Discrete Processes and their Continuous Limits

TL;DR

This paper explores how discrete processes can be approximated by continuous differential systems, highlighting benefits, limitations, and case studies across various fields.

Contribution

It provides a detailed analysis of multiple case studies illustrating successful and cautious applications of continuous limits to discrete processes.

Findings

Continuous limits can offer computational efficiency and theoretical insights.

Discrete processes may have richer dynamics than their continuous approximations.

Continuous limits can provide qualitative, but not always quantitative, understanding.

Abstract

The possibility that a discrete process can be fruitfully approximated by a continuous one, with the latter involving a differential system, is fascinating. Important theoretical insights, as well as significant computational efficiency gains may lie in store. A great success story in this regard are the Navier-Stokes equations, which model many phenomena in fluid flow rather well. Recent years saw many attempts to formulate more such continuous limits, and thus harvest theoretical and practical advantages, in diverse areas including mathematical biology, image processing, game theory, computational optimization, and machine learning. Caution must be applied as well, however. In fact, it is often the case that the given discrete process is richer in possibilities than its continuous differential system limit, and that a further study of the discrete process is practically rewarding. Furthermore, there are situations where the continuous limit process may provide important qualitative, but not quantitative, information about the actual discrete process. This paper considers several case studies of such continuous limits and demonstrates success as well as cause for caution. Consequences are discussed.