Learning nonlinear integral operators via Recurrent Neural Networks and its application in solving Integro-Differential Equations
This provides a faster solver for IDEs in fields like quantum physics, but it is incremental as it applies existing LSTM methods to a specific computational bottleneck.
The paper tackles solving nonlinear integro-differential equations (IDEs) by using LSTM-RNNs to learn integral operators, converting IDEs into ordinary differential equations for efficient solving, and reduces temporal cost from O(n_T^2) to O(n_T) for n_T-step trajectories.
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.