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.

翻译：神经算子已成为学习偏微分方程中函数空间映射的强大工具。然而，面对实际物理数据（通常具有高度非均匀分布特性）时，基于网格的技术（如FFT）难以直接应用。为此，我们提出非均匀神经算子（NUNO）——一个专为非均匀数据高效算子学习设计的综合框架。通过基于K-D树的区域分解，我们将非均匀数据转化为均匀网格，同时有效控制插值误差，从而在速度与精度上实现对非均匀数据学习的并行化。我们在二维弹性力学、（2+1）维通道流以及三维多物理场散热器上开展了广泛实验——据我们所知，这是首次探索具有复杂几何构型的三维偏微分方程问题。该框架将误差率降低高达60%，并将训练速度提升2倍至30倍。代码现已开源于https://github.com/thu-ml/NUNO。