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

TL;DR

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, suggesting a potential shift in methodological approaches for data-driven partial differential equation learning.

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.

Figures (5)

• Figure 1: Three different mesh topographies. (a) displays a structured mesh, which is believed to be advantageous for CV-based models due to its conformity with pixel images. (b) illustrates a graded mesh challenges the boundary between structured and unstructured meshes. In (c) depicts an unstructured mesh exemplifies the domain of graph-based models, optimized for irregular connectivity. This study evaluates whether CV-based or graph-based models are better suited for graded mesh topography.
• Figure 2: Input samples from the three datasets used in this article. (a) corresponds to a structured mesh, (b) to a graded mesh with a finer resolution towards the walls, and (c) to an unstructured mesh. See also Figure \ref{['fig:mesh_structures']} for reference.
• Figure 3: Ground-truth of the velocity$\lVert \mathbf{u} \rVert$ and the difference between ground-truth and the predictions of the best performing CV-based and graph-based model.
• Figure 4: Ground-truth of the pressure$p$ and the difference between ground-truth and the predictions of the best performing CV-based and graph-based model.
• Figure 5: Ground-truth of magnetic flux density$\mathbf{B}$ and difference between ground-truth and predictions of the best performing CV-based and graph-based model.