  One challenge of physics is to explain how collective properties arise from
microscopic interactions. Indeed, interactions form the building blocks of
almost all physical theories and are described by polynomial terms in the
action. The traditional approach is to derive these terms from elementary
processes and then use the resulting model to make predictions for the entire
system. But what if the underlying processes are unknown? Can we reverse the
approach and learn the microscopic action by observing the entire system? We
use invertible neural networks (INNs) to first learn the observed data
distribution. By the choice of a suitable nonlinearity for the neuronal
activation function, we are then able to compute the action from the weights of
the trained model; a diagrammatic language expresses the change of the action
from layer to layer. This process uncovers how the network hierarchically
constructs interactions via nonlinear transformations of pairwise relations. We
test this approach on simulated data sets of interacting theories. The network
consistently reproduces a broad class of unimodal distributions; outside this
class, it finds effective theories that approximate the data statistics up to
the third cumulant. We explicitly show how network depth and data quantity
jointly improve the agreement between the learned and the true model. This work
shows how to leverage the power of machine learning to transparently extract
microscopic models from data.
