Differential-Integral Neural Operator for Long-Term Turbulence Forecasting

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 Differential-Integral Neural Operator (DINO), a novel framework designed from a first-principles approach of operator decomposition. DINO 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 DINO with exceptional stability and robustness. Through extensive experiments on the challenging 2D Kolmogorov flow benchmark, we demonstrate that DINO significantly outperforms state-of-the-art models, achieving a 70% reduction in relative error for long-term forecasting (99 steps). Unlike baselines that suffer from spectral decay, DINO successfully suppresses error accumulation over hundreds of timesteps and accurately reproduces the theoretical k-3 energy spectrum. These results establish DINO as a new benchmark for physically consistent, long-range turbulence forecasting. Our codes are available at https://github.com/Alexander-wu/DINO.