Surrogate models are critical for accelerating computationally expensive simulations in science and engineering, particularly for solving parametric partial differential equations (PDEs). Developing practical surrogate models poses significant challenges, particularly in handling geometrically complex and variable domains, which are often discretized as point clouds. In this work, we systematically investigate the formulation of neural operators — maps between infinite-dimensional function spaces — on point clouds to better handle complex and variable geometries while mitigating discretization effects. We introduce the Point Cloud Neural Operator (PCNO), designed to efficiently approximate solution maps of parametric PDEs on such domains. We evaluate the performance of PCNO on a range of pedagogical PDE problems, focusing on aspects such as boundary layers, adaptively meshed point clouds, and variable domains with topological variations. Its practicality is further demonstrated through three-dimensional applications, such as predicting pressure loads on various vehicle types and simulating the inflation process of intricate parachute structures.

中文翻译：

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用于复杂和可变几何结构上的参数化 PDE 的点云神经运算符

代理模型对于加速科学和工程中计算成本高昂的仿真至关重要，尤其是对于求解参数偏微分方程 （PDE） 而言。开发实用的代理模型带来了重大挑战，尤其是在处理几何复杂和可变域时，这些域通常被离散化为点云。在这项工作中，我们系统地研究了点云上神经算子（无限维函数空间之间的映射）的公式，以更好地处理复杂和可变的几何形状，同时减轻离散化效应。我们介绍了点云神经算子 （PCNO），旨在有效地近似此类域上参数化偏微分方程的解图。我们评估了 PCNO 在一系列教学 PDE 问题上的性能，重点关注边界层、自适应网格划分的点云和具有拓扑变化的可变域等方面。它的实用性通过三维应用得到进一步证明，例如预测各种车辆类型的压力载荷和模拟复杂降落伞结构的充气过程。