From Structured to Unstructured:
A Comparative Analysis of Computer Vision and Graph Models in solving Mesh-based PDEs Thanks: This research has been funded by the Federal Ministry for Economic Affairs and Climate Action (BMWK) within the project ”KI-basierte Topologieoptimierung elektrischer Maschinen (KITE)” (19I21034C).

Partial differential equations (PDEs) are central to modeling a wide range of complex physical phenomena in various scientific and engineering disciplines, including fluid dynamics and electrodynamics. Traditionally, solving these equations in detailed simulations necessitates high-performance computing (HPC) resources [1, 2, 3]. However, the immense computational power of HPC systems leads to substantial energy consumption, raising environmental concerns due to the associated greenhouse gas emissions. In an era where sustainable computing is becoming increasingly crucial, our research explores a new data-driven approach to lower the computational demands of these simulations.

This article investigates the application of advanced deep learning models from the fields of computer vision (CV) and graph learning in solving mesh-based PDEs in an HPC context. Figure 1 provides illustrative examples of this study’s three distinct mesh topographies. We focus on six models of these two fields, tailored to three specific datasets: Darcy Flow [4] on a structured mesh (cf. Figure 1(a)), a U-bend [5] with a graded structured mesh (cf. Figure 1(b)) hereafter referred to as graded mesh, and an electric motor [6] with an unstructured mesh (cf. Figure 1(c)). We investigate the ability of CV-based architectures and graph-based models to accurately predict solutions and determine which model can offer the most energy-efficient alternatives to traditional HPC methods. CV-based models are known for their capabilities in analyzing structured data, while graph-based ones are effective in unstructured data scenarios. Although graded meshes have the inherent advantage of maintaining neighborhood relations of a structured mesh while also allowing for higher resolution in areas of interest, they have found limited attention in existing research. We fill this gap by determining whether CV-based or graph-based models are more suitable for these mesh topographies.

We detail the adaptation of CV-based and Graph-based models for these diverse datasets and evaluate their computational efficiency and performance. By showcasing the potential of these models to reduce both computational demands and environmental impact, our study advances sustainable HPC solutions for solving PDE with data-driven methods.

One popular and widely adopted way to reduce the number of evaluations needed for finding good-enough samples employs deep active design optimization (DADO) [7]. This advanced organic computing approach, effectively minimizes evaluation counts through strategic, self-aware acquisition methods. This method seeks to optimize computational efficiency by strategically choosing the most informative points for evaluation, utilizing a numerical simulation as an expert. This reduces the computational resources required by minimizing the total number of numerical simulations needed. Our investigation focuses on identifying a model that offers quick, efficient, and high-quality results suitable for incorporation into the DADO framework. This model serves as a surrogate for evaluating queries, offering evidently faster inference times than traditional numerical simulations.

Beyond design optimization, these surrogate models will find critical use in self-aware microsystems (SAM) [8], where the rapid inference capabilities of the surrogates are essential. By employing these surrogates within SAM, we embed knowledge about physical phenomena into the real-time decision-making processes of robots, a feature unachievable with time-consuming numerical simulations. This integration enables SAM to respond adaptively to environmental changes swiftly, bypassing the extensive computation times closely resembling the core idea to make decision at runtime [9]. The code ¹¹ 1 https://gitlab.uni-kassel.de/uk023894/cv˙vs˙graph used in our study is provided to facilitate further research based on our findings, along with configuration files to reproduce our experiments. We summarize our key contributions (C1-C4) as follows:

• C1: We incorporate state-of-the-art transformer based foundation models.

• C2: We apply the latest graph-based models designed for mesh-based PDEs.

• C3: We deliver an in-depth comparison between CV and graph-based models

• C4: We offer insights to model selection for surrogates in DADO and SAM.

The structure of this article is as follows: Section 2 reviews existing literature, focusing on the application of CV-based and graph-based models in solving physical simulations of PDEs. Section 3 details the six models and three datasets employed in our research and provides a comprehensive outline of our evaluation methodology. Section 4 presents our findings and discusses the implications and significance of our results. The article concludes with Section 5, which reflects our findings and proposes directions for future research in this evolving field.

