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的非线性模型降阶框架（GCA-ROM）。在降阶建模（ROM）背景下，研究目标是实现参数化偏微分方程（PDEs）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，但其更适用于处理具有快速Kolmogorov n-宽度衰减的线性和仿射模型。一方面，自编码器架构作为POD压缩过程的非线性推广，能够将主要信息编码至潜变量集合中，同时提取其关键特征。另一方面，图神经网络（GNNs）为研究定义在非结构化网格上的PDE解提供了自然框架。本文发展了一种非侵入式数据驱动非线性降阶方法，利用GNNs编码降阶流形，实现参数化PDEs的快速评估。针对物理与几何参数化场景中具有快/慢衰减特性的线性/非线性问题及标量/向量问题，我们展示了该方法的能力。本方法的核心特性包括：（i）即使在复杂工况下仍具备低数据体制下的强泛化能力；（ii）对通用非结构化网格的物理兼容性；（iii）利用池化与反池化操作从离散数据中学习。