Point Cloud Neural Operator for Parametric PDEs on Complex and Variable Geometries

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.