Sparsity in long-time control of neural ODEs

We consider the neural ODE and optimal control perspective of supervised learning, with ℓ1-control penalties, where rather than only minimizing a final cost (the empirical risk) for state, we integrate this over entire time horizon. prove that any (for cost) vanishes beyond some positive stopping time. When seen in discrete-time context, result entails an ordered sparsity pattern parameters associated residual network: sense these are all 0 certain layer. Furthermore, provide polynomial stability estimate risk respect to This can be as turnpike property, nonsmooth dynamics functionals ℓ1 without smallness assumptions on data, both which new literature.