document: Optimal control using the transport equation : the Liouville machine Transport theory describes the scattering behavior of physical particles such as photons . Here we show how to connect this theory to optimal control theory and to adaptive behavior of agents embedded in an environment . Environments and tasks are defined by physical boundary conditions . Given some task , we compute a set of probability densities on continuous state and action and time . From these densities we derive an optimal policy such that for all states the most likely action maximizes the probability of reaching a predefined goal state . Liouville 's conservation theorem tells us that the conditional density at time t , state s , and action a must equal the density at t + dt , s + ds , a + da . Discretization yields a linear system that can be solved directly and whose solution corresponds to an optimal policy . Discounted reward schemes are incorporated naturally by taking the Laplace transform of the equations . The Liouville machine quickly solves rather complex maze problems
doc_bio_tags: B I O O B I O O B I O O O O B I O B I O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O
extractive_keyphrases: optimal control transport equation liouville machine scattering behavior physical particles adaptive behavior
abstractive_keyphrases: embedded agents