Large-scale numerical simulations often come at the expense of daunting
computations. High-Performance Computing has enhanced the process, but adapting
legacy codes to leverage parallel GPU computations remains challenging.
Meanwhile, Machine Learning models can harness GPU computations effectively but
often struggle with generalization and accuracy. Graph Neural Networks (GNNs),
in particular, are great for learning from unstructured data like meshes but
are often limited to small-scale problems. Moreover, the capabilities of the
trained model usually restrict the accuracy of the data-driven solution. To
benefit from both worlds, this paper introduces a novel preconditioner
integrating a GNN model within a multi-level Domain Decomposition framework.
The proposed GNN-based preconditioner is used to enhance the efficiency of a
Krylov method, resulting in a hybrid solver that can converge with any desired
level of accuracy. The efficiency of the Krylov method greatly benefits from
the GNN preconditioner, which is adaptable to meshes of any size and shape, is
executed on GPUs, and features a multi-level approach to enforce the
scalability of the entire process. Several experiments are conducted to
validate the numerical behavior of the hybrid solver, and an in-depth analysis
of its performance is proposed to assess its competitiveness against a C++
legacy solver.

该研究介绍了一种新颖的预处理器，将 GNN 模型与多级域分解框架结合起来，以提高 Krylov 方法的效率，并产生一个混合求解器，可以以任何所需的精度收敛。

用于解决大规模问题的多级图神经网络预处理器

Multi-Level GNN Preconditioner for Solving Large Scale Problems

Large-scale numerical simulations are capable of generating data up to
terabytes or even petabytes. As a promising method of data reduction,
super-resolution (SR) has been widely studied in the scientific visualization
community. However, most of them are based on deep convolutional neural
networks (CNNs) or generative adversarial networks (GANs) and the scale factor
needs to be determined before constructing the network. As a result, a single
training session only supports a fixed factor and has poor generalization
ability. To address these problems, this paper proposes a Feature-Enhanced
Implicit Neural Representation (FFEINR) for spatio-temporal super-resolution of
flow field data. It can take full advantage of the implicit neural
representation in terms of model structure and sampling resolution. The neural
representation is based on a fully connected network with periodic activation
functions, which enables us to obtain lightweight models. The learned
continuous representation can decode the low-resolution flow field input data
to arbitrary spatial and temporal resolutions, allowing for flexible
upsampling. The training process of FFEINR is facilitated by introducing
feature enhancements for the input layer, which complements the contextual
information of the flow field.To demonstrate the effectiveness of the proposed
method, a series of experiments are conducted on different datasets by setting
different hyperparameters. The results show that FFEINR achieves significantly
better results than the trilinear interpolation method.

该论文提出了一种基于特征增强的隐式神经表示方法（FFEINR）来进行流场数据的时空超分辨率处理，通过引入特征增强来完善流场的上下文信息，通过设置不同的超参数，实验结果表明，FFEINR 方法相比三线性插值方法取得了显著改善的效果。