CV-based models such as Convolutional Neural Networks, with their roots in image processing, excel in identifying spatial patterns and relationships, making them inherently suited for structured mesh environments [10, 11]. This tendency arises because information in pixel images is evenly distributed, a characteristic also found in structured mesh configurations. Li et al. [12] introduced the idea of using transformers, which is built on self-attention, cross-attention, and multilayer perceptrons to solve PDEs. They also applied their model on structured (uniform) and unstructured (non-uniform) mesh datasets. In contrast to Li et al. we incorporate the Self-Distillation with No Labels [13] (DINOv2) model which leverages self-supervised learning to generate robust visual representations without relying on labeled data. By employing self-distillation, DINOv2 uncovers intricate patterns in images, enabling broad applicability in vision tasks without the extensive need for annotated datasets. This highlights DINOv2s as a promising approach to reduce reliance on large labeled datasets. In addition, the Data-Efficient Image Transformer [14] (DeiT) utilizes the transformer architecture to achieve impressive results with limited image data. DeiT’s novel training strategy, featuring distillation tokens, allows it to efficiently learn from smaller datasets, positioning it as a substantial development in data-efficient computing. DeiT demonstrates the potential of transformers to operate effectively in scenarios where data is scarce.

Graph-based models have been increasingly applied to solve PDEs in mesh-based simulations, demonstrating particular effectiveness in handling the complexities of unstructured meshes, which are pivotal for representing intricate geometric shapes [15, 16, 17]. Their flexibility in adapting to the variable nature of these meshes contrasts with the preference for structured meshes in numerical simulations (which are valued for their computational efficiency and precision), enabling fast and more accurate problem resolution. In this article, we intend to utilize three graph-based models. The first model is the Bi-Stride Multi-Scale Graph Neural Network (BSMS) [18], designed to handle varying scales and complexities in mesh topography. The second model is the Mesh Graph Net (MGN) [19], it focuses on capturing intricate relationships within mesh-based data. Lastly, the Mesh Graph Net + Tree (MGTN) [20], proposed by Ripken et al., is developed to process hierarchical tree edges efficiently.

Recent studies provide insights into the performance of neural operators across different mesh topographies. Ripken et al. [20] compared graph-based models against a U-Net and a Fourier neural operator on the Darcy Flow dataset from Takamoto et al. [4], revealing that graph-based models did not surpass the U-Net in scenarios involving a equidistant mesh resolution of 128x128 cells. Additionally, their exploration extended to the Electric Motor dataset with an unstructured mesh provided by Botache et al. [6], where CV-based models were not considered, emphasizing the reliance on graph-based models for such complex geometries. Their experiments demonstrate that the graph rewiring strategy in their model not only boosts the performance of baseline methods but also achieves state-of-the-art results in certain tasks, indicating a considerable advancement in the efficiency and performance of neural PDE operators.

A mesh topography that incorporate strong grading near boundaries enhances local resolution in these areas, as illustrated in Figure 1(b). This approach commonly observed in fluid dynamics, introduces additional challenges. These graded meshes, designed to capture steep gradients near walls, are exemplified by a dataset from Decke et al. [5], showcasing a U-bend Flow with variable mesh cell distribution to enhance resolution in critical areas. Such datasets underline the need for neural network models to adapt to non-equidistant mesh resolution, a challenge not addressed in current literature.

This research aims to evaluate advanced deep learning models from the field of CV and graph learning on structured, graded and unstructured meshes. By focusing on their adaptability and performance in learning numerical simulations, particularly in the less-explored area of graded meshes, the study seeks to provide insights for choosing suitable models. This could improve PDE solution strategies in computationally demanding areas and advance methodologies in this field. In this section, we first describe the models applied, then introduce the three datasets and outline our experimental setup.

The results from all experiments are summarized in Table 1, presenting the mean $\mu$ and standard deviation $\sigma$ of the RMSE for each model across three trials on various datasets. The analysis also includes the TTE and the number of trainable parameters for each model. Notably, for DeiT and DINOv2, the total number of parameters differs from the number of trainable parameters. The low standard deviation $\sigma$ observed across all models on every dataset indicates successful experiments, indicating that the results are meaningful and not merely due to random variance.

