Learning nonlinear integral operators via Recurrent Neural Networks and its application in solving Integro-Differential Equations

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.

Learning to predict synchronization of coupled oscillators on heterogeneous graphs

Suppose we are given a system of coupled oscillators on an arbitrary graph along with the trajectory of the system during some period. Can we predict whether the system will eventually synchronize? This is an important but analytically intractable question especially when the structure of the underlying graph is highly varied. In this work, we take an entirely different approach that we call "learning to predict synchronization" (L2PSync), by viewing it as a classification problem for sets of graphs paired with initial dynamics into two classes: `synchronizing' or `non-synchronizing'. Our conclusion is that, once trained on large enough datasets of synchronizing and non-synchronizing dynamics on heterogeneous sets of graphs, a number of binary classification algorithms can successfully predict the future of an unknown system with surprising accuracy. We also propose an "ensemble prediction" algorithm that scales up our method to large graphs by training on dynamics observed from multiple random subgraphs. We find that in many instances, the first few iterations of the dynamics are far more important than the static features of the graphs. We demonstrate our method on three models of continuous and discrete coupled oscillators -- The Kuramoto model, the Firefly Cellular Automata, and the Greenberg-Hastings model.