A real neural network is a huge cascade of chemical rewrites. So I can try my chemlambda with that task. From a programming point of view, the problem is to understand neural networks as a graph rewrite model of computation, together with a (yet undiscovered) discipline of using them.
Further are some pretty images showing the first tries. They are all made by filming real simulations obtained with chemlambda.
Before giving them, I tell you that this task seems hard and now I believe that an easier one would be to use the ideas of chemlambda in the frame of quantum computing. (Do I have to add that in a new way, different from what was proposed in the many graphical formalisms associated to category theory? Probably! All those formalisms fall into the family: topological changes do not compute. Wait and see.)