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

Jinsil Lee, Youngjoon Hong, Seungchan Ko, Jae Yong Lee

TL;DR

This work addresses solving singularly perturbed PDEs with sharp boundary and interior layers using a data-efficient operator-learning approach. It introduces eFEONet, an enriched Finite Element Operator Network that incorporates theory-derived corrector bases into a Galerkin framework, enabling accurate layer capture without large training datasets. A convergence analysis ties the neural predictions to the enriched FE solution, and extensive experiments on ODEs and a PDE demonstrate substantial accuracy gains over neural baselines, especially in data-scarce scenarios. The approach offers a practical, scalable path for reliable PDE surrogates in stiff regimes, with potential extensions to corner-layer effects and further parameter studies.

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.

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

TL;DR

This work addresses solving singularly perturbed PDEs with sharp boundary and interior layers using a data-efficient operator-learning approach. It introduces eFEONet, an enriched Finite Element Operator Network that incorporates theory-derived corrector bases into a Galerkin framework, enabling accurate layer capture without large training datasets. A convergence analysis ties the neural predictions to the enriched FE solution, and extensive experiments on ODEs and a PDE demonstrate substantial accuracy gains over neural baselines, especially in data-scarce scenarios. The approach offers a practical, scalable path for reliable PDE surrogates in stiff regimes, with potential extensions to corner-layer effects and further parameter studies.

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.
Paper Structure (13 sections, 1 theorem, 44 equations, 11 figures, 6 tables)

This paper contains 13 sections, 1 theorem, 44 equations, 11 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Proposed Method: Enriched Finite Element Operator Network
  3. Finite Element Method (FEM)
  4. Enriched FEONet with a Corrector Basis
  5. Convergence of eFEONet
  6. Numerical Experiments
  7. Experimental Setup and Implementation Details
  8. Ordinary Differential Equations with Boundary Layer
  9. Ordinary Differential Equations with Interior Layer
  10. Partial Differential Equations on Square
  11. Comparison with the original FEONet
  12. Conclusion
  13. Derivation of Corrector Basis Functions

Key Result

Theorem 2.3

Assume that f_ass holds. Then for given $\varepsilon>0$ and $h>0$, with probability $1$, we have that

Figures (11)

  • Figure 1: Representative solution profiles for singularly perturbed PDEs, illustrating the inherent stiffness of boundary and interior layers across various domains. The sharp gradients and rapid transitions depicted here highlight the intrinsic stiffness and associated computational challenges.
  • Figure 2: Examples of the boundary (left) and interior (right) layer phenomena and comparisons of the reference solution (True) and the predicted solutions using Standard FEM, PINN, and eFEONet (Ours). We set $\varepsilon = 10^{-5}$, the mesh size for the Standard FEM is $1/12000$ for the left case, while it is kept identical to those of the other methods for the right case.
  • Figure 3: Schematic illustration of eFEONet.
  • Figure 4: Comparison of predicted solutions $\widehat{u}_{\varepsilon}$ using FNO, ComFNO, and eFEONet with $\varepsilon=10^{-4}$ for the boundary layer problem. The external force input function is given by $f(x)=1.81 \sin(1.68x)+0.09\cos(-1.78x)$.
  • Figure 5: Visualization of 100 input functions $f$ (left), corresponding reference solutions (middle), and error plots (right) for boundary layer problem \ref{['eq:case1']} with $\varepsilon=10^{-5}$.
  • ...and 6 more figures

Theorems & Definitions (3)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3: Convergence of eFEONet