  Many important phenomena in chemistry and biology are realized via dynamical
features such as multi-stability, oscillations, and chaos. Construction of
novel chemical systems with such finely-tuned dynamics is a challenging problem
central to the growing field of synthetic biology. In this paper, we address
this problem by putting forward a molecular version of a recurrent artificial
neural network, which we call a recurrent neural chemical reaction network
(RNCRN). We prove that the RNCRN, with sufficiently many auxiliary chemical
species and suitable fast reactions, can be systematically trained to achieve
any dynamics. This approximation ability is shown to hold independent of the
initial conditions for the auxiliary species, making the RNCRN more
experimentally feasible. To demonstrate the results, we present a number of
relatively simple RNCRNs trained to display a variety of biologically-important
dynamical features.
