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 aCorrector'' 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.