  Particle dynamics and multi-agent systems provide accurate dynamical models
for studying and forecasting the behavior of complex interacting systems. They
often take the form of a high-dimensional system of differential equations
parameterized by an interaction kernel that models the underlying attractive or
repulsive forces between agents. We consider the problem of constructing a
data-based approximation of the interacting forces directly from noisy
observations of the paths of the agents in time. The learned interaction
kernels are then used to predict the agents behavior over a longer time
interval. The approximation developed in this work uses a randomized feature
algorithm and a sparse randomized feature approach. Sparsity-promoting
regression provides a mechanism for pruning the randomly generated features
which was observed to be beneficial when one has limited data, in particular,
leading to less overfitting than other approaches. In addition, imposing
sparsity reduces the kernel evaluation cost which significantly lowers the
simulation cost for forecasting the multi-agent systems. Our method is applied
to various examples, including first-order systems with homogeneous and
heterogeneous interactions, second order homogeneous systems, and a new sheep
swarming system.
