The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at https://github.com/thu-ml/NUNO.

本文介绍了一种基于K-D树的非均匀神经算子 (Non-Uniform Neural Operator, NUNO) 框架，用于在高度非均匀分布的实际物理数据上进行有效操作学习，可将数据转换为均匀格点，实现了在非均匀数据上的相当速度和准确度，但同时能够控制插值误差。作者在2D弹性、(2+1)D通道流和3D多物理热沉等实验中对该框架进行了广泛实验，得到60%以上的误差率降低和2x到30x的训练速度提升。

NUNO: 使用非均匀数据学习参数化偏微分方程的通用框架