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Imaging Anisotropic Conductivity from Internal Measurements with Mixed Least-Squares Deep Neural Networks

Siyu Cen, Bangti Jin, Xiyao Li, Zhi Zhou

TL;DR

This work introduces MLS-DNN, a mixed least-squares deep neural network framework to recover anisotropic conductivity tensors $A(x)$ from internal measurements of solutions to elliptic PDEs, using a dual representation with fluxes $\sigma=A\nabla u$. The method solves a residual-minimization problem for the pair $(A,\sigma)$ under a projection $P_{\mathcal{K}}$ to enforce admissible conductivities, and provides rigorous error bounds for both population and empirical losses via canonical source-type conditions and statistical learning tools. The authors demonstrate 2D and 3D reconstructions with internal data subject to noise up to $10\%$, including partial data scenarios, and compare against Galerkin FEM and PINN baselines, highlighting robustness, mesh-free scalability, and applicability to challenging inverse problems. Overall, the MLS-DNN framework delivers provable convergence properties and practical accuracy for recovering anisotropic conductivities in both Neumann and Dirichlet settings, with clear potential for high-dimensional PDE inverse imaging.

Abstract

In this work we develop a novel algorithm, termed as mixed least-squares deep neural network (MLS-DNN), to recover an anisotropic conductivity tensor from the internal measurements of the solutions. It is based on applying the least-squares formulation to the mixed form of the elliptic problem, and approximating the internal flux and conductivity tensor simultaneously using deep neural networks. We provide error bounds on the approximations obtained via both population and empirical losses. The analysis relies on the canonical source condition, approximation theory of deep neural networks and statistical learning theory. We also present multiple numerical experiments to illustrate the performance of the method, and conduct a comparative study with the standard Galerkin finite element method and physics informed neural network. The results indicate that the method can accurately recover the anisotropic conductivity in both two- and three-dimensional cases, up to 10\% noise in the data.

Imaging Anisotropic Conductivity from Internal Measurements with Mixed Least-Squares Deep Neural Networks

TL;DR

This work introduces MLS-DNN, a mixed least-squares deep neural network framework to recover anisotropic conductivity tensors from internal measurements of solutions to elliptic PDEs, using a dual representation with fluxes . The method solves a residual-minimization problem for the pair under a projection to enforce admissible conductivities, and provides rigorous error bounds for both population and empirical losses via canonical source-type conditions and statistical learning tools. The authors demonstrate 2D and 3D reconstructions with internal data subject to noise up to , including partial data scenarios, and compare against Galerkin FEM and PINN baselines, highlighting robustness, mesh-free scalability, and applicability to challenging inverse problems. Overall, the MLS-DNN framework delivers provable convergence properties and practical accuracy for recovering anisotropic conductivities in both Neumann and Dirichlet settings, with clear potential for high-dimensional PDE inverse imaging.

Abstract

In this work we develop a novel algorithm, termed as mixed least-squares deep neural network (MLS-DNN), to recover an anisotropic conductivity tensor from the internal measurements of the solutions. It is based on applying the least-squares formulation to the mixed form of the elliptic problem, and approximating the internal flux and conductivity tensor simultaneously using deep neural networks. We provide error bounds on the approximations obtained via both population and empirical losses. The analysis relies on the canonical source condition, approximation theory of deep neural networks and statistical learning theory. We also present multiple numerical experiments to illustrate the performance of the method, and conduct a comparative study with the standard Galerkin finite element method and physics informed neural network. The results indicate that the method can accurately recover the anisotropic conductivity in both two- and three-dimensional cases, up to 10\% noise in the data.

Paper Structure

This paper contains 11 sections, 5 theorems, 59 equations, 18 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Reconstruction algorithm: mixed least-squares DNN
  3. Deep neural networks
  4. MLS-DNN in the Neumann case
  5. MLS-DNN in the Dirichlet case
  6. Error analysis
  7. Error estimate for the population loss
  8. Error estimate for the empirical loss
  9. Numerical experiments and discussions
  10. The Neumann problem
  11. The Dirichlet problem

Key Result

Lemma 2.1

Let $s\in\mathbb{N}\cup\{0\}$ and $p\in[1,\infty]$ be fixed, and $v\in W^{k,p} (\Omega)$ with $\mathbb{N}\ni k\geq s+1$. Then for any $\epsilon>0$, there exists at least one $\theta\in\Theta$ with depth $O(\log(d+k))$ and number of nonzero weights $O (\epsilon^{-\frac{d}{k-s-\mu (s=2)}})$ and weight

Figures (18)

  • Figure 1: The reconstructions for Example \ref{['exam:neu2d1']} with exact data in (b) and noisy data $(\delta=10\%)$ in (d). From the top to bottom, the results are for $A_{11}$, $A_{12}$ and $A_{22}$, respectively.
  • Figure 2: The evolution of the loss $($top$)$ and the reconstruction error $e(\hat{A})$$($bottom$)$ during the training process for Example \ref{['exam:neu2d1']} at different noise levels.
  • Figure 3: The reconstructions for Example \ref{['exam:neu2d2']} with exact data in (b) and noisy data $(\delta=10\%)$ in (d). From the top to bottom, the results are for $A_{11}$, $A_{12}$ and $A_{22}$, respectively.
  • