Modeling real-world problems with partial differential equations (PDEs) is a
prominent topic in scientific machine learning. Classic solvers for this task
continue to play a central role, e.g. to generate training data for deep
learning analogues. Any such numerical solution is subject to multiple sources
of uncertainty, both from limited computational resources and limited data
(including unknown parameters). Gaussian process analogues to classic PDE
simulation methods have recently emerged as a framework to construct fully
probabilistic estimates of all these types of uncertainty. So far, much of this
work focused on theoretical foundations, and as such is not particularly data
efficient or scalable. Here we propose a framework combining a discretization
scheme based on the popular Finite Volume Method with complementary numerical
linear algebra techniques. Practical experiments, including a spatiotemporal
tsunami simulation, demonstrate substantially improved scaling behavior of this
approach over previous collocation-based techniques.

应用偏微分方程模型到现实世界问题是科学机器学习的重要课题。本文提出一种结合基于有限体积法的离散化方案和数值线性代数技术的框架，通过实验验证在空间与时间方面的海啸模拟中该方法相比之前的基于插值法的技术有显著的性能提升。

使用结构化体积信息放大概率 PDE 模拟器

Scaling up Probabilistic PDE Simulators with Structured Volumetric  Information

Computational fluid dynamics (CFD) simulation is an irreplaceable modelling
step in many engineering designs, but it is often computationally expensive.
Some graph neural network (GNN)-based CFD methods have been proposed. However,
the current methods inherit the weakness of traditional numerical simulators,
as well as ignore the cell characteristics in the mesh used in the finite
volume method, a common method in practical CFD applications. Specifically, the
input nodes in these GNN methods have very limited information about any object
immersed in the simulation domain and its surrounding environment. Also, the
cell characteristics of the mesh such as cell volume, face surface area, and
face centroid are not included in the message-passing operations in the GNN
methods. To address these weaknesses, this work proposes two novel geometric
representations: Shortest Vector (SV) and Directional Integrated Distance
(DID). Extracted from the mesh, the SV and DID provide global geometry
perspective to each input node, thus removing the need to collect this
information through message-passing. This work also introduces the use of
Finite Volume Features (FVF) in the graph convolutions as node and edge
attributes, enabling its message-passing operations to adjust to different
nodes. Finally, this work is the first to demonstrate how residual training,
with the availability of low-resolution data, can be adopted to improve the
flow field prediction accuracy. Experimental results on two datasets with five
different state-of-the-art GNN methods for CFD indicate that SV, DID, FVF and
residual training can effectively reduce the predictive error of current
GNN-based methods by as much as 41%.

本研究拟定了两种新的几何表示方法：最短向量（SV）和方向综合距离（DID），并提出了有限体积特征（FVF）和剩余训练的使用，可以有效地降低当前基于图神经网络的方法的预测误差高达 41％。