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Solving Partial Differential Equations in Different Domains by Operator Learning method Based on Boundary Integral Equations

Bin Meng, Yutong Lu, Ying Jiang

TL;DR

This work addresses solving PDEs on arbitrary domains without retraining by integrating boundary integral equations with neural operator learning. It proposes two models, BI-DeepONet and BI-TDONet, that use boundary data and boundary-integral representations to learn operators mapping boundary information to PDE solutions across diverse geometries. BI-TDONet leverages a singular value expansion-inspired architecture and employs trigonometric coefficients to capture oscillatory signals, achieving faster convergence and lower error than BI-DeepONet in multiple test cases. Across LBVPs, potential flow, elastostatics, and acoustic scattering, the BI-TDONet framework provides accurate solutions with very fast per-sample inference, highlighting the practical potential of boundary-focused neural operators for domain-general PDE solving. The approach lays groundwork for scalable, fast, and geometry-robust PDE solvers with prospects for 3D extensions and nonlinear PDE applications.

Abstract

This article explores operator learning models that can deduce solutions to partial differential equations (PDEs) on arbitrary domains without requiring retraining. We introduce two innovative models rooted in boundary integral equations (BIEs): the Boundary Integral Type Deep Operator Network (BI-DeepONet) and the Boundary Integral Trigonometric Deep Operator Neural Network (BI-TDONet), which are crafted to address PDEs across diverse domains. Once fully trained, these BIE-based models adeptly predict the solutions of PDEs in any domain without the need for additional training. BI-TDONet notably enhances its performance by employing the singular value decomposition (SVD) of bounded linear operators, allowing for the efficient distribution of input functions across its modules. Furthermore, to tackle the issue of function sampling values that do not effectively capture oscillatory and impulse signal characteristics, trigonometric coefficients are utilized as both inputs and outputs in BI-TDONet. Our numerical experiments robustly support and confirm the efficacy of this theoretical framework.

Solving Partial Differential Equations in Different Domains by Operator Learning method Based on Boundary Integral Equations

TL;DR

This work addresses solving PDEs on arbitrary domains without retraining by integrating boundary integral equations with neural operator learning. It proposes two models, BI-DeepONet and BI-TDONet, that use boundary data and boundary-integral representations to learn operators mapping boundary information to PDE solutions across diverse geometries. BI-TDONet leverages a singular value expansion-inspired architecture and employs trigonometric coefficients to capture oscillatory signals, achieving faster convergence and lower error than BI-DeepONet in multiple test cases. Across LBVPs, potential flow, elastostatics, and acoustic scattering, the BI-TDONet framework provides accurate solutions with very fast per-sample inference, highlighting the practical potential of boundary-focused neural operators for domain-general PDE solving. The approach lays groundwork for scalable, fast, and geometry-robust PDE solvers with prospects for 3D extensions and nonlinear PDE applications.

Abstract

This article explores operator learning models that can deduce solutions to partial differential equations (PDEs) on arbitrary domains without requiring retraining. We introduce two innovative models rooted in boundary integral equations (BIEs): the Boundary Integral Type Deep Operator Network (BI-DeepONet) and the Boundary Integral Trigonometric Deep Operator Neural Network (BI-TDONet), which are crafted to address PDEs across diverse domains. Once fully trained, these BIE-based models adeptly predict the solutions of PDEs in any domain without the need for additional training. BI-TDONet notably enhances its performance by employing the singular value decomposition (SVD) of bounded linear operators, allowing for the efficient distribution of input functions across its modules. Furthermore, to tackle the issue of function sampling values that do not effectively capture oscillatory and impulse signal characteristics, trigonometric coefficients are utilized as both inputs and outputs in BI-TDONet. Our numerical experiments robustly support and confirm the efficacy of this theoretical framework.
Paper Structure (12 sections, 1 theorem, 49 equations, 18 figures, 10 tables)

This paper contains 12 sections, 1 theorem, 49 equations, 18 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Optimizing Operator Learning Frameworks Through the Integration of Boundary Integral Equations
  4. Boundary integral type DeepOnet (BI-DeepONet)
  5. Boundary Integral Trigonometric Deep Operator Network (BI-TDONet)
  6. Generate datasets
  7. Experiments
  8. LBVPs
  9. Potential flow
  10. Elastostatic problems
  11. Obstacle scattering problem for acoustic waves
  12. Conclusion

Key Result

Lemma 1

Let $X$ and $Y$ be real separable Hilbert spaces and let $T : X \to Y$ be a bounded linear operator. Then there exist a Borel space $(M,\mathcal{A}, \mu)$, isometries $V:L^2(M,\mathcal{A}, \mu)\to X$, $U:L^2(M,\mathcal{A}, \mu)\to Y$, and an essentially bounded measurable function $\sigma:M \to \mat where $V^{-1}$ is the generalized inverse of $V$ and $m_{\sigma}$ is the multiplication operator de

Figures (18)

  • Figure 1: The network structure of BI-DeepONet.
  • Figure 2: The network structure of BI-TDONet.
  • Figure 3: Partial generated boundary data. (a) and (b) are generated by with $\rho=0.1$ and $\rho=0.4$, respectively. (c) generated by with $\rho=0.1$ and limiting maximum curvature to less than 10.
  • Figure 4: (a) The diversity of boundaries in the boundary dataset. (b) The function $\varphi_n$ generated with $m=5$.
  • Figure 5: The logarithmic loss trajectories for BI-DeepONet and BI-TDONet.
  • ...and 13 more figures

Theorems & Definitions (1)

  • Lemma