Abstract
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 long-time dynamics. This to some extend avoids the curse-of-dimensionality for estimating large dynamical systems under symmetry of the particle interaction.
Highlights
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 is able to estimate the interaction potential parameter with optimal rate of convergence simultaneously in mean-field limit and in long-time dynamics
This to some extend avoids the curse-ofdimensionality for estimating large dynamical systems under symmetry of the particle interaction
Dynamical systems of interacting particles have a wide range of applications on modeling collective behaviors in physics [5], biology [14, 19], social science [15], and more recently in machine learning as useful tools to understand the stochastic gradient descent (SGD) dynamics on neural networks [13]
Summary
Dynamical systems of interacting particles have a wide range of applications on modeling collective behaviors in physics [5], biology [14, 19], social science [15], and more recently in machine learning as useful tools to understand the stochastic gradient descent (SGD) dynamics on neural networks [13]. 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 [1, 10, 11]. 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
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