Moment Closure - A Brief Review

TL;DR

Moment closure methods are widely used in modeling complex systems to simplify large sets of equations by approximating higher-order moments, with ongoing challenges in rigorous justification despite practical success.

Contribution

This review highlights the diverse applications of moment closure methods and offers a geometric perspective on the difficulties in their rigorous validation.

Findings

Moment closure methods are prevalent across scientific disciplines.

A geometric explanation sheds light on the justification challenges.

Despite theoretical difficulties, moment closure works well in practice.

Abstract

Moment closure methods appear in myriad scientific disciplines in the modelling of complex systems. The goal is to achieve a closed form of a large, usually even infinite, set of coupled differential (or difference) equations. Each equation describes the evolution of one "moment", a suitable coarse-grained quantity computable from the full state space. If the system is too large for analytical and/or numerical methods, then one aims to reduce it by finding a moment closure relation expressing "higher-order moments" in terms of "lower-order moments". In this brief review, we focus on highlighting how moment closure methods occur in different contexts. We also conjecture via a geometric explanation why it has been difficult to rigorously justify many moment closure approximations although they work very well in practice.