Realization Theory of Recurrent Neural ODEs using Polynomial System Embeddings

In this paper we show that neural ODE analogs of recurrent (ODE-RNN) and Long Short-Term Memory (ODE-LSTM) networks can be algorithmically embedded into a class of polynomial systems. This embedding preserves input–output behavior and can suitably be extended to other neural differential equation (neural DE) architectures. We then use realization theory of polynomial systems to provide necessary conditions for an input–output map to be realizable by an ODE-LSTM and sufficient conditions for minimality of such systems. These results represent the first steps towards realization theory of recurrent neural ODE architectures, which is expected to be useful for model reduction and learning algorithm analysis of recurrent neural ODEs.