We introduce the Transformed Generative Pre-Trained Physics-Informed Neural
Networks (TGPT-PINN) for accomplishing nonlinear model order reduction (MOR) of
transport-dominated partial differential equations in an MOR-integrating PINNs
framework. Building on the recent development of the GPT-PINN that is a
network-of-networks design achieving snapshot-based model reduction, we design
and test a novel paradigm for nonlinear model reduction that can effectively
tackle problems with parameter-dependent discontinuities. Through incorporation
of a shock-capturing loss function component as well as a parameter-dependent
transform layer, the TGPT-PINN overcomes the limitations of linear model
reduction in the transport-dominated regime. We demonstrate this new capability
for nonlinear model reduction in the PINNs framework by several nontrivial
parametric partial differential equations.

我们介绍了转换生成式预训练物理信息神经网络 (TGPT-PINN)，用于在 MOR 整合 PINNs 框架中完成非线性模型降阶的任务。通过引入用于捕捉震荡的损失函数组件以及参数相关的转换层，TGPT-PINN 克服了传输主导区域中线性模型降阶的限制。我们通过多个复杂的参数化偏微分方程在 PINNs 框架中展示了这一新的非线性模型降阶能力。

TGPT-PINN: 基于转换后 GPT-PINNs 的非线性模型简化

TGPT-PINN: Nonlinear model reduction with transformed GPT-PINNs

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.

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，利用图神经网络（GNNs）对无结构网格上的非参数 PDE 解进行编码降维，实现对物理和几何参数化设置下具有快速 / 缓慢衰减的线性 / 非线性和标量 / 矢量问题的快速评估和高度通用的数据驱动非线性降阶方法。