mlsanityData to 7 Oct 2026

Neural Operator Learning for Long-Time Integration in Dynamical Systems with Recurrent Neural Networks

arXiv 2303.02243 ↗First posted 3 Mar 2023cs.LGcited by 32

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Abstract (arXiv)

Deep neural networks are an attractive alternative for simulating complex dynamical systems, as in comparison to traditional scientific computing methods, they offer reduced computational costs during inference and can be trained directly from observational data. Existing methods, however, cannot extrapolate accurately and are prone to error accumulation in long-time integration. Herein, we address this issue by combining neural operators with recurrent neural networks, learning the operator mapping, while offering a recurrent structure to capture temporal dependencies. The integrated framework is shown to stabilize the solution and reduce error accumulation for both interpolation and extrapolation of the Korteweg-de Vries equation.

Key facts. Cited by 32 papers in the map (data through 7 Oct 2026). Builds on 15 of its 25 references that are in the map. First posted on arXiv 3 Mar 2023.

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15 of 25 references.

References · 15

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 748
  4. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 292
  5. 2019An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applicationscited by 118
  6. 2021A physics-informed variational DeepONet for predicting the crack path in brittle materialscited by 98
  7. 2022Learning Operators with Coupled Attentioncited by 88
  8. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 87
  9. 2022Deep transfer operator learning for partial differential equations under conditional shiftcited by 83
  10. 2022Learning two-phase microstructure evolution using neural operators and autoencoder architecturescited by 72
  11. 2022Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modelingcited by 65
  12. 2022Reliable extrapolation of deep neural operators informed by physics or sparse observationscited by 62
3 more
  1. 2023Learning stiff chemical kinetics using extended deep neural operatorscited by 30
  2. 2022Neural operator learning of heterogeneous mechanobiological insults contributing to aortic aneurysmscited by 14
  3. 2023Learning bias corrections for climate models using deep neural operatorscited by 9

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 3; 2024: 7; 2025: 16; 2026: 6.

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  1. 2024DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Trainingcited by 137
  2. 2024State-space models are accurate and efficient neural operators for dynamical systemscited by 27
  3. 2024Basis-to-Basis Operator Learning Using Function Encoderscited by 17
  4. 2025Recurrent Neural Operators: Stable Long-Term PDE Predictioncited by 15
  5. 2025TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolationcited by 14
  6. 2025Binned Spectral Power Loss for Improved Prediction of Chaotic Systemscited by 10
  7. 2025Mixture-of-Experts Operator Transformer for Large-Scale PDE Pre-Trainingcited by 10
  8. 2024Neural Operators for Predictor Feedback Control of Nonlinear Delay Systemscited by 9
  9. 2025Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEscited by 9
  10. 2023Understanding and Mitigating Extrapolation Failures in Physics-Informed Neural Networkscited by 8
  11. 2025Time Marching Neural Operator FE Coupling: AI Accelerated Physics Modelingcited by 8
  12. 2025Temporal Neural Operator for Modeling Time-Dependent Physical Phenomenacited by 8
20 more
  1. 2025PIORF: Physics-Informed Ollivier-Ricci Flow for Long-Range Interactions in Mesh Graph Neural Networkscited by 5
  2. 2025Physics-Informed Time-Integrated DeepONet: Temporal Tangent Space Operator Learning for High-Accuracy Inferencecited by 5
  3. 2025The Seismic Wavefield Common Task Frameworkcited by 5
  4. 2024Symplectic Neural Flows for Modeling and Discoverycited by 4
  5. 2023Evaluation of Deep Neural Operator Models toward Ocean Forecastingcited by 3
  6. 2024Towards Efficient Modelling of String Dynamics: A Comparison of State Space and Koopman based Deep Learning Methodscited by 3
  7. 2026A Score Filter Enhanced Data Assimilation Framework for Data-Driven Dynamical Systemscited by 3
  8. 2026Geometric Neural Operators via Lie Group-Constrained Latent Dynamicscited by 2
  9. 2023B-LSTM-MIONet: Bayesian LSTM-based Neural Operators for Learning the Response of Complex Dynamical Systems to Length-Variant Multiple Input Functionscited by 1
  10. 2024A Novel Neural Filter to Improve Accuracy of Neural Network Models of Dynamic Systemscited by 1
  11. 2025Physics-informed Multiple-Input Operators for efficient dynamic response prediction of structurescited by 1
  12. 2025PMNO: A novel physics guided multi-step neural operator predictor for partial differential equationscited by 1
  13. 2025Application of Reduced-Order Models for Temporal Multiscale Representations in the Prediction of Dynamical Systemscited by 1
  14. 2025Assumed Density Filtering and Smoothing with Neural Network Surrogate Modelscited by 1
  15. 2025A Neural Operator based on Dynamic Mode Decompositioncited by 0
  16. 2025ANCHOR: Error-Controlled Adaptive Numerical Correction for Neural Operator Time Marchingcited by 0
  17. 2026Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEscited by 0
  18. 2026HAMNO: A Hierarchical Adaptive Multi-scale Neural Operator with Physics-Informed Learning for Dynamical Systemscited by 0
  19. 2026SpectONet: A Physics-Guided Spectral Deep Operator Network for Euler-Bernoulli Beam Dynamicscited by 0
  20. 2026LSR-Net: Learning the Forward Evolution Operator for Nonlinear Fluid Dynamicscited by 0