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On reaction network implementations of neural networks.

Authors
  • Anderson, David F1
  • Joshi, Badal2
  • Deshpande, Abhishek1
  • 1 Department of Mathematics, University of Wisconsin-Madison, Madison, WI, USA.
  • 2 Department of Mathematics, California State University San Marcos, San Marcos, CA, USA.
Type
Published Article
Journal
Journal of The Royal Society Interface
Publisher
The Royal Society
Publication Date
Apr 01, 2021
Volume
18
Issue
177
Pages
20210031–20210031
Identifiers
DOI: 10.1098/rsif.2021.0031
PMID: 33849332
Source
Medline
Keywords
Language
English
License
Unknown

Abstract

This paper is concerned with the utilization of deterministically modelled chemical reaction networks for the implementation of (feed-forward) neural networks. We develop a general mathematical framework and prove that the ordinary differential equations (ODEs) associated with certain reaction network implementations of neural networks have desirable properties including (i) existence of unique positive fixed points that are smooth in the parameters of the model (necessary for gradient descent) and (ii) fast convergence to the fixed point regardless of initial condition (necessary for efficient implementation). We do so by first making a connection between neural networks and fixed points for systems of ODEs, and then by constructing reaction networks with the correct associated set of ODEs. We demonstrate the theory by constructing a reaction network that implements a neural network with a smoothed ReLU activation function, though we also demonstrate how to generalize the construction to allow for other activation functions (each with the desirable properties listed previously). As there are multiple types of 'networks' used in this paper, we also give a careful introduction to both reaction networks and neural networks, in order to disambiguate the overlapping vocabulary in the two settings and to clearly highlight the role of each network's properties.

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