Green-Routed Neural Operators：让物理方程决定网络在哪里读取特征

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Abstract: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.

Subjects:	Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
Cite as:	arXiv:2610.05337 [cs.LG]
	(or arXiv:2610.05337v2 [cs.LG] for this version)
	https://doi.org/10.48550/arXiv.2610.05337 arXiv-issued DOI via DataCite

Submission history

From: Chenhao Si [view email]
[v1] Sun, 4 Oct 2026 16:01:18 UTC (8,324 KB)
[v2] Fri, 9 Oct 2026 10:12:53 UTC (8,325 KB)