Inferring microscale properties of interacting systems from macroscale observations

Emergent dynamics of complex systems are observed throughout nature and society. The coordinated motion of bird flocks, the spread of opinions, fashions and fads, or the dynamics of an epidemic, are all examples of complex macroscale phenomena that arise from fine-scale interactions at the individual level. In many scenarios, observations of the system can only be made at the macroscale, while we are interested in creating and fitting models of the microscale dynamics. This creates a challenge for inference as a formal mathematical link between the microscale and macroscale is rarely available. Here, we develop an inferential framework that bypasses the need for a formal link between scales and instead uses sparse Gaussian process regression to learn the drift and diffusion terms of an empirical Fokker-Planck equation, which describes the time evolution of the probability density of a macroscale variable. This gives us access to the likelihood of the microscale properties of the physical system and a second Gaussian process is then used to emulate the log-likelihood surface, allowing us to define a fast, adaptive MCMC sampler, which iteratively refines the emulator when needed. We illustrate the performance of our method by applying it to a simple model of collective motion.