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A Parameter-Driven Physics-Informed Neural Network Framework for Solving Two-Parameter Singular Perturbation Problems Involving Boundary Layers

Pradanya Boro, Aayushman Raina, Srinivasan Natesan

TL;DR

This work tackles two-parameter singular perturbation problems with boundary-layer phenomena by extending parameter-asymptotic PINNs (PA-PINNs) to 1D and 2D elliptic and parabolic PDEs. It introduces a mesh-free PA-PINN framework that first captures the smooth component at large perturbations and then progressively reduces the perturbation parameters, guided by adaptive collocation point sampling and a weighted residual loss. Numerical experiments on a variety of 1D/2D problems demonstrate that PA-PINNs achieve higher accuracy and robustness than standard PINNs, Upwind FDM, and FEM, across different regimes determined by the ratio of $\varepsilon_1$ and $\varepsilon_2$, including complex boundary and corner layers. The approach eliminates the need for a priori knowledge of boundary-layer locations and widths, offering a scalable, architecture-agnostic method that can outperform traditional mesh-based methods in challenging layer-dominated regimes.

Abstract

In this article, our goal is to solve two-parameter singular perturbation problems (SPPs) in one- and two-dimensions using an adapted Physics-Informed Neural Networks (PINNs) approach. Such problems are of major importance in engineering and sciences as it appears in control theory, fluid and gas dynamics, financial modelling and so on. Solutions of such problems exhibit boundary and/or interior layers, which make them difficult to handle. It has been validated in the literature that standard PINNs have low accuracy and can't handle such problems efficiently. Recently Cao et. al \cite{cao2023physics} proposed a new parameter asymptotic PINNs (PA-PINNs) to solve one-parameter singularly perturbed convection-dominated problems. It was observed that PA-PINNs works better than standard PINNs and gPINNs in terms of accuracy, convergence and stability. In this article, for the first time robustness of PA-PINNs will be validated for solving two-parameter SPPs.

A Parameter-Driven Physics-Informed Neural Network Framework for Solving Two-Parameter Singular Perturbation Problems Involving Boundary Layers

TL;DR

This work tackles two-parameter singular perturbation problems with boundary-layer phenomena by extending parameter-asymptotic PINNs (PA-PINNs) to 1D and 2D elliptic and parabolic PDEs. It introduces a mesh-free PA-PINN framework that first captures the smooth component at large perturbations and then progressively reduces the perturbation parameters, guided by adaptive collocation point sampling and a weighted residual loss. Numerical experiments on a variety of 1D/2D problems demonstrate that PA-PINNs achieve higher accuracy and robustness than standard PINNs, Upwind FDM, and FEM, across different regimes determined by the ratio of and , including complex boundary and corner layers. The approach eliminates the need for a priori knowledge of boundary-layer locations and widths, offering a scalable, architecture-agnostic method that can outperform traditional mesh-based methods in challenging layer-dominated regimes.

Abstract

In this article, our goal is to solve two-parameter singular perturbation problems (SPPs) in one- and two-dimensions using an adapted Physics-Informed Neural Networks (PINNs) approach. Such problems are of major importance in engineering and sciences as it appears in control theory, fluid and gas dynamics, financial modelling and so on. Solutions of such problems exhibit boundary and/or interior layers, which make them difficult to handle. It has been validated in the literature that standard PINNs have low accuracy and can't handle such problems efficiently. Recently Cao et. al \cite{cao2023physics} proposed a new parameter asymptotic PINNs (PA-PINNs) to solve one-parameter singularly perturbed convection-dominated problems. It was observed that PA-PINNs works better than standard PINNs and gPINNs in terms of accuracy, convergence and stability. In this article, for the first time robustness of PA-PINNs will be validated for solving two-parameter SPPs.
Paper Structure (16 sections, 1 theorem, 45 equations, 14 figures, 15 tables, 1 algorithm)

This paper contains 16 sections, 1 theorem, 45 equations, 14 figures, 15 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model Problems
  3. One-dimensional two-parameter singular perturbation problem
  4. Time-independent case:
  5. Time-dependent case:
  6. Two-dimensional two-parameter singular perturbation problem
  7. Elliptic boundary-value problem
  8. 2D Parabolic initial-boundary-value problem
  9. Modified Physics Informed Neural Network
  10. Fully connected neural network: Mathematical background
  11. Parameter Asymptotic PINNs (PA-PINNs) Methodology cao2023physics
  12. Algorithm for PA-PINNs
  13. Numerical Experiments
  14. Accuracy of PA-PINNs with different width, depth and training points
  15. Comparison with some standard numerical methods
  16. ...and 1 more sections

Key Result

Theorem 2.1

Assume that the solution $\mathfrak{v}$ of elliptic_problem has the following decomposition where $\mathscr{S}$ is the smooth part while $\mathscr{E}_L ,\, \mathscr{E}_R ,\, \mathscr{E}_B ,\, \mathscr{E}_T$ are the boundary layer parts near the domain boundary$\mathcal{D}_{10},\, \mathcal{D}_{11},\, \mathcal{D}_{20},\, \mathcal{D}_{21}$ respectively and $\mathscr{E}_{LB} ,\, \mathscr{E}_{ Let $\b

Figures (14)

  • Figure 1: A basic feed-forward neural network.
  • Figure 2: PA-PINNs Algorithm flowchart.
  • Figure 3: Comparison between Exact and PA-PINNs solution for different values of perturbation parameter $\varepsilon_1, \varepsilon_2$ for Example \ref{['ch2exmp1']}.
  • Figure 4: Comparison between Exact and PINNs solution for different values of perturbation parameter $\varepsilon_1, \varepsilon_2$ for Example \ref{['ch2exmp2']}.
  • Figure 5: Comparison between Exact and PA-PINNs solution for different values of perturbation parameter $\varepsilon_1, \varepsilon_2$ for Example \ref{['ex2D']}.
  • ...and 9 more figures

Theorems & Definitions (8)

  • Theorem 2.1
  • Remark 3.1
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • Example 4.4
  • Example 4.5
  • Example 4.6