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.

翻译：使用偏微分方程（PDEs）对现实世界问题进行建模是科学机器学习中的一个重要课题。针对此任务的经典求解器继续发挥着核心作用，例如为深度学习类比模型生成训练数据。任何此类数值解都会受到多种不确定性来源的影响，这些不确定性既来自有限的计算资源，也来自有限的数据（包括未知参数）。作为经典PDE模拟方法类比的高斯过程最近已成为一个框架，用于构建所有这些类型不确定性的完全概率估计。迄今为止，这类工作大多侧重于理论基础，因此数据效率或可扩展性并不突出。本文提出一个框架，将基于流行有限体积法的离散化方案与互补的数值线性代数技术相结合。包括时空海啸模拟在内的实际实验表明，与之前基于配置点的方法相比，该方法的可扩展性得到了显著改善。