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

Jianqiao Zeng, Ruocheng Wang, Yanzhi Liu, Hao Xiong, Junchi Yan
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:153614-153640, 2026.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-zeng26r, title = {{C}o{E}vol-{NO}: State and Coordinate Co-Evolution with an Error-Driven Predictor-Corrector Paradigm for Neural Operator Transformer}, author = {Zeng, Jianqiao and Wang, Ruocheng and Liu, Yanzhi and Xiong, Hao and Yan, Junchi}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {153614--153640}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/zeng26r/zeng26r.pdf}, url = {https://proceedings.mlr.press/v306/zeng26r.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T CoEvol-NO: State and Coordinate Co-Evolution with an Error-Driven Predictor-Corrector Paradigm for Neural Operator Transformer %A Jianqiao Zeng %A Ruocheng Wang %A Yanzhi Liu %A Hao Xiong %A Junchi Yan %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-zeng26r %I PMLR %P 153614--153640 %U https://proceedings.mlr.press/v306/zeng26r.html %V 306 %X 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.
APA
Zeng, J., Wang, R., Liu, Y., Xiong, H. & Yan, J.. (2026). CoEvol-NO: State and Coordinate Co-Evolution with an Error-Driven Predictor-Corrector Paradigm for Neural Operator Transformer. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:153614-153640 Available from https://proceedings.mlr.press/v306/zeng26r.html.

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