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Physics-Informed Tailored Finite Point Operator Network for Parametric Interface Problems

Ting Du, Xianliang Xu, Wang Kong, Ye Li, Zhongyi Huang

TL;DR

The paper tackles parametric PDEs with interface-induced non-smooth solutions by proposing PI-TFPONet, which learns coefficients of local basis functions rather than direct solutions, enabling unsupervised training without PDE residual losses. It leverages a coordinate transformation and local Green's-function-based bases to reconstruct the solution from f to local coefficients, guaranteeing convergence as the local mesh is refined and loss minimized. Theoretical results establish both standard and uniform convergence in singular perturbation regimes, while extensive 1D and 2D numerical experiments demonstrate competitive or superior accuracy to supervised models, particularly in problems with singularities or high contrast. The approach reduces the need for labeled data, handles complex interface behavior robustly, and offers a practical pathway for efficient, physics-consistent operator learning in challenging interface problems.

Abstract

Learning operators for parametric partial differential equations (PDEs) using neural networks has gained significant attention in recent years. However, standard approaches like Deep Operator Networks (DeepONets) require extensive labeled data, and physics-informed DeepONets encounter training challenges. In this paper, we introduce a novel physics-informed tailored finite point operator network (PI-TFPONet) method to solve parametric interface problems without the need for labeled data. Our method fully leverages the prior physical information of the problem, eliminating the need to include the PDE residual in the loss function, thereby avoiding training challenges. The PI-TFPONet is specifically designed to address certain properties of the problem, allowing us to naturally obtain an approximate solution that closely matches the exact solution. Our method is theoretically proven to converge if the local mesh size is sufficiently small and the training loss is minimized. Notably, our approach is uniformly convergent for singularly perturbed interface problems. Extensive numerical studies show that our unsupervised PI-TFPONet is comparable to or outperforms existing state-of-the-art supervised deep operator networks in terms of accuracy and versatility.

Physics-Informed Tailored Finite Point Operator Network for Parametric Interface Problems

TL;DR

The paper tackles parametric PDEs with interface-induced non-smooth solutions by proposing PI-TFPONet, which learns coefficients of local basis functions rather than direct solutions, enabling unsupervised training without PDE residual losses. It leverages a coordinate transformation and local Green's-function-based bases to reconstruct the solution from f to local coefficients, guaranteeing convergence as the local mesh is refined and loss minimized. Theoretical results establish both standard and uniform convergence in singular perturbation regimes, while extensive 1D and 2D numerical experiments demonstrate competitive or superior accuracy to supervised models, particularly in problems with singularities or high contrast. The approach reduces the need for labeled data, handles complex interface behavior robustly, and offers a practical pathway for efficient, physics-consistent operator learning in challenging interface problems.

Abstract

Learning operators for parametric partial differential equations (PDEs) using neural networks has gained significant attention in recent years. However, standard approaches like Deep Operator Networks (DeepONets) require extensive labeled data, and physics-informed DeepONets encounter training challenges. In this paper, we introduce a novel physics-informed tailored finite point operator network (PI-TFPONet) method to solve parametric interface problems without the need for labeled data. Our method fully leverages the prior physical information of the problem, eliminating the need to include the PDE residual in the loss function, thereby avoiding training challenges. The PI-TFPONet is specifically designed to address certain properties of the problem, allowing us to naturally obtain an approximate solution that closely matches the exact solution. Our method is theoretically proven to converge if the local mesh size is sufficiently small and the training loss is minimized. Notably, our approach is uniformly convergent for singularly perturbed interface problems. Extensive numerical studies show that our unsupervised PI-TFPONet is comparable to or outperforms existing state-of-the-art supervised deep operator networks in terms of accuracy and versatility.
Paper Structure (42 sections, 4 theorems, 38 equations, 15 figures, 3 tables)

This paper contains 42 sections, 4 theorems, 38 equations, 15 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Works
  4. Data-based PDE solver.
  5. Physics-informed PDE solver.
  6. Interface problem solver.
  7. Preliminary
  8. Interface Problems
  9. Prior Physical information
  10. One dimension.
  11. Two dimension and above.
  12. Methods
  13. Definition of PI-TFPONet.
  14. Network Architecture.
  15. Loss Function.
  16. ...and 27 more sections

Key Result

Theorem 1

The error estimate holds with a constant $C$ independent of $h$, $f$, $g_D$, and $g_N$.

Figures (15)

  • Figure 1: A sketch of the domain $\Omega$ and interface $\Gamma$ from wu2024solving. Here $\Gamma$ divide $\Omega$ into two disjoint subdomains $\Omega_1,\Omega_2$.
  • Figure 2: Architecture of PI-TFPNet.
  • Figure 3: Illustration of the point sets $X^C$, $X^B$, and $X^J$.
  • Figure 4: [1D smooth] (a)PI-TFPONet’s refinement prediction. (b)Error distribution across 200 test examples. Solid line: median error, shaded area: min to max error range.
  • Figure 5: [1D singular] (a)PI-TFPONet’s refinement prediction. (b)Error distribution across 200 test examples. Solid line: median error, shaded area: min to max error range.
  • ...and 10 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof