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
this https URL

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

NUNO: A General Framework for Learning Parametric PDEs with Non-Uniform  Data

Classification and segmentation of 3D point clouds are important tasks in
computer vision. Because of the irregular nature of point clouds, most of the
existing methods convert point clouds into regular 3D voxel grids before they
are used as input for ConvNets. Unfortunately, voxel representations are highly
insensitive to the geometrical nature of 3D data. More recent methods encode
point clouds to higher dimensional features to cover the global 3D space.
However, these models are not able to sufficiently capture the local structures
of point clouds.
Therefore, in this paper, we propose a method that exploits both local and
global contextual cues imposed by the k-d tree. The method is designed to learn
representation vectors progressively along the tree structure. Experiments on
challenging benchmarks show that the proposed model provides discriminative
point set features. For the task of 3D scene semantic segmentation, our method
significantly outperforms the state-of-the-art on the Stanford Large-Scale 3D
Indoor Spaces Dataset (S3DIS).

本文提出了一种基于 k-d 树的方法，旨在同时利用点云的局部和全局背景信息，通过沿树结构逐步学习表征向量，从而生成区分度强的点集特征。实验证明，该方法在 3D 场景语义分割等任务上明显优于现有的方法。

3DContextNet：利用局部和全局上下文线索，通过 K-d 树引导的分层点云学习

3DContextNet: K-d Tree Guided Hierarchical Learning of Point Clouds  Using Local and Global Contextual Cues

The k-d tree was one of the first spatial data structures proposed for
nearest neighbor search. Its efficacy is diminished in high-dimensional spaces,
but several variants, with randomization and overlapping cells, have proved to
be successful in practice. We analyze three such schemes. We show that the
probability that they fail to find the nearest neighbor, for any data set and
any query point, is directly related to a simple potential function that
captures the difficulty of the point configuration. We then bound this
potential function in two situations of interest: the first, when data come
from a doubling measure, and the second, when the data are documents from a
topic model.

通过分析三种 k-d 树的变体，我们确定了一个简单的潜在函数，该函数能直接反映数据集和查询点难度，并在正增长测度和主题模型中限制了该潜在函数。