On the identifiability of interaction functions in systems of interacting particles

Li, Zhongyang; Lu, Fei; Maggioni, Mauro; Tang, Sui; Zhang, Cheng

doi:10.48550/arxiv.1912.11965

Cited by 4 publications

(12 citation statements)

References 30 publications

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“…Second, the rate of convergence in (11) is parametric (and thus rate-optimal) in both N and t. Specifically, for fixed d, we can obtain from Theorem 3.1 the large N (mean-field limit) and large t (long-time dynamics) asymptotics as:…”

Section: Rate Of Convergencementioning

confidence: 98%

“…In particular, if θ j is closer to zero, then the log-likelihood ratio becomes flatter and thus larger t is necessary to see the information from samples of the stationary distribution. On the other hand, if the processes start from the stationary distribution, then this trajectory time lower bound is not needed to obtain (11).…”

Section: Rate Of Convergencementioning

confidence: 99%

“…Dynamical systems of interacting particles have a wide range of applications on modeling collective behaviors in physics [5], biology [16,21], social science [17], and more recently in machine learning as useful tools to understand the stochastic gradient descent (SGD) dynamics on neural networks [15]. Due to the large number of particles, such dynamical systems are high-dimensional even for single-trajectory data from each particle, and statistical learning problems for lower-dimensional interaction functionals from data are usually challenging [2,11,13]. In this paper, we propose a likelihood based inference to estimate the parameters of the interaction function induced by a potential energy in an N interacting particle system based on their observed (continuous-time) trajectories, and establish its statistical guarantees.…”

Section: Introductionmentioning

confidence: 99%

“…Existing literature. Learnability (i.e., identifiability) of interaction functions in interacting particle systems under the coercivity condition were studied in [13,11]. In the noiseless setting, estimation of the interaction kernel, a scalar-valued function of pairwise distance between particles in the system, was first studied in [2] for single-trajectory data in the mean-field limit, where the rate of convergence is no faster than N −1/d .…”

Section: Introductionmentioning

confidence: 99%

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Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data

Chen

2020

Preprint

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This paper concerns the parameter estimation problem for the quadratic potential energy in interacting particle systems from continuous-time and single-trajectory data. Even though such dynamical systems are high-dimensional, we show that the vanilla maximum likelihood estimator (without regularization) is able to estimate the interaction potential parameter with optimal rate of convergence simultaneously in mean-field limit and in longtime dynamics. This to some extend avoids the curse-of-dimensionality for estimating large dynamical systems under symmetry of the particle interaction.

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Section: Rate Of Convergencementioning

confidence: 98%

Section: Rate Of Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data

Chen

2020

Preprint

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“…In [13], a convergence study of learning unknown interaction kernels from observation of first-order models of homogeneous agents was done for increasing N , the number of agents. The estimation problem with N fixed, but the number of trajectories M varying, for first-order and second-order models of heterogeneous agents was numerically studied in [52] and learning theory on these first-order models was developed in [51,48]. Further extension of the model and algorithm to more complicated second-order systems, with particular emphasis on emergent collective behaviors, was discussed in [83].…”

Section: Introductionmentioning

confidence: 99%

Learning Theory for Inferring Interaction Kernels in Second-Order Interacting Agent Systems

Miller¹,

Tang²,

Zhong³

et al. 2020

Preprint

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Modeling the complex interactions of systems of particles or agents is a fundamental scientific and mathematical problem that is studied in diverse fields, ranging from physics and biology, to economics and machine learning. In this work, we describe a very general second-order, heterogeneous, multivariable, interacting agent model, with an environment, that encompasses a wide variety of known systems. We describe an inference framework that uses nonparametric regression and approximation theory based techniques to efficiently derive estimators of the interaction kernels which drive these dynamical systems. We develop a complete learning theory which establishes strong consistency and optimal nonparametric min-max rates of convergence for the estimators, as well as provably accurate predicted trajectories. The estimators exploit the structure of the equations in order to overcome the curse of dimensionality and we describe a fundamental coercivity condition on the inverse problem which ensures that the kernels can be learned and relates to the minimal singular value of the learning matrix. The numerical algorithm presented to build the estimators is parallelizable, performs well on high-dimensional problems, and is demonstrated on complex dynamical systems.

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Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories

Maggioni

Tang

2020

Preprint

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We consider stochastic systems of interacting particles or agents, with dynamics determined by an interaction kernel which only depends on pairwise distances. We study the problem of inferring this interaction kernel from observations of the positions of the particles, in either continuous or discrete time, along multiple independent trajectories. We introduce a nonparametric inference approach to this inverse problem, based on a regularized maximum likelihood estimator constrained to suitable hypothesis spaces adaptive to data. We show that a coercivity condition enables us to control the condition number of this problem and prove the consistency of our estimator, and that in fact it converges at a near-optimal learning rate, equal to the min-max rate of 1dimensional non-parametric regression. In particular, this rate is independent of the dimension of the state space, which is typically very high. We also analyze the discretization errors in the case of discrete-time observations, showing that it is of order 1/2 in terms of the time gaps between observations. This term, when large, dominates the sampling error and the approximation error, preventing convergence of the estimator. Finally, we exhibit an efficient parallel algorithm to construct the estimator from data, and we demonstrate the effectiveness of our algorithm with numerical tests on prototype systems including stochastic opinion dynamics and a Lennard-Jones model.

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On the identifiability of interaction functions in systems of interacting particles

Cited by 4 publications

References 30 publications

Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data

Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data

Learning Theory for Inferring Interaction Kernels in Second-Order Interacting Agent Systems

Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories

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