Data-Free Asymptotics-Informed Operator Networks for Singularly Perturbed PDEs

TL;DR

This paper introduces eFEONet, a novel operator network that leverages asymptotic theory and enrichment basis functions to accurately solve singularly perturbed PDEs with minimal data, addressing challenges of sharp layers and high computational cost.

Contribution

The paper presents a data-efficient, asymptotics-informed operator network tailored for singularly perturbed PDEs, incorporating classical theory for improved accuracy and convergence.

Findings

eFEONet accurately captures sharp boundary and interior layers.

The method achieves convergence without extensive training data.

Experimental results validate the approach on representative problems.

Abstract

Recent advances in machine learning (ML) have opened new possibilities for solving partial differential equations (PDEs), yet robust performance in challenging regimes remains limited. In particular, singularly perturbed differential equations exhibit sharp boundary or interior layers with rapid transitions, where standard ML surrogates often fail without extensive resolution. Generating training data for such problems is also costly, as accurate reference solutions typically require massive adaptive mesh refinement. In this work, we propose eFEONet, an enriched Finite Element Operator Network tailored to singularly perturbed problems. Guided by classical singular perturbation theory, eFEONet augments the operator-learning framework with specialized enrichment basis functions that encode the asymptotic structure of layer solutions. This design enables accurate approximation of sharp transitions without relying on large datasets, and can operate with minimal supervision-or even in a data-free manner under appropriate settings. We further provide a rigorous convergence analysis of the proposed method and demonstrate its effectiveness through extensive experiments on representative problems featuring both boundary and interior layers.