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BIAN: A Deep Learning Method to Solve Inverse Problems Using Only Boundary Information

Feng Chen, Kegan Li, Yiran Meng, Zhiyi Xiao, Pengqi Wu

TL;DR

This work introduces BIAN, a boundary-informed neural network framework for solving coefficient identification inverse problems in PDEs using only boundary information. By reformulating indeterminate coefficients as an equivalent source term and enforcing energy conservation via Green's theorem, BIAN links boundary energy flux to interior distributions without requiring internal measurements. A collaborative trio of networks—the approximator for the medium, the generator for the solution, and the discriminator for accuracy—learns the unknown coefficients and PDE fields, with a convergence theory showing data-efficient improvements as boundary data grow. Numerical results on Poisson, Laplace, piecewise uniform media, and a high-dimensional 3D diffusion problem demonstrate superior accuracy and faster convergence compared with PINN, DRM, and WAN, highlighting practical impact for boundary-rich inverse problems where internal data are unavailable.

Abstract

Over the past years, inverse problems in partial differential equations have garnered increasing interest among scientists and engineers. However, due to the lack of conventional stability, nonlinearity and non-convexity, these problems are quite challenging and difficult to solve. In this work, we propose a new kind of neural network to solve the coefficient identification problems with only the boundary information. In this work, three networks has been utilized as an approximator, a generator and a discriminator, respectively. This method is particularly useful in scenarios where the coefficients of interest have a complicated structure or are difficult to represent with traditional models. Comparative analysis against traditional coefficient estimation techniques demonstrates the superiority of our approach, not only handling highdimensional data and complex coefficient distributions adeptly by incorporating neural networks but also eliminating the necessity for extensive internal information due to the relationship between the energy distribution within the domain to the energy flux on the boundary. Several numerical examples have been presented to substantiate the merits of this algorithm including solving the Poisson equation and Helmholtz equation with spatially varying and piecewise uniform medium.

BIAN: A Deep Learning Method to Solve Inverse Problems Using Only Boundary Information

TL;DR

This work introduces BIAN, a boundary-informed neural network framework for solving coefficient identification inverse problems in PDEs using only boundary information. By reformulating indeterminate coefficients as an equivalent source term and enforcing energy conservation via Green's theorem, BIAN links boundary energy flux to interior distributions without requiring internal measurements. A collaborative trio of networks—the approximator for the medium, the generator for the solution, and the discriminator for accuracy—learns the unknown coefficients and PDE fields, with a convergence theory showing data-efficient improvements as boundary data grow. Numerical results on Poisson, Laplace, piecewise uniform media, and a high-dimensional 3D diffusion problem demonstrate superior accuracy and faster convergence compared with PINN, DRM, and WAN, highlighting practical impact for boundary-rich inverse problems where internal data are unavailable.

Abstract

Over the past years, inverse problems in partial differential equations have garnered increasing interest among scientists and engineers. However, due to the lack of conventional stability, nonlinearity and non-convexity, these problems are quite challenging and difficult to solve. In this work, we propose a new kind of neural network to solve the coefficient identification problems with only the boundary information. In this work, three networks has been utilized as an approximator, a generator and a discriminator, respectively. This method is particularly useful in scenarios where the coefficients of interest have a complicated structure or are difficult to represent with traditional models. Comparative analysis against traditional coefficient estimation techniques demonstrates the superiority of our approach, not only handling highdimensional data and complex coefficient distributions adeptly by incorporating neural networks but also eliminating the necessity for extensive internal information due to the relationship between the energy distribution within the domain to the energy flux on the boundary. Several numerical examples have been presented to substantiate the merits of this algorithm including solving the Poisson equation and Helmholtz equation with spatially varying and piecewise uniform medium.
Paper Structure (12 sections, 3 theorems, 45 equations, 16 figures, 1 table, 1 algorithm)

This paper contains 12 sections, 3 theorems, 45 equations, 16 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Algorithm
  3. Mathematical Model of coefficient Identification Problem
  4. Mathematical Principles underlying BIAN
  5. The convergence analysis of BIAN
  6. Numerical Examples
  7. Evaluation of Computational Feasibility
  8. The convergence analysis
  9. Dealing with Piecewise uniform medium distribution
  10. Dealing with high-dimensional problem
  11. Conclusion
  12. Proof of Theorem 1

Key Result

Theorem 1

Suppose Assumption 3.1 holds. Let $m_b$ and $m_i$ be the number of iid samples from $\mu_b$ and $\mu_i$, respectively. With probability at least, $(1-\sqrt{m_{b}}(1-1/\sqrt{m_{b}})^{m_{b}})(1-\sqrt{m_{i}}(1-1/\sqrt{m_{i}})^{m_{i}})$, we have here $C'_1=3 \frac{C_{b}C_{i}}{c_b c_i}m_b^{0.5} m_i^{0.5} \sqrt d ^{2d-1}$ and $C_{max} = max\{ 2[n]^2_{a,U}d^{a+0.5d-0.5} c_b^{-\frac{2a+d-1}{d-1}}\\, 2[z]

Figures (16)

  • Figure 1: The structure of the neural network used in this work. Two residual blocks and two extra fully connected layers are employed.
  • Figure 2: The relationships between the three neural networks.
  • Figure 3: The solution and the medium distribution of the Laplace problem.
  • Figure 4: The solution and the medium distribution of the Laplace problem obtained from WAN.
  • Figure 5: The solution and the medium distribution of the Laplace problem obtained from DRM.
  • ...and 11 more figures

Theorems & Definitions (7)

  • Definition 1: Hölder Continuity
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof