Data-Driven Self-Supervised Learning for the Discovery of Solution Singularity for Partial Differential Equations

2025-06-29

AI Summary

Accurately locating singularities remains a fundamental challenge in the numerical solution of partial differential equations (PDEs), severely limiting the design and efficiency of adaptive methods. This paper introduces the first purely data-driven, self-supervised framework for singularity detection—requiring no labeled data and operating solely on mesh vertex coordinates. We propose a pre-task combining k-nearest-neighbor graph construction and kernel density estimation filtering to guide either graph neural networks or multilayer perceptrons in learning robust singularity representations. To our knowledge, this is the first application of self-supervised learning to singularity detection, significantly enhancing resilience to data perturbations and label noise. The method accurately identifies complex singularity patterns—including interior circular singularities, boundary layers, and concentric semicircular structures. Experiments demonstrate substantial improvements in localization accuracy and stability over unfiltered baselines, with clear practical value in reducing downstream numerical computation costs.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Data Mining & Knowledge Management: Anomaly/Outlier Detection

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Web data integration and cleaning

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.

Problem

• Detect singularities in PDE solutions with unknown locations
• Develop data-driven self-supervised learning for singularity detection
• Improve numerical scheme accuracy by filtering raw input data

Innovation

• Self-supervised learning for singularity detection
• Filtering methods using k-nearest neighbors
• Kernel density estimation for data preprocessing