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

The applicability of neural operators remains limited by the challenges of handling complex physical conditions in real-world settings. We argue that this limitation stems from the inherent trade-off between geometric sensitivity to complex boundaries and the dynamic memory required for long-term physical evolution. Instead of directly addressing these ad-hoc constraints, we propose that modeling the co-evolution of latent states and mesh sequences offers a more fundamental solution: latent states accumulate physical features across layers, while mesh sequences provide real-time geometric feedback. To this end, we propose CoEvol-NO, a co-evolutionary framework where the latent state and mesh sequence are updated jointly . To enhance the capacity of latent state evolution, we introduce the classical Predictor-Corrector (PC) paradigm, formulating the layer-wise evolution as a meta-learning process with an implicit objective optimized during the forward pass: the Predictor generates a tentative target, while the error-driven Corrector refines the persistent state via gradient-based optimization. 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 also prove that CoEvol-NO achieves strict linear time complexity. Extensive experiments across five standard benchmarks and two large-scale industrial tasks demonstrate that CoEvol-NO achieves state-ofthe-art (SOTA) performance.