Title: Data-Driven Self-Supervised Learning for the Discovery of Solution Singularity for Partial Differential Equations Show Chinese title

Title: 数据驱动的自监督学习用于偏微分方程解奇点的发现

Abstract: The appearance of singularities in the function of interest constitutes a fundamental challenge in scientific computing. It can significantly undermine the effectiveness of numerical schemes for function approximation, numerical integration, and the solution of partial differential equations (PDEs), etc. The problem becomes more sophisticated if the location of the singularity is unknown, which is often encountered in solving PDEs. Detecting the singularity is therefore critical for developing efficient adaptive methods to reduce computational costs in various applications. In this paper, we consider singularity detection in a purely data-driven setting. Namely, the input only contains given data, such as the vertex set from a mesh. To overcome the limitation of the raw unlabeled data, we propose a self-supervised learning (SSL) framework for estimating the location of the singularity. A key component is a filtering procedure as the pretext task in SSL, where two filtering methods are presented, based on $k$ nearest neighbors and kernel density estimation, respectively. We provide numerical examples to illustrate the potential pathological or inaccurate results due to the use of raw data without filtering. Various experiments are presented to demonstrate the ability of the proposed approach to deal with input perturbation, label corruption, and different kinds of singularities such interior circle, boundary layer, concentric semicircles, etc. Show Chinese abstract

Abstract: 在感兴趣的函数中出现奇点构成了科学计算中的一个基本挑战。 它可能会显著削弱函数逼近、数值积分和偏微分方程（PDEs）求解等数值方案的有效性。 如果奇点的位置未知，问题会变得更加复杂，这在求解PDEs时经常遇到。 因此，检测奇点对于开发高效的自适应方法以减少各种应用中的计算成本至关重要。 在本文中，我们考虑在一个纯粹的数据驱动设置中的奇点检测。 即，输入仅包含给定的数据，例如来自网格的顶点集。 为了克服原始未标记数据的局限性，我们提出了一种自监督学习（SSL）框架来估计奇点的位置。 一个关键组成部分是作为SSL中预训练任务的过滤过程，其中提出了两种基于$k$最近邻和核密度估计的过滤方法。 我们提供了数值例子来说明在没有过滤的情况下使用原始数据可能导致的病态或不准确的结果。 进行了各种实验，以展示所提出方法处理输入扰动、标签损坏以及不同类型奇点（如内部圆、边界层、同心半圆等）的能力。