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。