Dynamic Network Functional Comparison via Approximate-bisimulation

It is generally unknown how to formally determine whether different neural networks have a similar behaviour. This question intimately relates to the problem of finding a suitable similarity measure to identify bounds on the input-output response distances of neural networks, which has several interesting theoretical and computational implications. For example, it can allow one to speed up the learning processes by restricting the network parameter space, or to test the robustness of a network with respect to parameter variation. In this paper we develop a procedure that allows for comparing neural structures among them. In particular, we consider dynamic networks composed of neural units, characterised by non-linear differential equations, described in terms of autonomous continuous dynamic systems. The comparison is established by importing and adapting from the formal verification setting the concept of ?-approximate bisimulations techniques for non-linear systems. We have positively tested the proposed approach over continuous time recurrent neural networks (CTRNNs).