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.

基于图神经网络的代理模型在模拟时变偏微分方程方面具有计算效率和泛化能力，并有效替代传统的复杂求解器。

基于深度学习的参数化偏微分方程代理模型：通过图神经网络处理几何变化

Deep Learning-based surrogate models for parametrized PDEs: handling  geometric variability through graph neural networks

Mesh-based simulations are central to modeling complex physical systems in
many disciplines across science and engineering. Mesh representations support
powerful numerical integration methods and their resolution can be adapted to
strike favorable trade-offs between accuracy and efficiency. However,
high-dimensional scientific simulations are very expensive to run, and solvers
and parameters must often be tuned individually to each system studied. Here we
introduce MeshGraphNets, a framework for learning mesh-based simulations using
graph neural networks. Our model can be trained to pass messages on a mesh
graph and to adapt the mesh discretization during forward simulation. Our
results show it can accurately predict the dynamics of a wide range of physical
systems, including aerodynamics, structural mechanics, and cloth. The model's
adaptivity supports learning resolution-independent dynamics and can scale to
more complex state spaces at test time. Our method is also highly efficient,
running 1-2 orders of magnitude faster than the simulation on which it is
trained. Our approach broadens the range of problems on which neural network
simulators can operate and promises to improve the efficiency of complex,
scientific modeling tasks.

MeshGraphNets 是一种利用图神经网络学习网格化仿真的框架，具有自适应性和高效性，能够应用于广泛的物理系统仿真任务。