Data-driven self-supervised learning for the discovery of solution singularity for partial differential equations

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 handle the raw unlabeled data, we propose a self-supervised learning (SSL) framework for learning the equation that describes the unknown singularity. We show that filtering is critical for obtaining desired detection and propose two filtering options - one based on kernel density estimation, another based on k nearest neighbors - as the pretext task in SSL. We provide numerical examples to illustrate the potential pathological or inaccurate results due to the use of raw data without filtering. The framework can be easily integrated with point cloud reconstruction methods to improve the reconstruction quality and speed for noisy data. Extensive experiments are presented to demonstrate the ability of the proposed approach to deal with input noise, label corruption, and different kinds of singularities such interior and boundary layers, concentric semicircles, multiple disconnected components. Applications to three dimensional noisy point cloud reconstruction are presented with comparison to radial basis function and Poisson surface reconstructions to demonstrate the approximation quality, flexibility, and computational efficiency of the proposed framework. Both visual and quantitative results are reported.