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

Deep Learning Moment Closure Approximations using Dynamic Boltzmann Distributions