  This paper is concerned with the utilization of deterministically modeled
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" utilized 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.
