The Mori-Zwanzig formulation of deep learning

TL;DR

This paper introduces a novel deep learning framework based on the Mori-Zwanzig formalism, enabling the analysis and reduction of neural networks through operator equations and memory decay properties.

Contribution

It develops a new operator-theoretic formulation of deep learning using Mori-Zwanzig, highlighting neural network memory and providing conditions for network simplification.

Findings

Derived exact linear operator equations for neural network propagation.

Identified conditions under which neural network memory decays with depth.

Provided methods to transform deep networks into shallow ones using memory decay.

Abstract

We develop a new formulation of deep learning based on the Mori-Zwanzig (MZ) formalism of irreversible statistical mechanics. The new formulation is built upon the well-known duality between deep neural networks and discrete dynamical systems, and it allows us to directly propagate quantities of interest (conditional expectations and probability density functions) forward and backward through the network by means of exact linear operator equations. Such new equations can be used as a starting point to develop new effective parameterizations of deep neural networks, and provide a new framework to study deep-learning via operator theoretic methods. The proposed MZ formulation of deep learning naturally introduces a new concept, i.e., the memory of the neural network, which plays a fundamental role in low-dimensional modeling and parameterization. By using the theory of contraction mappings, we develop sufficient conditions for the memory of the neural network to decay with the number of layers. This allows us to rigorously transform deep networks into shallow ones, e.g., by reducing the number of neurons per layer (using projection operators), or by reducing the total number of layers (using the decay property of the memory operator).