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文章背景与核心概要

偏微分方程（PDEs）的求解在科学计算和工程领域具有核心地位，而神经算子作为近年来备受瞩目的替代方法，展现出了巨大的潜力。然而，在进行长时序自回归预测时，神经算子往往会遭遇累积误差的困扰，具体表现为谱不一致、相位错位以及均值漂移等现象。

为了克服这些局限性，本文提出了几何感知增量神经算子（GeoIncNO）。该方法通过预测用于残差推进的潜在增量、利用轻量级低秩投影器调节通道耦合，并引入均值-涨落解耦重建机制，在1D、2D和3D动力学系统中实现了卓越的 rollout（滚动预测）稳定性、更高的谱保真度以及强大的预测准确性。

Geometry-aware Incremental Neural Operator for Long-Horizon PDE prediction

arXiv: 2608.11237 [cs.AI]
Submitted: 31 July 2026
Authors: Jiaquan Zhang, Shuxu Chen, Haifan Meng, Yi Lu, Zhihan Lyu, Fan Mo, Wei Dong, Yang Yang, Chaoning Zhang

Executive Summary

神经算子为学习偏微分方程（PDE）的解算子提供了一种极具前景的方法。然而，长时序自回归预测往往会遭受累积误差的困扰，这些误差表现为谱不一致、相位错位和均值漂移。

为了克服这些局限性，本文引入了几何感知增量神经算子（GeoIncNO）。通过预测用于残差推进的潜在增量、利用轻量级低秩投影器调节通道耦合，以及引入均值-涨落解耦重建机制，GeoIncNO在1D、2D和3D动力学系统中实现了卓越的滚动预测稳定性、更高的谱保真度和强大的预测准确性。

Neural operators offer a promising approach for learning solution operators of partial differential equations (PDEs). However, long-horizon autoregressive predictions often suffer from cumulative errors manifested as spectral inconsistency, phase misalignment, and mean drift.

To overcome these limitations, this paper introduces the Geometry-aware Incremental Neural Operator (GeoIncNO). By predicting latent increments for residual advancement, regulating channel coupling using lightweight low-rank projectors, and introducing a mean–fluctuation decoupled reconstruction mechanism, GeoIncNO achieves superior rollout stability, higher spectral fidelity, and strong prediction accuracy across 1D, 2D, and 3D dynamical systems.

Abstract

神经算子在学习偏微分方程（PDE）的解算子方面展现出强大的潜力。然而，长时序自回归预测依然极具挑战性：局部误差会累积表现为谱不一致、相位错位或均值漂移。现有方法主要集中于改进状态表示和算子骨干网络，而对被反复应用的潜在跃迁增量（latent transition increment）的结构设计较弱，从而导致谱误差和不稳定的通道耦合在滚动预测过程中不断累积。

为了解决这些问题，我们提出了一种用于稳定长时序PDE预测的几何感知增量神经算子（GeoIncNO）。GeoIncNO通过预测用于残差推进的潜在增量，并利用轻量级低秩投影器，在由增量谱能量分布导出的有效频带内调节通道耦合。为了减少物理空间的重建误差，GeoIncNO进一步引入了均值-涨落解耦重建机制，其中稳定的均值结构和动态涨落被分别融合，且相位校正仅应用于零均值的涨落分量。

在涵盖1D、2D和3D动力学系统的六个PDE基准测试上的广泛实验表明，与竞争性的神经算子基线相比，GeoIncNO始终能够实现出色的预测准确性、更高的滚动预测稳定性和更好的谱保真度。

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.

Key Features of GeoIncNO

  • 增量预测： 专注于预测用于残差推进的潜在增量，而不是原始的绝对状态，从而最大限度地减少局部误差传播。
  • 谱能量正则化： 利用轻量级低秩投影器，在通过增量谱能量分布确定的有效频带内管理通道耦合。
  • 解耦重建： 实现了均值-涨落解耦重建机制，独立处理稳定的均值结构和动态涨落，仅将相位校正应用于零均值的涨落分量。
  • Incremental Prediction: Focuses on predicting latent increments for residual advancement rather than raw absolute states to minimize local error propagation.
  • Spectral Energy Regularization: Utilizes lightweight low-rank projectors to manage channel coupling within active frequency bands determined via the increment spectral energy distribution.
  • Decoupled Reconstruction: Implements a mean–fluctuation decoupled reconstruction mechanism that processes stable mean structures and dynamic fluctuations independently, applying phase correction solely to the zero-mean fluctuation component.

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