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Conformal mapping Coordinates Physics-Informed Neural Networks (CoCo-PINNs): learning neural networks for designing neutral inclusions

Daehee Cho, Hyeonmin Yun, Jaeyong Lee, Mikyoung Lim

TL;DR

This work tackles the inverse problem of designing neutral inclusions in 2D with imperfect interfaces by merging conformal-mapping coordinates with Physics-Informed Neural Networks (PINNs). The authors introduce CoCo-PINNs, which use a Fourier-series representation p^{(n)} of the boundary interface and exterior conformal sampling to enforce neutrality for general-shaped inclusions; they prove that training with a single linear background field can yield neutrality for all directions under a mild condition. Empirically, CoCo-PINNs outperform classical PINNs in credibility, consistency, and stability, providing reliable forward solves and robust inverse parameter identification without additional training data. The approach is grounded in complex geometric function theory (Faber polynomials, Grunsky coefficients) and offers a path toward explainable, reliable design of neutral inclusions with arbitrary shapes in 2D, with potential extensions to multiple inclusions and three dimensions.

Abstract

We focus on designing and solving the neutral inclusion problem via neural networks. The neutral inclusion problem has a long history in the theory of composite materials, and it is exceedingly challenging to identify the precise condition that precipitates a general-shaped inclusion into a neutral inclusion. Physics-informed neural networks (PINNs) have recently become a highly successful approach to addressing both forward and inverse problems associated with partial differential equations. We found that traditional PINNs perform inadequately when applied to the inverse problem of designing neutral inclusions with arbitrary shapes. In this study, we introduce a novel approach, Conformal mapping Coordinates Physics-Informed Neural Networks (CoCo-PINNs), which integrates complex analysis techniques into PINNs. This method exhibits strong performance in solving forward-inverse problems to construct neutral inclusions of arbitrary shapes in two dimensions, where the imperfect interface condition on the inclusion's boundary is modeled by training neural networks. Notably, we mathematically prove that training with a single linear field is sufficient to achieve neutrality for untrained linear fields in arbitrary directions, given a minor assumption. We demonstrate that CoCo-PINNs offer enhanced performances in terms of credibility, consistency, and stability.

Conformal mapping Coordinates Physics-Informed Neural Networks (CoCo-PINNs): learning neural networks for designing neutral inclusions

TL;DR

This work tackles the inverse problem of designing neutral inclusions in 2D with imperfect interfaces by merging conformal-mapping coordinates with Physics-Informed Neural Networks (PINNs). The authors introduce CoCo-PINNs, which use a Fourier-series representation p^{(n)} of the boundary interface and exterior conformal sampling to enforce neutrality for general-shaped inclusions; they prove that training with a single linear background field can yield neutrality for all directions under a mild condition. Empirically, CoCo-PINNs outperform classical PINNs in credibility, consistency, and stability, providing reliable forward solves and robust inverse parameter identification without additional training data. The approach is grounded in complex geometric function theory (Faber polynomials, Grunsky coefficients) and offers a path toward explainable, reliable design of neutral inclusions with arbitrary shapes in 2D, with potential extensions to multiple inclusions and three dimensions.

Abstract

We focus on designing and solving the neutral inclusion problem via neural networks. The neutral inclusion problem has a long history in the theory of composite materials, and it is exceedingly challenging to identify the precise condition that precipitates a general-shaped inclusion into a neutral inclusion. Physics-informed neural networks (PINNs) have recently become a highly successful approach to addressing both forward and inverse problems associated with partial differential equations. We found that traditional PINNs perform inadequately when applied to the inverse problem of designing neutral inclusions with arbitrary shapes. In this study, we introduce a novel approach, Conformal mapping Coordinates Physics-Informed Neural Networks (CoCo-PINNs), which integrates complex analysis techniques into PINNs. This method exhibits strong performance in solving forward-inverse problems to construct neutral inclusions of arbitrary shapes in two dimensions, where the imperfect interface condition on the inclusion's boundary is modeled by training neural networks. Notably, we mathematically prove that training with a single linear field is sufficient to achieve neutrality for untrained linear fields in arbitrary directions, given a minor assumption. We demonstrate that CoCo-PINNs offer enhanced performances in terms of credibility, consistency, and stability.
Paper Structure (24 sections, 4 theorems, 35 equations, 12 figures, 6 tables, 1 algorithm)

This paper contains 24 sections, 4 theorems, 35 equations, 12 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Inverse problem of neutral inclusions with imperfect conditions
  3. Series solution for the governing equation via conformal mapping
  4. Series representation of the interface function
  5. The proposed method: CoCo-PINNs
  6. Model design for the forward-inverse problem
  7. Neutral inclusion effects for untrained linear fields
  8. Experiments
  9. Neutral inclusion
  10. Credibility of classical PINNs and CoCo-PINNs
  11. Consistency of classical PINNs and CoCo-PINNs
  12. Stability of classical PINNs and CoCo-PINNs
  13. Conclusion
  14. Notations
  15. Geometric function theory
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $F_m(z)$ be the Faber polynomials associated with $\Omega$ and the applied field is given by $H(z) = \hbox{Re}\left(\sum_{m=1}^\infty \alpha_m F_m(z)\right)$, the solution $u$ satisfies where $\bm{\alpha}$ is a diagonal matrix whose entries are $\alpha_m$, and $\hbox{Re}(\cdot)$ denotes the real part of a complex number. The matrices $\widetilde{A}_1$ and $\widetilde{A}_2$ are determined by t

Figures (12)

  • Figure 1.1: Inclusion makes perturbation.
  • Figure 1.2: Neutral inclusions with circular shapes.
  • Figure 3.1: Credibility scheme. $u_{\text{NN}},p_\text{NN},p^{(n)}$ are trained results by PINNs.
  • Figure 4.1: The shaped inclusions: square, fish, kite, and spike.
  • Figure 4.2: Neutral inclusion effect appeared after training. For the 'square' and 'spike' shaped inclusions, the interface function $p^{(n)}$ is separately trained using a single background solution $H(x)=x_1$ via CoCo-PINNs.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Definition 1
  • Theorem 2.1: Analytic solution formula
  • Theorem 3.1
  • Remark 1
  • Theorem 3.2
  • Remark 2
  • Theorem B.1: Riemann mapping theorem
  • Remark 3