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Solving parametric elliptic interface problems via interfaced operator network

Sidi Wu, Aiqing Zhu, Yifa Tang, Benzhuo Lu

TL;DR

This paper proposes an interfaced operator network (IONet) to solve parametric elliptic interface PDEs, where different coefficients, source terms, and boundary conditions are considered as input features and tailored physics-informed loss of IONet is proposed to ensure physical consistency.

Abstract

Learning operators mapping between infinite-dimensional Banach spaces via neural networks has attracted a considerable amount of attention in recent years. In this paper, we propose an interfaced operator network (IONet) to solve parametric elliptic interface PDEs, where different coefficients, source terms, and boundary conditions are considered as input features. To capture the discontinuities in both the input functions and the output solutions across the interface, IONet divides the entire domain into several separate subdomains according to the interface and uses multiple branch nets and trunk nets. Each branch net extracts latent representations of input functions at a fixed number of sensors on a specific subdomain, and each trunk net is responsible for output solutions on one subdomain. Additionally, tailored physics-informed loss of IONet is proposed to ensure physical consistency, which greatly reduces the training dataset requirement and makes IONet effective without any paired input-output observations inside the computational domain. Extensive numerical studies demonstrate that IONet outperforms existing state-of-the-art deep operator networks in terms of accuracy and versatility.

Solving parametric elliptic interface problems via interfaced operator network

TL;DR

This paper proposes an interfaced operator network (IONet) to solve parametric elliptic interface PDEs, where different coefficients, source terms, and boundary conditions are considered as input features and tailored physics-informed loss of IONet is proposed to ensure physical consistency.

Abstract

Learning operators mapping between infinite-dimensional Banach spaces via neural networks has attracted a considerable amount of attention in recent years. In this paper, we propose an interfaced operator network (IONet) to solve parametric elliptic interface PDEs, where different coefficients, source terms, and boundary conditions are considered as input features. To capture the discontinuities in both the input functions and the output solutions across the interface, IONet divides the entire domain into several separate subdomains according to the interface and uses multiple branch nets and trunk nets. Each branch net extracts latent representations of input functions at a fixed number of sensors on a specific subdomain, and each trunk net is responsible for output solutions on one subdomain. Additionally, tailored physics-informed loss of IONet is proposed to ensure physical consistency, which greatly reduces the training dataset requirement and makes IONet effective without any paired input-output observations inside the computational domain. Extensive numerical studies demonstrate that IONet outperforms existing state-of-the-art deep operator networks in terms of accuracy and versatility.
Paper Structure (12 sections, 1 theorem, 54 equations, 15 figures, 5 tables)

This paper contains 12 sections, 1 theorem, 54 equations, 15 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Learning operators with neural networks
  3. Interfaced operator network
  4. Network architecture of IONet
  5. Loss function of IONet
  6. Numerical Results
  7. Parametric elliptic interface problems in one dimension
  8. Parametric elliptic interface problems in two dimensions
  9. Parametric elliptic interface problems in three dimensions
  10. Parametric elliptic interface problems in six dimensions
  11. Conclusions
  12. Proof of Theorem \ref{['thm:approximation of ino']}

Key Result

Theorem 1

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain, $\Omega_{i}$ with $i=1,\cdots, I-1$ be disjoint open domains and $\Omega_{I}=\Omega\setminus \bigcup_{i=1}^{I-1}\Omega_{i}$. Assume $\mathcal{G}: \bigcap_{i=1}^{I} C(\Omega_{i})\bigcap L^{\infty}(\Omega) \rightarrow X(\Omega)$ is a continuous op where $\mathcal{S}$ is the summation of all the components of a vector, and $\odot$ is the Hadamard

Figures (15)

  • Figure 1: Domain $\Omega$, its subdomains $\Omega_1$, $\Omega_2$. The interface $\Gamma$ divides $\Omega$ into two disjoint subdomains.
  • Figure 2: A schematic diagram of the IONet for solving the parametric elliptic interface problem by minimizing the physics-informed loss function. Here, the input function is the coefficient $a(\mathbf{x})$.
  • Figure 3: The mean and one standard deviation of relative $L^2$ error for PI-IONet with different weights in the physics-informed loss function \ref{['eq: empirical loss function']}. Left: $\lambda_1=1$ and $\lambda_3=\lambda_4=100$. Middle: $\lambda_2=1$ and $\lambda_3=\lambda_4=100$. Right: $\lambda_1=\lambda_2=1$.
  • Figure 4: Left column: Five input functions randomly selected from the test set (distinguished by different colors). Second and third columns: The reference solutions (solid lines) versus the numerical solutions (dashed lines) of DD-DeepONet, DD-IONet, PI-DeepONet, and PI-IONet. Fourth column: The mean and one standard deviation of the numerical solutions, averaged over these 5 test examples.
  • Figure 5: Left: Five input functions randomly selected from the test set (distinguished by different colors). Middle: Reference solutions (solid lines) versus the numerical solutions (dashed lines) of PI-IONet. Right: Absolute point-wise errors over the whole domain. The gray point-dashed lines represent the location of the interfaces.
  • ...and 10 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Remark 1
  • proof