# Differential-Integral Neural Operator for Long-Term Turbulence Forecasting

arXiv: https://arxiv.org/abs/2509.21196 · HTML: https://mlsanity.com/paper/2509.21196/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 57 references that are in the map.
- First posted on arXiv 25 Sep 2025.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Measuring and Relieving the Over-smoothing Problem for Graph Neural Networks from the Topological View](https://mlsanity.com/paper/1909.03211/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [SimVP: Simpler yet Better Video Prediction](https://mlsanity.com/paper/2206.05099/) (2022)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [Revisiting Over-smoothing in Deep GCNs](https://mlsanity.com/paper/2003.13663/) (2020)
- [PastNet: Introducing Physical Inductive Biases for Spatio-temporal Video Prediction](https://mlsanity.com/paper/2305.11421/) (2023)
- [OneForecast: A Universal Framework for Global and Regional Weather Forecasting](https://mlsanity.com/paper/2502.00338/) (2025)
- [Physically Constrained Generative Adversarial Networks for Improving Precipitation Fields from Earth System Models](https://mlsanity.com/paper/2209.07568/) (2022)
- [NeuralOM: Neural Ocean Model for Subseasonal-to-Seasonal Simulation](https://mlsanity.com/paper/2505.21020/) (2025)
- [Breaking the Discretization Barrier of Continuous Physics Simulation Learning](https://mlsanity.com/paper/2509.17955/) (2025)

## Built on by (most cited first, 4 of 4)

- [PnP-Corrector: A Universal Correction Framework for Coupled Spatiotemporal Forecasting](https://mlsanity.com/paper/2605.08935/) (2026)
- [TIDE: A Physically Diverse 3D Turbulence Benchmark Dataset for Advancing Scientific Machine Learning](https://mlsanity.com/paper/2608.04222/) (2026)
- [Geometry-aware Incremental Neural Operator for Long-Horizon PDE prediction](https://mlsanity.com/paper/2608.11237/) (2026)
- [Dynamics-Aware Weighting for Deep Learning Forecasts of Chaotic Systems](https://mlsanity.com/paper/2608.22277/) (2026)

## Abstract (arXiv)

Accurately forecasting the long-term evolution of turbulence represents a grand challenge in scientific computing and is crucial for applications ranging from climate modeling to aerospace engineering. Existing deep learning methods, particularly neural operators, often fail in long-term autoregressive predictions, suffering from catastrophic error accumulation and a loss of physical fidelity. This failure stems from their inability to simultaneously capture the distinct mathematical structures that govern turbulent dynamics: local, dissipative effects and global, non-local interactions. In this paper, we propose the textbfunderlineDifferential-textbfunderlineIntegral textbfunderlineNeural textbfunderlineOperator (method), a novel framework designed from a first-principles approach of operator decomposition. method explicitly models the turbulent evolution through parallel branches that learn distinct physical operators: a local differential operator, realized by a constrained convolutional network that provably converges to a derivative, and a global integral operator, captured by a Transformer architecture that learns a data-driven global kernel. This physics-based decomposition endows method with exceptional stability and robustness. Through extensive experiments on the challenging 2D Kolmogorov flow benchmark, we demonstrate that method significantly outperforms state-of-the-art models in long-term forecasting. It successfully suppresses error accumulation over hundreds of timesteps, maintains high fidelity in both the vorticity fields and energy spectra, and establishes a new benchmark for physically consistent, long-range turbulence forecast.