  Continuous-time Markov chains are used to model stochastic systems where
transitions can occur at irregular times, e.g., birth-death processes, chemical
reaction networks, population dynamics, and gene regulatory networks. We
develop a method to learn a continuous-time Markov chain's transition rate
functions from fully observed time series. In contrast with existing methods,
our method allows for transition rates to depend nonlinearly on both state
variables and external covariates. The Gillespie algorithm is used to generate
trajectories of stochastic systems where propensity functions (reaction rates)
are known. Our method can be viewed as the inverse: given trajectories of a
stochastic reaction network, we generate estimates of the propensity functions.
While previous methods used linear or log-linear methods to link transition
rates to covariates, we use neural networks, increasing the capacity and
potential accuracy of learned models. In the chemical context, this enables the
method to learn propensity functions from non-mass-action kinetics. We test our
method with synthetic data generated from a variety of systems with known
transition rates. We show that our method learns these transition rates with
considerably more accuracy than log-linear methods, in terms of mean absolute
error between ground truth and predicted transition rates. We also demonstrate
an application of our methods to open-loop control of a continuous-time Markov
chain.
