Mesh-based simulations play a key role when modeling complex physical systems that, in many disciplines across science and engineering, require the solution of parametrized time-dependent nonlinear partial differential equations (PDEs). In this context, full order models (FOMs), such as those relying on the finite element method, can reach high levels of accuracy, however often yielding intensive simulations to run. For this reason, surrogate models are developed to replace computationally expensive solvers with more efficient ones, which can strike favorable trade-offs between accuracy and efficiency. This work explores the potential usage of graph neural networks (GNNs) for the simulation of time-dependent PDEs in the presence of geometrical variability. In particular, we propose a systematic strategy to build surrogate models based on a data-driven time-stepping scheme where a GNN architecture is used to efficiently evolve the system. With respect to the majority of surrogate models, the proposed approach stands out for its ability of tackling problems with parameter dependent spatial domains, while simultaneously generalizing to different geometries and mesh resolutions. We assess the effectiveness of the proposed approach through a series of numerical experiments, involving both two- and three-dimensional problems, showing that GNNs can provide a valid alternative to traditional surrogate models in terms of computational efficiency and generalization to new scenarios. We also assess, from a numerical standpoint, the importance of using GNNs, rather than classical dense deep neural networks, for the proposed framework.

翻译：基于网格的模拟在模拟复杂物理系统中扮演关键角色，这些系统在科学与工程的众多学科中需要求解参数化的时间依赖非线性偏微分方程。在此背景下，全阶模型（如依赖有限元方法的模型）能够达到高精度，但通常会导致计算密集的模拟。因此，人们开发代理模型来用更高效的求解器替代计算昂贵的求解器，从而在精度与效率之间取得有利平衡。本研究探索了图神经网络在存在几何变异性情况下模拟时间依赖偏微分方程的潜在应用。具体而言，我们提出了一种系统性策略，基于数据驱动的时间步进方案构建代理模型，其中使用图神经网络架构高效演化系统。与大多数代理模型相比，所提方法的核心优势在于能够处理具有参数依赖空间域的问题，同时泛化到不同几何形状和网格分辨率。通过一系列涉及二维和三维问题的数值实验，我们评估了所提方法的有效性，结果表明图神经网络在计算效率和对新场景的泛化能力方面可作为传统代理模型的有效替代方案。我们还从数值角度评估了在所提框架中使用图神经网络而非经典密集深度神经网络的重要性。