发表机构
School of Information and Software Engineering, UESTC; Electronics and Information Convergence Engineering, KHU; School of Computer Science and Engineering, UESTC; Department of Mathematical Sciences, UOL; School of Computer Science and Technology, XDU; School of Mechanical and Electrical Engineering, UESTC; College of Computer and Information Engineering, XAUAT（电子科技大学信息与软件工程学院; 韩国庆熙大学电子与信息融合工程系; 电子科技大学计算机科学与工程学院; 利物浦大学数学科学系; 西安电子科技大学计算机科学与技术学院; 电子科技大学机械与电气工程学院; 西安建筑科技大学计算机与信息工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出几何感知增量神经算子GeoIncNO，通过调节通道耦合、均值-波动解耦重构等机制，在6个PDE基准测试中实现了更优的长时序PDE预测精度与稳定性。
AI 中文摘要
神经算子在学习偏微分方程（PDE）的解算子方面展现出强大潜力，但长时序自回归预测仍具挑战性：局部误差会累积为频谱不一致、相位错位或均值漂移。现有方法主要改进状态表示和算子骨干，而反复应用的隐态转移增量结构薄弱，导致推演过程中频谱误差和不稳定通道耦合不断累积。为解决这些问题，本文提出几何感知增量神经算子（GeoIncNO）用于稳定的长时序PDE预测。GeoIncNO预测用于残差推进的隐态增量，并采用轻量级低秩投影器，根据增量频谱能量分布调节活跃频带内的通道耦合；为降低物理空间重构误差，GeoIncNO进一步引入均值-波动解耦的重构机制，分别融合稳定的均值结构与动态波动，并仅对零均值波动分量应用相位校正。在涵盖1D、2D和3D动力学系统的6个PDE基准测试上开展的大量实验表明，与竞争性神经算子基线相比，GeoIncNO始终展现出强劲的预测精度、提升的推演稳定性及更优的频谱保真度。
英文摘要
Neural operators have shown strong potential for learning solution operators of partial differential equations (PDEs). However, long-horizon autoregressive prediction remains challenging: local errors accumulate as spectral inconsistency, phase misalignment, or mean drift. Existing methods mainly improve state representations and operator backbones, while leaving the repeatedly applied latent transition increment weakly structured, allowing spectral errors and unstable channel couplings to accumulate during rollout. To address these issues, we propose a geometry-aware incremental neural operator (GeoIncNO) for stable long-horizon PDE prediction. GeoIncNO predicts latent increments for residual advancement and uses lightweight low-rank projectors to regulate channel coupling within active frequency bands derived from the increment spectral energy distribution. To reduce physical-space reconstruction errors, GeoIncNO further introduces a mean--fluctuation decoupled reconstruction mechanism, where stable mean structures and dynamic fluctuations are fused separately, and phase correction is applied only to the zero-mean fluctuation component. Extensive experiments on six PDE benchmarks, covering 1D, 2D, and 3D dynamical systems, show that GeoIncNO achieves consistently strong prediction accuracy, improved rollout stability, and better spectral fidelity compared with competitive neural-operator baselines.
