  Complex behaviour in many systems arises from the stochastic interactions of
spatially distributed particles or agents. Stochastic reaction-diffusion
processes are widely used to model such behaviour in disciplines ranging from
biology to the social sciences, yet they are notoriously difficult to simulate
and calibrate to observational data. Here we use ideas from statistical physics
and machine learning to provide a solution to the inverse problem of learning a
stochastic reaction-diffusion process from data. Our solution relies on a
non-trivial connection between stochastic reaction-diffusion processes and
spatio-temporal Cox processes, a well-studied class of models from
computational statistics. This connection leads to an efficient and flexible
algorithm for parameter inference and model selection. Our approach shows
excellent accuracy on numeric and real data examples from systems biology and
epidemiology. Our work provides both insights into spatio-temporal stochastic
systems, and a practical solution to a long-standing problem in computational
modelling.
