  Fluid approximations have seen great success in approximating the macro-scale
behaviour of Markov systems with a large number of discrete states. However,
these methods rely on the continuous-time Markov chain (CTMC) having a
particular population structure which suggests a natural continuous state-space
endowed with a dynamics for the approximating process. We construct here a
general method based on spectral analysis of the transition matrix of the CTMC,
without the need for a population structure. Specifically, we use the popular
manifold learning method of diffusion maps to analyse the transition matrix as
the operator of a hidden continuous process. An embedding of states in a
continuous space is recovered, and the space is endowed with a drift vector
field inferred via Gaussian process regression. In this manner, we construct an
ODE whose solution approximates the evolution of the CTMC mean, mapped onto the
continuous space (known as the fluid limit).
