Learning Collective Behaviors from Observation

We present a review of a series of learning methods used to identify the structure of dynamical systems, aiming to understand emergent behaviors in complex systems of interacting agents. These methods not only offer theoretical guarantees of convergence but also demonstrate computational efficiency in handling high-dimensional observational data. They can manage observation data from both first- and second-order dynamical systems, accounting for observation/stochastic noise, complex interaction rules, missing interaction features, and real-world observations of interacting agent systems. The essence of developing such a series of learning methods lies in designing appropriate loss functions using the variational inverse problem approach, which inherently provides dimension reduction capabilities to our learning methods.