we develop WSINDy for the second-order IPS to learn equations for communities of cells. Our approach learns the directional interaction rules for each individual cell that in aggregate govern the dynamics of a heterogeneous population of migrating cells. To sort a cell according to the active classes present in its model, we also develop a novel ad hoc classification scheme (which accounts for the fact that some cells do not have enough evidence to accurately infer a model). Aggregated models are then constructed hierarchically to simultaneously identify different species of cells present in the population and determine best-fit models for each species. We demonstrate the efficiency and proficiency of the method on several test scenarios, motivated by common cell migration experiments.