Realization Theory Of Recurrent Neural ODEs Using Polynomial System Embeddings
This work provides foundational steps towards realization theory for recurrent neural ODEs, which could be useful for model reduction and learning algorithm analysis in this domain.
The paper tackles the problem of embedding recurrent neural ODE architectures, such as ODE-RNN and ODE-LSTM, into polynomial systems while preserving input-output behavior, and uses realization theory to provide necessary conditions for realizability and sufficient conditions for minimality.
In this paper we show that neural ODE analogs of recurrent (ODE-RNN) and Long Short-Term Memory (ODE-LSTM) networks can be algorithmically embeddeded into the class of polynomial systems. This embedding preserves input-output behavior and can suitably be extended to other 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 is expected be useful for model reduction and learning algorithm analysis of recurrent neural ODEs.