PhyLoopNO: Looped Neural Operator with Physics-Gated Dense Expert Layers

Low-rank Transformer neural operators solve partial differential equations (PDEs) by modeling global interactions through compact latent representations. Slicing introduces an explicit latent bottleneck; global interactions are then realized through latent-token attention followed by deslicing, or through deslicing decoupled from slicing. We observe that slicing and deslicing are the main sources of differentiation across layers. Assigning layer-specific parameters to most other components offers relatively limited improvements in model performance, resulting in low parameter efficiency. Inspired by looped Transformers in large language models, we introduce PhyLoopNO, a parameter-efficient neural operator built on LinearNO. It reuses operator and feed-forward transformations across loop iterations while retaining independent slicing and deslicing, preserving spatial interactions that vary with depth. However, parameter sharing alone reduces parameter capacity and can limit predictive accuracy. To address this trade-off, we replace the pointwise feed-forward network with physics-gated dense experts. Inspired by domain decomposition, a learned gate softly groups points according to their physical features and combines the responses of all parallel experts at each point. Experiments show the best results on five of six standard PDE benchmarks in our comparison and competitive performance on industrial aerodynamic tasks. Increasing depth from 8 to 40 layers further reduces its errors on Airfoil, Darcy, and Elasticity overall. Selective parameter reuse lowers parameter growth on Airfoil and Darcy, enabling parameter-efficient depth scaling.