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 direct substitution and 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.