The top-performing model on each dataset is highlighted in bold. Specifically, within the domain of computer vision (CV) models attributing our contribution C1, the U-Net model surpasses both DINOv2 and DeiT across all datasets, arranged by RMSE performance as U-Net $<$ DINOv2 $<$ DeiT. A similar consistent pattern is noted among the graph models, for contribution C2, with RMSE performance ranked as BSMS $<$ MGTN $<$ MGN. When comparing the highest-performing graph model (BSMS) against the top CV-based model (U-Net), to account our contribution C3, it’s observed that U-Net outperforms in the Darcy Flow and U-bend Flow datasets, while BSMS leads on the Electric Motor dataset. The observed dominance of the U-Net model in processing the structured Darcy Flow dataset, alongside a superior performance of the BSMS model on the unstructured mesh of the Electric Motor dataset, aligns with our initial expectations. Nevertheless, the notable performance of U-Net over BSMS in evaluating the U-bend Flow dataset, which includes graded meshes, is particularly engaging. This result implies that converting graded mesh cells into a 2D array, thus considering them as a regular mesh, notably helps in reducing the RMSE in model performance compared to treating them as graph data. Figure 3 displays the results of the velocity vector $\lVert\mathbf{u}\rVert$ for a sample from the U-bend Flow dataset. Figure 3(a) shows the ground-truth, which is generated through time-intensive numerical simulations.

Figures 3(b) and 3(c) present the absolut difference that illustrate the discrepancies between the models’ predictions and the ground-truth for the U-net and the BSMS model, respectively. We use a grayscale colorbar where darker areas indicate larger errors in this specific cell location. In Figure 3(b), it is observed that the U-net model exhibits a very low error across large areas of the solution space. The only exception is in the wake on the inner side of the channel, where flow separation would occur, and a vortex forms; here, the model slightly mispredicts. For the BSMS model in Figure 3(c), considerably larger errors are evident throughout the entire wake, as well as noticeable issues on the outer side at the inlet transition, highlighting the BSMS model’s challenges in these areas.

Utilizing the U-bend Flow dataset, we addressed two solution variables: in addition to the velocity $\lVert\mathbf{u}\rVert$, we also solved for the pressure $p$. In Figure 4 the ground-truth (cf. Figure 4(a)) and the two difference images in accordance to Figure 3 can be seen.

The observations made for Figure 3 are equally applicable to Figure 4, with the findings being even more pronounced. The U-net model (cf. Figure 4(b)) provides more uniform estimation with lower error values. In addition to deviations at the transition point, discrepancies are also recorded on the outer side of the channel’s wake. On the other hand, the deviations in the BSMS model (cf. Figure 4(c)) appear vastly more irregular and thus less systematic or interpretable. Notably, the inlet is characterized by substantial deviations.

Contrary to expectations, the slight difference in performance between BSMS and U-Net on the Electric Motor dataset is unexpected, revealing that our relatively straightforward strategy for managing unstructured mesh data with CV-based models can produce promising results. Figure 5 presents three images from a sample of the Electric Motor dataset. Figure 5(a) illustrates the ground-truth for the magnetic flux density $\mathbf{B}$. Figure 5(b) displays the difference image, highlighting discrepancies between the ground-truth and the U-net model’s prediction. Meanwhile, Figure 5(c) depicts the difference corresponding to the BSMS model’s prediction. It is observable that there are no clear patterns for discrepancies within the solution space. Notably, the deviations of the U-Net are slightly larger than those of the BSMS model, a conclusion that was previously drawn from the analysis in Table 1 and is now further illustrated visually. To enhance the performance and applicability of predictions for electric motors, the computational mesh cells in the air gap between the rotor and stator are crucial for calculating motor torque for example. Employing a weighted loss function could underscore the importance of these cells in the mesh, thereby encouraging the model to focus on achieving more precise estimations in these critical areas. We omit a further evaluation of the Darcy Flow dataset since all results in this dataset yielded the anticipated results. Further experimental findings, along with the corresponding code, can be accessed via the link mentioned above.