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<p>The use of neural networks to solve partial differential equations (PDEs) has gained significant interest recently due to their simplicity and flexibility. In practical applications, these equations are often solved on highly complex physical domains, yet many current methods overlook these irregularities. This paper introduces the feature diffusion enhanced neural operator transformer, designed to solve PDEs on general geometries. It employs local attention to approximate the PDE mappings, while geometric operators are used to compute diffused features that enhance latent encodings. The learnable time diffusion automatically adapts to various input physical domains, efficiently capturing intricate geometric features. These geometry-aware encodings are decoded via recurrent feedforward networks to obtain solutions at each time step. Experimental results on both 2D and 3D benchmarks show that DNOT outperforms state-of-the-art methods in terms of accuracy when solving PDEs in irregularly shaped domains. Furthermore, ablation studies demonstrate that the model maintains consistent performance across scales, highlighting its potential for real-world applications.</p>

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Dnot: feature diffusion enhanced neural operator transformer for PDEs on general geometries

  • 期刊论文

  • 发表日期：

    2026年3月17日

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  • Hang Yu,
  • Juan Cao,
  • Xiangrong Liu,
  • Zhonggui Chen

摘要

The use of neural networks to solve partial differential equations (PDEs) has gained significant interest recently due to their simplicity and flexibility. In practical applications, these equations are often solved on highly complex physical domains, yet many current methods overlook these irregularities. This paper introduces the feature diffusion enhanced neural operator transformer, designed to solve PDEs on general geometries. It employs local attention to approximate the PDE mappings, while geometric operators are used to compute diffused features that enhance latent encodings. The learnable time diffusion automatically adapts to various input physical domains, efficiently capturing intricate geometric features. These geometry-aware encodings are decoded via recurrent feedforward networks to obtain solutions at each time step. Experimental results on both 2D and 3D benchmarks show that DNOT outperforms state-of-the-art methods in terms of accuracy when solving PDEs in irregularly shaped domains. Furthermore, ablation studies demonstrate that the model maintains consistent performance across scales, highlighting its potential for real-world applications.