Learning Dynamics from Data: Discovering Interaction Laws of Particle and Agent Systems

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Particle and agent-based systems are ubiquitous in science, for example, particle systems in fundamental physics, agent-based systems that model opinion dynamics under the social influence, prey-predator dynamics, flocking and swarming, and phototaxis in cell dynamics. To understand the mechanism of these systems, a fundamental challenge is to infer the laws of interaction between the particles and agents from observational data. This project aims to develop mathematical and statistical theory and computationally efficient algorithms for learning these interaction laws from observations in the form of trajectories of the systems. The theory provides performance guarantees and uncertainty quantification in the estimations, therefore providing foundations for model selection and for optimal data collection. The algorithms are scalable to large data sets, avoiding the curse of dimensionality, and are applicable to a wide variety of systems from Physics, Biology, Ecology and Social Sciences.The interaction laws vary largely for different systems, and there is no analytical form in general. The PIs propose non-parametric statistical inference approaches for learning the interaction laws, with no reference or assumption on their analytical form. The research will develop a systematical learning theory for the non-parametric regression of the interaction kernels, whose values are not observed and can not be computed from the data of trajectories of the particles or agents in the system. The theory will study the identifiability of the interaction kernels, the consistency of the estimators, and the optimal choice of hypothesis spaces to achieve optimal rate of convergence of the estimators. With the guidance from the learning theory, we will design computationally efficient algorithms with the following features: (i) avoiding the curse of dimensionality by focusing on the intrinsic dimension of the interaction kernels; (ii) with theoretical guarantee and uncertainty quantification which can be used for model selection and optimal data collection; (iii) scalable to large data sets by implementing in parallel.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.