Learning actions from data using invertible neural networks

Many problems in physics can be cast into the form of a polynomial action, of which the coefficients determine physical properties. A typical approach is to derive these coefficients from a theory of microscopic interactions. However, this may not always be possible, or a microscopic theory may not be known. We here use invertible neural networks (INNs) trained in an unsupervised manner to describe data distributions. We choose a nonlinearity for which the coefficients of the corresponding action can be computed from the trained weights. A diagrammatic language expresses the change in the action from one layer of the INN to the next. Inverting the network allows us to extract coefficients of the data distribution and to trace how the INN parameters shape the interaction terms in its action. We test this formalism on a reduced model of Ising spins.