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Recurrent neural networks (RNN) as used in machine learning are commonlyformulated in discrete time, i.e. as recursive maps. This brings a lot ofadvantages for training models on data, e.g. for the purpose of time seriesprediction or dynamical systems identification, as powerful and efficientinference algorithms exist for discrete time systems and numerical integrationof differential equations is not necessary. On the other hand, mathematicalanalysis of dynamical systems inferred from data is often more convenient andenables additional insights if these are formulated in continuous time, i.e. assystems of ordinary (or partial) differential equations (ODE). Here we show howto perform such a translation from discrete to continuous time for a particularclass of ReLU-based RNN. We prove three theorems on the mathematicalequivalence between the discrete and continuous time formulations under avariety of conditions, and illustrate how to use our mathematical results ondifferent machine learning and nonlinear dynamical systems examples.

Transformation of ReLU-based recurrent neural networks from discrete-time to continuous-time