mlsanityData to 7 Oct 2026

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

arXiv 2310.09434 ↗First posted 13 Oct 2023cs.LG math.DS physics.comp-phcited by 5

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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.

Key facts. Cited by 5 papers in the map (data through 7 Oct 2026). Builds on 17 of its 44 references that are in the map. First posted on arXiv 13 Oct 2023.

Builds on

17 of 44 references.

References · 17

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5 more
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Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 2; 2025: 1; 2026: 2.

5 papers.

  1. 2024PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problemscited by 3
  2. 2024Scalable learning of potentials to predict time-dependent Hartree-Fock dynamicscited by 1
  3. 2025MINPO: Memory-Informed Neural Pseudo-Operator to Resolve Nonlocal Spatiotemporal Dynamicscited by 1
  4. 2026ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidancecited by 1
  5. 2026Capturing non-Markovian dynamics in non-equilibrium stochastic systems using flow matchingcited by 1