  Mathematical methods are developed to characterize the asymptotics of
recurrent neural networks (RNN) as the number of hidden units, data samples in
the sequence, hidden state updates, and training steps simultaneously grow to
infinity. In the case of an RNN with a simplified weight matrix, we prove the
convergence of the RNN to the solution of an infinite-dimensional ODE coupled
with the fixed point of a random algebraic equation. The analysis requires
addressing several challenges which are unique to RNNs. In typical mean-field
applications (e.g., feedforward neural networks), discrete updates are of
magnitude $\mathcal{O}(\frac{1}{N})$ and the number of updates is
$\mathcal{O}(N)$. Therefore, the system can be represented as an Euler
approximation of an appropriate ODE/PDE, which it will converge to as $N
\rightarrow \infty$. However, the RNN hidden layer updates are
$\mathcal{O}(1)$. Therefore, RNNs cannot be represented as a discretization of
an ODE/PDE and standard mean-field techniques cannot be applied. Instead, we
develop a fixed point analysis for the evolution of the RNN memory states, with
convergence estimates in terms of the number of update steps and the number of
hidden units. The RNN hidden layer is studied as a function in a Sobolev space,
whose evolution is governed by the data sequence (a Markov chain), the
parameter updates, and its dependence on the RNN hidden layer at the previous
time step. Due to the strong correlation between updates, a Poisson equation
must be used to bound the fluctuations of the RNN around its limit equation.
These mathematical methods give rise to the neural tangent kernel (NTK) limits
for RNNs trained on data sequences as the number of data samples and size of
the neural network grow to infinity.
