  Analyzing the motion of multiple biological agents, be it cells or individual
animals, is pivotal for the understanding of complex collective behaviors. With
the advent of advanced microscopy, detailed images of complex tissue formations
involving multiple cell types have become more accessible in recent years.
However, deciphering the underlying rules that govern cell movements is far
from trivial. Here, we present a novel deep learning framework to estimate the
underlying equations of motion from observed trajectories, a pivotal step in
decoding such complex dynamics. Our framework integrates graph neural networks
with neural differential equations, enabling effective prediction of two-body
interactions based on the states of the interacting entities. We demonstrate
the efficacy of our approach through two numerical experiments. First, we used
a simulated data from a toy model to tune the hyperparameters. Based on the
obtained hyperparameters, we then applied this approach to a more complex model
that describes interacting cells of cellular slime molds. Our results show that
the proposed method can accurately estimate the function of two-body
interactions, thereby precisely replicating both individual and collective
behaviors within these systems.
