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

arXiv: https://arxiv.org/abs/2506.23344 · HTML: https://mlsanity.com/paper/2506.23344/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 77 references that are in the map.
- First posted on arXiv 29 Jun 2025.

## Builds on (most cited first, 18 of 18)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
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- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [Bootstrap your own latent: A new approach to self-supervised Learning](https://mlsanity.com/paper/2006.07733/) (2020)
- [wav2vec 2.0: A Framework for Self-Supervised Learning of Speech Representations](https://mlsanity.com/paper/2006.11477/) (2020)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs](https://mlsanity.com/paper/1603.09320/) (2016)
- [data2vec: A General Framework for Self-supervised Learning in Speech, Vision and Language](https://mlsanity.com/paper/2202.03555/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Using Self-Supervised Learning Can Improve Model Robustness and Uncertainty](https://mlsanity.com/paper/1906.12340/) (2019)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [A Cookbook of Self-Supervised Learning](https://mlsanity.com/paper/2304.12210/) (2023)
- [Vision Models Are More Robust And Fair When Pretrained On Uncurated Images Without Supervision](https://mlsanity.com/paper/2202.08360/) (2022)
- [Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control](https://mlsanity.com/paper/2105.14094/) (2021)
- [Posterior Covariance Structures in Gaussian Processes](https://mlsanity.com/paper/2408.07379/) (2024)
- [CPAFT: A Consistent Parallel Advancing Front Technique for Unstructured Triangular/Tetrahedral Mesh Generation](https://mlsanity.com/paper/2405.20618/) (2024)

## Abstract (arXiv)

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.