Computer Validation of Neural Network Dynamics: A First Case Study

A large number of current machine learning methods rely upon deep neural networks. Yet, viewing neural networks as nonlinear dynamical systems, it becomes quickly apparent that mathematically rigorously establishing certain patterns generated by the nodes in the network is extremely difficult. Indeed, it is well-understood in the nonlinear dynamics of complex systems that, even in low-dimensional models, analytical techniques rooted in pencil-and-paper approaches reach their limits quickly. In this work, we propose a completely different perspective via the paradigm of rigorous numerical methods of nonlinear dynamics. The idea is to use computer-assisted proofs to validate mathematically the existence of nonlinear patterns in neural networks. As a case study, we consider a class of recurrent neural networks, where we prove via computer assistance the existence of several hundred Hopf bifurcation points, their non-degeneracy, and hence also the existence of several hundred periodic orbits. Our paradigm has the capability to rigorously verify complex nonlinear behaviour of neural networks, which provides a first step to explain the full abilities, as well as potential sensitivities, of machine learning methods via computer-assisted proofs