绿色路由神经算子：物理决定网络读取位置
Green-Routed Neural Operators:\\Physics Determines Where the Network Reads
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- The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
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中文总结 AI 辅助
针对偏微分方程中输运场在神经算子中仅作输入而物理上应决定读取位置的不匹配，提出绿色路由神经算子（GRNO），利用控制方程构造读取坐标，在五个PDE系统上多数取得最低误差，验证了方程指定读取坐标作为长时程预测结构先验的有效性。
中文摘要 AI 辅助
我们发现许多偏微分方程中输运场的物理作用与其在神经算子中的通常作用之间存在不匹配：偏微分方程利用输运场来选择读取坐标，而神经算子通常仅将其视为输入值。我们通过绿色路由神经算子（GRNO）解决这一不匹配问题，该算子利用控制方程来确定潜在特征在何处被采样。一个无参数方程适配器评估诊断关系并构建一个出发图，其值为读取坐标。一个多尺度编码器-解码器结合潜在特征的中心读取和路由读取，以学习完整的有限时间更新。在五个二维和三维偏微分方程系统中，GRNO在少于40步的自回归评估中，在四个系统上取得了最低的平均最终相对$L^2$误差，并在Keller-Segel系统上保持竞争力。固定权重路由干预显示，在四个系统中对方向和空间对齐有强烈依赖，而在Keller-Segel系统中依赖较弱。在独立训练的消融实验中，与仅将输运场作为输入特征提供的变体、用学习到的位移替代方程指定路由的变体、或应用空间错位路由的变体相比，GRNO在所有五个系统中均实现了更低的平均误差。它还显著优于直接平流物理状态并学习剩余更新的方法，表明方程指定的读取坐标为长时程偏微分方程预测提供了有效的结构先验。
英文摘要
We identify a mismatch between the physical role of transport fields in many PDEs and their usual role in neural operators: PDEs use them to select read coordinates, whereas neural operators typically treat them only as input values. We address this mismatch with the Green-Routed Neural Operator (GRNO), which uses the governing equation to determine where latent features are sampled. A parameter-free equation adapter evaluates the diagnostic relation and constructs a departure map whose values are the read coordinates. A multiscale encoder-decoder combines centered and routed reads of latent features to learn the complete finite-time update. Across five two- and three-dimensional PDE systems, GRNO achieves the lowest mean final relative $L^2$ error on four under 40-step autoregressive evaluation and remains competitive on Keller-Segel. Fixed-weight route interventions reveal strong dependence on direction and spatial alignment in four systems, with weak dependence in Keller-Segel. In independently trained ablations, GRNO achieves lower mean errors than variants that supply the transport field only as an input feature, substitute a learned displacement for the equation-specified route, or apply the route with a spatial misalignment, across all five systems. It also substantially outperforms directly advecting the physical state and learning the remaining update, indicating that equation-specified read coordinates provide an effective structural prior for long-horizon PDE forecasting.
