A recurrent challenge in biology is the development of predictive quantitative models because most molecular and cellular parameters have unknown values and realistic models are analytically intractable. While the dynamics of the system can be analyzed via computer simulations, substantial computational resources are often required given uncertain parameter values resulting in large numbers of parameter combinations, especially when realistic biological features are included. Simulation alone also often does not yield the kinds of intuitive insights from analytical solutions. Here we introduce a general framework combining stochastic/mechanistic simulation of reaction systems and machine learning of the simulation data to generate computationally efficient predictive models and interpretable parameter-phenotype maps. We applied our approach to investigate stochastic gene expression propagation in biological networks, which is a contemporary challenge in the quantitative modeling of single-cell heterogeneity. We found that accurate, predictive machine-learning models of stochastic simulation results can be constructed. Even in the simplest networks existing analytical schemes generated significantly less accurate predictions than our approach, which revealed interesting insights when applied to more complex circuits, including the extensive tunability of information propagation enabled by feedforward circuits and how even single negative feedbacks can utilize stochastic fluctuations to generate robust oscillations. Our approach is applicable beyond biology and opens up a new avenue for exploring complex dynamical systems.