HiLNO improves efficiency and accuracy in solving complex PDEs
HiLNO: A Hierarchical Latent Neural Operator with Multi-Scale Supervision for PDEs on General Geometries
Machine Learning
Summary
Partial differential equations (PDEs) describe many natural and engineered systems but can be hard to solve quickly. The authors developed HiLNO, a new neural network method that keeps important details at multiple scales to better predict PDE solutions while using fewer computing resources. HiLNO also works well with complicated shapes and can handle different levels of detail it hasn’t seen before. This approach reduces the size and time needed for computations without losing accuracy.
What this means in practice
- •For engineering simulation teams: Simulate aerodynamic flows more efficiently on complex vehicle shapes using accurate multiscale PDE predictions.
- •For computational fluid dynamics developers: Build faster PDE solvers that maintain accuracy when switching between different spatial resolutions or mesh sizes.
Authors
Zhicheng Hu, Jiacheng Li, Min Yang
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.