Learning moment closure in reaction-diffusion systems with spatial dynamic Boltzmann distributions.

Many physical systems are described by probability distributions that evolve in both time and space. Modeling these systems is often challenging due to their large state space and analytically intractable or computationally expensive dynamics. To address these problems, we study a machine-learning approach to model reduction based on the Boltzmann machine. Given the form of the reduced model Boltzmann distribution, we introduce an autonomous differential equation system for the interactions appearing in the energy function. The reduced model can treat systems in continuous space (described by continuous random variables), for which we formulate a variational learning problem using the adjoint method to determine the right-hand sides of the differential equations. This approach can be used to enforce a reduced physical model by a suitable parametrization of the differential equations. The parametrization we employ uses the basis functions from finite-element methods, which can be used to model any physical system. One application domain for such physics-informed learning algorithms is to modeling reaction-diffusion systems. We study a lattice version of the Rössler chaotic oscillator, which illustrates the accuracy of the moment closure approximation made by the method and its dimensionality reduction power.

Learning dynamic Boltzmann distributions as reduced models of spatial chemical kinetics.

Finding reduced models of spatially distributed chemical reaction networks requires an estimation of which effective dynamics are relevant. We propose a machine learning approach to this coarse graining problem, where a maximum entropy approximation is constructed that evolves slowly in time. The dynamical model governing the approximation is expressed as a functional, allowing a general treatment of spatial interactions. In contrast to typical machine learning approaches which estimate the interaction parameters of a graphical model, we derive Boltzmann-machine like learning algorithms to estimate directly the functionals dictating the time evolution of these parameters. By incorporating analytic solutions from simple reaction motifs, an efficient simulation method is demonstrated for systems ranging from toy problems to basic biologically relevant networks. The broadly applicable nature of our approach to learning spatial dynamics suggests promising applications to multiscale methods for spatial networks, as well as to further problems in machine learning.