Linking Microscopic and Macroscopic Models for Evolution: Markov Chain Network Training and Conservation Law Approximations

TL;DR

This paper introduces a framework connecting neural network training dynamics modeled by Markov chains with conservation law equations, enabling efficient numerical approximations and analysis of stability and consistency.

Contribution

It establishes a novel link between microscopic Markov chain models of neural networks and macroscopic conservation laws, providing a unified analysis approach.

Findings

Markov chain models can approximate conservation laws effectively

Numerical methods derived from network models are computationally efficient

Stability and consistency conditions for these models are discussed

Abstract

In this paper, a general framework for the analysis of a connection between the training of artificial neural networks via the dynamics of Markov chains and the approximation of conservation law equations is proposed. This framework allows us to demonstrate an intrinsic link between microscopic and macroscopic models for evolution via the concept of perturbed generalized dynamic systems. The main result is exemplified with a number of illustrative examples where efficient numerical approximations follow directly from network-based computational models, viewed here as Markov chain approximations. Finally, stability and consistency conditions of such computational models are discussed.