# Recurrent Neural Operators: Stable Long-Term PDE Prediction

arXiv: https://arxiv.org/abs/2505.20721 · HTML: https://mlsanity.com/paper/2505.20721/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 29 references that are in the map.
- First posted on arXiv 27 May 2025.

## Builds on (most cited first, 24 of 24)

- [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)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022)
- [Towards Stability of Autoregressive Neural Operators](https://mlsanity.com/paper/2306.10619/) (2023)
- [Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations](https://mlsanity.com/paper/1804.07010/) (2018)
- [LNO: Laplace Neural Operator for Solving Differential Equations](https://mlsanity.com/paper/2303.10528/) (2023)
- [DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation](https://mlsanity.com/paper/2012.02681/) (2020)
- [Mitigating spectral bias for the multiscale operator learning](https://mlsanity.com/paper/2210.10890/) (2022)
- [MgNO: Efficient Parameterization of Linear Operators via Multigrid](https://mlsanity.com/paper/2310.19809/) (2023)
- [Neural Operator Learning for Long-Time Integration in Dynamical Systems with Recurrent Neural Networks](https://mlsanity.com/paper/2303.02243/) (2023)
- [Transfer Learning Enhanced DeepONet for Long-Time Prediction of Evolution Equations](https://mlsanity.com/paper/2212.04663/) (2022)
- [State-space models are accurate and efficient neural operators for dynamical systems](https://mlsanity.com/paper/2409.03231/) (2024)
- [Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Laws](https://mlsanity.com/paper/2401.01783/) (2024)
- [FBSJNN: A Theoretically Interpretable and Efficiently Deep Learning method for Solving Partial Integro-Differential Equations](https://mlsanity.com/paper/2412.11010/) (2024)
- [A MgNO Method for Multiphase Flow in Porous Media](https://mlsanity.com/paper/2407.02505/) (2024)
- [Finite Operator Learning: Bridging Neural Operators and Numerical Methods for Efficient Parametric Solution and Optimization of PDEs](https://mlsanity.com/paper/2407.04157/) (2024)

## Built on by (most cited first, 16 of 16)

- [Uni-Flow: a unified autoregressive-diffusion model for complex multiscale flows](https://mlsanity.com/paper/2602.15592/) (2026)
- [Latent Generative Solvers for Generalizable Long-Term Physics Simulation](https://mlsanity.com/paper/2602.11229/) (2026)
- [Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling](https://mlsanity.com/paper/2510.00233/) (2025)
- [Geometric Neural Operators via Lie Group-Constrained Latent Dynamics](https://mlsanity.com/paper/2602.16209/) (2026)
- [Autoregression-Free Neural Operators for Time-Dependent PDEs](https://mlsanity.com/paper/2605.25413/) (2026)
- [Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators](https://mlsanity.com/paper/2605.10154/) (2026)
- [HERO: History-Enriched Rollout Training for Long-Horizon Autoregressive Neural Operators](https://mlsanity.com/paper/2607.29135/) (2026)
- [Learning Heat-based Equations in Self-similar variables](https://mlsanity.com/paper/2602.00872/) (2026)
- [Adaptive recurrent flow map operator learning for reaction diffusion dynamics](https://mlsanity.com/paper/2602.09487/) (2026)
- [Scale-Aware Learning of Chaotic Dynamics on Unstructured Meshes via Binned Spectral Losses](https://mlsanity.com/paper/2607.19387/) (2026)
- [No Free Lunch in Flow Surrogates under Time-Varying Boundary Conditions: A Two-Regime Study](https://mlsanity.com/paper/2607.23667/) (2026)
- [One-Step Evolution for Long-Time Extrapolation: An Error-Bound-Informed and Prior-Guided Neural Residual Framework for Autonomous PDEs](https://mlsanity.com/paper/2608.22026/) (2026)
- [Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2608.23546/) (2026)
- [Large Distant Gradients Need Not Be Reliable: reliability-weighted credit assignment for long-horizon autoregressive forecasting](https://mlsanity.com/paper/2609.12890/) (2026)
- [Staying on the Attractor: Supervising Neural Surrogates of 3D Turbulence Where They Leave It](https://mlsanity.com/paper/2609.32864/) (2026)
- [Physics is the Best Teacher: Consistency Learning for Time-Invariant Operators of Chaotic Dynamics](https://mlsanity.com/paper/2610.04108/) (2026)

## Abstract (arXiv)

Neural operators have emerged as powerful tools for learning solution operators of partial differential equations. However, in time-dependent problems, standard training strategies such as teacher forcing introduce a mismatch between training and inference, leading to compounding errors in long-term autoregressive predictions. To address this issue, we propose Recurrent Neural Operators (RNOs)-a novel framework that integrates recurrent training into neural operator architectures. Instead of conditioning each training step on ground-truth inputs, RNOs recursively apply the operator to their own predictions over a temporal window, effectively simulating inference-time dynamics during training. This alignment mitigates exposure bias and enhances robustness to error accumulation. Theoretically, we show that recurrent training can reduce the worst-case exponential error growth typical of teacher forcing to linear growth. Empirically, we demonstrate that recurrently trained Multigrid Neural Operators significantly outperform their teacher-forced counterparts in long-term accuracy and stability on standard benchmarks. Our results underscore the importance of aligning training with inference dynamics for robust temporal generalization in neural operator learning.