From Structured to Unstructured:A Comparative Analysis of Computer Vision and Graph Models in solving Mesh-based PDEs

TL;DR

This study compares computer vision and graph models for solving mesh-based PDEs, finding that vision models like U-Net outperform graph models in efficiency and accuracy across various mesh types, including unstructured meshes.

Contribution

It provides a comprehensive comparison of vision and graph models for PDEs on different mesh topographies, highlighting the effectiveness of vision models in unstructured mesh scenarios.

Findings

Vision models outperform graph models in structured and graded meshes.

Computer vision models show unexpected effectiveness on unstructured meshes.

Deep learning offers a promising alternative for high-performance PDE solutions.

Abstract

This article investigates the application of computer vision and graph-based models in solving mesh-based partial differential equations within high-performance computing environments. Focusing on structured, graded structured, and unstructured meshes, the study compares the performance and computational efficiency of three computer vision-based models against three graph-based models across three data\-sets. The research aims to identify the most suitable models for different mesh topographies, particularly highlighting the exploration of graded meshes, a less studied area. Results demonstrate that computer vision-based models, notably U-Net, outperform the graph models in prediction performance and efficiency in two (structured and graded) out of three mesh topographies. The study also reveals the unexpected effectiveness of computer vision-based models in handling unstructured meshes, suggesting a potential shift in methodological approaches for data-driven partial differential equation learning. The article underscores deep learning as a viable and potentially sustainable way to enhance traditional high-performance computing methods, advocating for informed model selection based on the topography of the mesh.