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

arXiv: https://arxiv.org/abs/2310.09434 · HTML: https://mlsanity.com/paper/2310.09434/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 44 references that are in the map.
- First posted on arXiv 13 Oct 2023.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [LSTM Fully Convolutional Networks for Time Series Classification](https://mlsanity.com/paper/1709.05206/) (2017)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise](https://mlsanity.com/paper/1906.02355/) (2019)
- [Infinitely Deep Bayesian Neural Networks with Stochastic Differential Equations](https://mlsanity.com/paper/2102.06559/) (2021)
- [SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs](https://mlsanity.com/paper/2102.13037/) (2021)
- [Machine Learning for Prediction with Missing Dynamics](https://mlsanity.com/paper/1910.05861/) (2019)
- [Multilingual Sentiment Analysis: An RNN-Based Framework for Limited Data](https://mlsanity.com/paper/1806.04511/) (2018)
- [Learning Stochastic Dynamics with Statistics-Informed Neural Network](https://mlsanity.com/paper/2202.12278/) (2022)
- [Analyzing and predicting non-equilibrium many-body dynamics via dynamic mode decomposition](https://mlsanity.com/paper/2107.09635/) (2021)

## Built on by (most cited first, 5 of 5)

- [PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems](https://mlsanity.com/paper/2409.01899/) (2024)
- [ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidance](https://mlsanity.com/paper/2601.20227/) (2026)
- [Scalable learning of potentials to predict time-dependent Hartree-Fock dynamics](https://mlsanity.com/paper/2408.04765/) (2024)
- [MINPO: Memory-Informed Neural Pseudo-Operator to Resolve Nonlocal Spatiotemporal Dynamics](https://mlsanity.com/paper/2512.17273/) (2025)
- [Capturing non-Markovian dynamics in non-equilibrium stochastic systems using flow matching](https://mlsanity.com/paper/2606.06658/) (2026)

## Abstract (arXiv)

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(nT) from O(nT2) if a nT-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.