【SIST】Lift-and-Embed Learning Methods for Partial Differential Equations with Singularities

Physics-Informed neural networks have emerged as a powerful tool in scientific machine learning, providing efficient numerical solutions for broad classes of partial differential equations. However, conventional learning approaches often struggle to resolve problems involving singular behaviors, such as discontinuities in hyperbolic equations or singularities in Green’s functions. In this talk, lift-and-embed learning methods are introduced to address these challenges, which comprise three innovative components: (1) incorporating domain-specific prior knowledge into the solution ansatz by including an augmented variable; (2) utilizing neural networks to handle the increased dimensionality and address both linear and nonlinear problems within a unified mesh-free learning framework; (3) projecting the trained model back onto the original physical domain to obtain the approximate solution. With collocation points sampled only on piecewise surfaces rather than fulfilling the whole lifted space, we demonstrate through numerical benchmarks that our methods can efficiently resolve solution singularities in both hyperbolic and elliptic problems.