  In this paper, we propose using LSTM-RNNs (Long Short-Term Memory-Recurrent
Neural Networks) to learn and represent nonlinear integral operators that
appear in nonlinear integro-differential equations (IDEs). The LSTM-RNN
representation of the nonlinear integral operator allows us to turn a system of
nonlinear integro-differential equations into a system of ordinary differential
equations for which many efficient solvers are available. Furthermore, because
the use of LSTM-RNN representation of the nonlinear integral operator in an IDE
eliminates the need to perform a numerical integration in each numerical time
evolution step, the overall temporal cost of the LSTM-RNN-based IDE solver can
be reduced to $O(n_T)$ from $O(n_T^2)$ if a $n_T$-step trajectory is to be
computed. We illustrate the efficiency and robustness of this LSTM-RNN-based
numerical IDE solver with a model problem. Additionally, we highlight the
generalizability of the learned integral operator by applying it to IDEs driven
by different external forces. As a practical application, we show how this
methodology can effectively solve the Dyson's equation for quantum many-body
systems.
