The present work proposes a framework for nonlinear model order reduction based on a Graph Convolutional Autoencoder (GCA-ROM). In the reduced order modeling (ROM) context, one is interested in obtaining real-time and many-query evaluations of parametric Partial Differential Equations (PDEs). Linear techniques such as Proper Orthogonal Decomposition (POD) and Greedy algorithms have been analyzed thoroughly, but they are more suitable when dealing with linear and affine models showing a fast decay of the Kolmogorov n-width. On one hand, the autoencoder architecture represents a nonlinear generalization of the POD compression procedure, allowing one to encode the main information in a latent set of variables while extracting their main features. On the other hand, Graph Neural Networks (GNNs) constitute a natural framework for studying PDE solutions defined on unstructured meshes. Here, we develop a non-intrusive and data-driven nonlinear reduction approach, exploiting GNNs to encode the reduced manifold and enable fast evaluations of parametrized PDEs. We show the capabilities of the methodology for several models: linear/nonlinear and scalar/vector problems with fast/slow decay in the physically and geometrically parametrized setting. The main properties of our approach consist of (i) high generalizability in the low-data regime even for complex regimes, (ii) physical compliance with general unstructured grids, and (iii) exploitation of pooling and un-pooling operations to learn from scattered data.

翻译：本文提出了一种基于图卷积自编码器（GCA-ROM）的非线性模型降阶框架。在降阶建模（ROM）背景下，主要目标是实现参数化偏微分方程（PDE）的实时与多查询评估。诸如本征正交分解（POD）和贪婪算法等线性技术已被深入分析，但更适用于处理具有快速Kolmogorov n-宽度衰减的线性及仿射模型。一方面，自编码器架构代表了POD压缩过程的非线性推广，能够将主要信息编码至潜在变量集合中并提取其特征。另一方面，图神经网络（GNN）构成了研究定义于非结构化网格上PDE解的自然框架。本文开发了一种非侵入式数据驱动非线性降阶方法，利用GNN编码降阶流形以实现参数化PDE的快速评估。我们通过多个算例验证了该方法的能力：包括物理域和几何域参数化框架下具有快/慢衰减特性的线性/非线性、标量/矢量问题。该方法的主要特性包括：（i）在低数据量场景下仍具备高度泛化能力，尤其适用于复杂情形；（ii）对通用非结构化网格的物理相容性；（iii）利用池化与逆池化操作从散射数据中学习。