Data-Driven Model Selections of Second-Order Particle Dynamics via
  Integrating Gaussian Processes with Low-Dimensional Interacting Structures
Jinchao Feng Charles Kulick Sui Tang
In this paper, we focus on the data-driven discovery of a general
second-order particle-based model that contains many state-of-the-art models
for modeling the aggregation and collective behavior of interacting agents of
similar size and body type. This model takes the form of a high-dimensional
system of ordinary differential equations parameterized by two interaction
kernels that appraise the alignment of positions and velocities. We propose a
Gaussian Process-based approach to this problem, where the unknown model
parameters are marginalized by using two independent Gaussian Process (GP)
priors on latent interaction kernels constrained to dynamics and observational
data. This results in a nonparametric model for interacting dynamical systems
that accounts for uncertainty quantification. We also develop acceleration
techniques to improve scalability. Moreover, we perform a theoretical analysis
to interpret the methodology and investigate the conditions under which the
kernels can be recovered. We demonstrate the effectiveness of the proposed
approach on various prototype systems, including the selection of the order of
the systems and the types of interactions. In particular, we present
applications to modeling two real-world fish motion datasets that display
flocking and milling patterns up to 248 dimensions. Despite the use of small
data sets, the GP-based approach learns an effective representation of the
nonlinear dynamics in these spaces and outperforms competitor methods.