  The moments of spatial probabilistic systems are often given by an infinite
hierarchy of coupled differential equations. Moment closure methods are used to
approximate a subset of low order moments by terminating the hierarchy at some
order and replacing higher order terms with functions of lower order ones. For
a given system, it is not known beforehand which closure approximation is
optimal, i.e. which higher order terms are relevant in the current regime.
Further, the generalization of such approximations is typically poor, as higher
order corrections may become relevant over long timescales. We have developed a
method to learn moment closure approximations directly from data using dynamic
Boltzmann distributions (DBDs). The dynamics of the distribution are
parameterized using basis functions from finite element methods, such that the
approach can be applied without knowing the true dynamics of the system under
consideration. We use the hierarchical architecture of deep Boltzmann machines
(DBMs) with multinomial latent variables to learn closure approximations for
progressively higher order spatial correlations. The learning algorithm uses a
centering transformation, allowing the dynamic DBM to be trained without the
need for pre-training. We demonstrate the method for a Lotka-Volterra system on
a lattice, a typical example in spatial chemical reaction networks. The
approach can be applied broadly to learn deep generative models in applications
where infinite systems of differential equations arise.
