HiLNO: A Hierarchical Latent Neural Operator with Multi-Scale Supervision for PDEs on General Geometries

2026-09-16

AI Summary

为解决PDE解的多尺度结构信息丢失问题，提出HiLNO方法，通过构建细到粗再到细的潜在空间，并引入多尺度监督和各向异性高斯注意力机制。

Abstract

Latent neural operators improve the efficiency of operator learning for partial differential equations (PDEs) by performing the main computation on compact latent representations. However, directly compressing the input representation to obtain such compact representations may discard solution-relevant spatial information, especially for PDE solutions with multiscale structures. To address this problem, we propose HiLNO, a hierarchical latent neural operator that constructs a fine-to-coarse-to-fine latent space and further introduces multi-scale supervision (MSS) and anisotropic Gaussian attention. The hierarchy mitigates potential information loss during compression, while MSS aligns intermediate predictions with downsampled target fields, encouraging solution-relevant structures to be captured across multiple spatial scales. Anisotropic Gaussian attention enables feature transfer across the hierarchy, making HiLNO applicable to general geometries. Experiments on representative PDE benchmarks and a large-scale automotive aerodynamics task show that HiLNO achieves competitive predictive accuracy, while reducing the parameter count by an average of 84.4% and FLOPs by an average of 69.2% compared with LinearNO. Additional experiments demonstrate effective generalization to unseen spatial resolutions. Code is available at https://github.com/JcLimath/HiLNO.

Problem

• latent neural operators
• partial differential equations
• multi-scale structures
• information loss

Innovation

• Hierarchical Latent Neural Operator
• Multi-Scale Supervision
• Anisotropic Gaussian Attention
• General Geometries
• Efficient PDE Solving