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Solving Poisson Problems in Polygonal Domains with Singularity Enriched Physics Informed Neural Networks

Tianhao Hu, Bangti Jin, Zhi Zhou

TL;DR

This work develops SEPINN, a singularity enriched PINN framework for Poisson problems in polygonal domains, by decomposing the solution into a regular part $w$ and a singular part $S$ that captures corner and edge effects via explicit singular functions. In 2D, SEPINN employs a singular function representation $u=S+w$ with corner-based components; in 3D, it handles edge singularities using a similar decomposition with Fourier-augmented representations. Two implementation paths are proposed: SEPINN-C (cutoff, truncating the singular expansion) and SEPINN-N (neural-network-based approximation of the singular part), both optimized with a two-stage path-following strategy to stabilize training and enforce boundary conditions progressively. The paper provides error analyses and demonstrates extensive numerical experiments, showing substantial accuracy gains over standard PINNs and related methods, and extends the framework to the modified Helmholtz equation and eigenvalue problems, indicating broad applicability to PDEs with geometric singularities.

Abstract

Physics-Informed Neural Networks (PINNs) are a powerful class of numerical solvers for partial differential equations, employing deep neural networks with successful applications across a diverse set of problems. However, their effectiveness is somewhat diminished when addressing issues involving singularities, such as point sources or geometric irregularities, where the approximations they provide often suffer from reduced accuracy due to the limited regularity of the exact solution. In this work, we investigate PINNs for solving Poisson equations in polygonal domains with geometric singularities and mixed boundary conditions. We propose a novel singularity enriched PINN (SEPINN), by explicitly incorporating the singularity behavior of the analytic solution, e.g., corner singularity, mixed boundary condition and edge singularities, into the ansatz space, and present a convergence analysis of the scheme. We present extensive numerical simulations in two and three-dimensions to illustrate the efficiency of the method, and also a comparative study with several existing neural network based approaches.

Solving Poisson Problems in Polygonal Domains with Singularity Enriched Physics Informed Neural Networks

TL;DR

This work develops SEPINN, a singularity enriched PINN framework for Poisson problems in polygonal domains, by decomposing the solution into a regular part and a singular part that captures corner and edge effects via explicit singular functions. In 2D, SEPINN employs a singular function representation with corner-based components; in 3D, it handles edge singularities using a similar decomposition with Fourier-augmented representations. Two implementation paths are proposed: SEPINN-C (cutoff, truncating the singular expansion) and SEPINN-N (neural-network-based approximation of the singular part), both optimized with a two-stage path-following strategy to stabilize training and enforce boundary conditions progressively. The paper provides error analyses and demonstrates extensive numerical experiments, showing substantial accuracy gains over standard PINNs and related methods, and extends the framework to the modified Helmholtz equation and eigenvalue problems, indicating broad applicability to PDEs with geometric singularities.

Abstract

Physics-Informed Neural Networks (PINNs) are a powerful class of numerical solvers for partial differential equations, employing deep neural networks with successful applications across a diverse set of problems. However, their effectiveness is somewhat diminished when addressing issues involving singularities, such as point sources or geometric irregularities, where the approximations they provide often suffer from reduced accuracy due to the limited regularity of the exact solution. In this work, we investigate PINNs for solving Poisson equations in polygonal domains with geometric singularities and mixed boundary conditions. We propose a novel singularity enriched PINN (SEPINN), by explicitly incorporating the singularity behavior of the analytic solution, e.g., corner singularity, mixed boundary condition and edge singularities, into the ansatz space, and present a convergence analysis of the scheme. We present extensive numerical simulations in two and three-dimensions to illustrate the efficiency of the method, and also a comparative study with several existing neural network based approaches.
Paper Structure (23 sections, 13 theorems, 73 equations, 10 figures, 8 tables, 1 algorithm)

This paper contains 23 sections, 13 theorems, 73 equations, 10 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Our contributions
  4. Preliminaries
  5. Deep neural networks
  6. Physics informed neural networks
  7. Two existing methods
  8. Singularity enriched PINN
  9. Two-dimensional problem
  10. Singular function representation
  11. Singularity enriched physics-informed neural network
  12. Three-dimensional problem
  13. Singular function representation
  14. SEPINN -- Cutoff approximation
  15. SEPINN -- Neural networks approximation
  16. ...and 8 more sections

Key Result

Theorem 1

\newlabeltheorem:3d split0 For any fixed $f\in L^2(\Omega)$, let $u\in H^1(\Omega)$ be the unique weak solution to problem problem. Then there exist unique functions $\Psi_{j,i}\in H^{1-\lambda_{j,i}}(0,l)$ of the variable $z$ such that $u$ can be split into a sum of a regular part $w \in H^2(\Ome where the functions $T_j$ are fixed Poisson's kernels, and the symbol $*$ denotes the convolution in

Figures (10)

  • Figure 1: \newlabelfig:sketch0 The sketch of the local domain $G_j$ in two- and three-dimensions. In the 3D case,$G_{j,0}$ is defined by $G_{j,0}=\{(x_j,y_j)\in\Omega_0:0<r_j<R_j,0<\theta_j<\omega_j\}$.
  • Figure 1: The numerical approximations for Example \ref{['exam:2d-lshape']} by SEPINN, SAPINN, FIPINN, PINN and DRM. From top to bottom: DNN approximation and pointwise error.
  • Figure 1: \newlabelfig:exam:helmholtz0 2D problem with SEPINN (top) and 3D problem with SEPINN-C (middle) and SEPINN-N (bottom) for Example \ref{['exam:helmholtz']}, with slice at $z=\frac{1}{2}$.
  • Figure 2: \newlabelfig:exam:2d-lshape compare0 Training dynamics for SEPINN and benchmark methods: (a) the decay of the empirical loss $\widehat{\mathcal{L}}$ versus the iteration index $i$ (counted along the path-following trajectory) and (b) the error $e$ versus $i$.
  • Figure 3: The numerical approximations of Example \ref{['exam:2d-mix']} by the proposed SEPINN (error: $\text{4.62e-3}$), SAPINN (error: $\text{4.40e-2}$), FIPINN (error: $\text{3.65e-2}$), PINN (error: $\text{7.33e-2}$) and DRM (error: $\text{1.86e-1}$). From top to bottom: DNN approximation and pointwise error.
  • ...and 5 more figures

Theorems & Definitions (28)

  • Remark 3.1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proof 1
  • Lemma 1
  • Lemma 2
  • Proof 2
  • Remark 4.1
  • Lemma 3
  • ...and 18 more