Title: Data-free Asymptotics-Informed Operator Networks for Singularly Perturbed PDEs
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: scientific machine learning, operator networks, singularly perturbed PDEs, spectral bias, boundary and interior layer

Abstract: Recent advancements in machine learning (ML) have shown promise in solving partial differential equations (PDEs), but significant challenges remain, particularly in handling complex scenarios. Singularly perturbed differential equations present unique computational difficulties due to rapid transitions within thin boundary or interior layers, where ML methods often struggle. Moreover, these problems require massive adaptive mesh refinement, making dataset generation computationally expensive. In this paper, we introduce eFEONet, an enriched Finite Element Operator Network designed to overcome these challenges. By leveraging singular perturbation analysis from PDE theory, eFEONet incorporates special basis functions that capture the asymptotic behavior of solutions, enabling accurate modeling of sharp transitions. Our approach is highly data-efficient, requiring minimal training data or even functioning without a dataset. Furthermore, we provide a rigorous convergence analysis and empirically validate eFEONet across various boundary and interior layer problems.

TL;DR: Data free operator networks for solving singularly perturbed differential equations using finite element space.