# Geometric fluid approximation for general continuous-time Markov chains

## TL;DR

This paper introduces a novel geometric fluid approximation method for continuous-time Markov chains that leverages spectral analysis and manifold learning to model macro-scale dynamics without requiring specific population structures.

## Contribution

The authors develop a general approach using diffusion maps and Gaussian process regression to approximate CTMC behavior in a continuous space, independent of population structure assumptions.

## Key findings

- Effective embedding of CTMC states into continuous space.
- Accurate approximation of CTMC mean evolution via derived ODE.
- Method applicable to a wide class of Markov systems.

## Abstract

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).