Inferring Particle Interaction Physical Models and Their Dynamical Properties

Learning Interaction Dynamics from Particle Trajectories and Density Evolution

“…It is additionally assumed that ρ, m, u are compactly supported. Substituting in (7), we obtain the (d+1) compressible Euler equations on [t 0 , t f ] × D (see also [12]):…”

“…An energy-based approach is given in [7], where it is shown that the Cucker-Smale model ( 1) is equivalent to a fully connected N-dimensional network of generalized mass-springdampers with appropriately defined Hamiltonian functions, that can be written in an input-state-output port-Hamiltonian form [19]:…”

“…being the vectors of relative distances and momenta between each pair of particles, and the quantities J = −J T , H and R are appropriately defined. The dependence of (57) on the interaction function ψ is introduced by the resistive term R = R(ψ) [7], and is modeled as an artificial neural network with a single hidden layer. The parameters are represented by a vector θ and the learning process is formulated as a least-squares optimization problem…”

“…We note, however, that there is a potential advantage in using the proposed interaction function model ( 17), (18), even in learning the interaction dynamics of a swarm from particle trajectories. First, the number of parameters to be estimated is greatly reduced, compared to a general regression function such as a neural network [7], or a mixture of Gaussians [11], which reduces the amount of data required for convergence. Secondly, every update in the optimization algorithm improves the estimate of the interaction function over the entire domain D, and not only over a small subset D o ⊂ D where the distances between each pair of interacting particles happen to be observed.…”

“…Extracting the laws of interaction between agents of general networked systems finds applications in a wide range of fields, from power systems and chemical reaction networks, to social networks and UAV swarms [3]- [5], [7]- [9]. Statistical [10], and model-based [2], [3], [7], [11] learning approaches have been used to learn the interaction rules between agents.…”