Green-Routed Neural Operators:\\Physics Determines Where the Network Reads

2026-10-04

AI Summary

This study addresses the physical role mismatch in neural operators, where transport fields are treated merely as input values rather than readout coordinates. To overcome this limitation, we propose the Governing-equation-defined Readout Neural Operator (GRNO) framework. Its core innovation lies in utilizing governing equations to directly define network readout coordinates as a structural prior, departing from conventional paradigms that treat transport fields as pure inputs or learn displacements implicitly. Methodologically, GRNO determines feature sampling locations via governing equations and constructs mappings through parameter-free adapters, integrated with a multi-scale encoder-decoder architecture and autoregressive evaluation for efficient PDE prediction. Experiments across five PDE systems demonstrate that GRNO achieves the lowest error on four benchmarks, significantly outperforming baseline methods and validating the effectiveness of equation-specified readout coordinates.

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.

Problem

• Neural Operators
• Partial Differential Equations
• Transport Fields
• Read Coordinates
• PDE Forecasting

Innovation

• Neural Operators
• Physics-informed routing
• Transport fields
• Partial Differential Equations
• Multiscale encoder-decoder