CoEvol-NO: State and Coordinate Co-Evolution with an Error-Driven Predictor-Corrector Paradigm for Neural Operator Transformer

Despite the fast progress in neural operator learning, long-sequence modeling still is a standing challenge whereby latent states have been introduced with techniques well derived. Diverging from existing methods that treat latent states as transient variables or decoupled representations, CoEvol-NO introduces a persistent state to establish a co-evolutionary framework, where the latent state and mesh sequence are updated jointly and bidirectionally. Inspired by classical numerical methods, we model the layer-wise state evolution as a Predictor-Corrector (PC) process. Specifically, a "Predictor'' generates a tentative target, followed by a "Corrector'' that refines the persistent state via an {error-driven update mechanism}. Furthermore, our theoretical analysis reveals that the widely used \textit{direct substitution} and \textit{residual update} paradigms are essentially {first-order approximations} of this error-driven correction under different loss assumptions. We theoretically prove that CoEvol-NO achieves strict linear time complexity. Extensive experiments on five standard benchmarks and two large-scale industrial design tasks demonstrate that CoEvol-NO consistently achieves state-of-the-art (SOTA) performance.

Despite rapid progress in using artificial intelligence to solve physics problems, accurately modeling long sequences of physical changes remains difficult. Existing approaches typically treat the working memory of the system as temporary or disconnected from the physical layout. We propose a new method that keeps a running record of the physical state and updates it together with the spatial structure in an integrated way. Drawing on classical numerical techniques, our approach works in two stages: it first generates a preliminary estimate of the physical state, then refines that estimate by directly responding to the error. Our analysis shows that two common update strategies—directly replacing the old state and adding a small adjustment—are actually special cases of this broader error-based refinement under different conditions. Tests on multiple standard problems and large engineering tasks show that our method consistently achieves better accuracy than current alternatives.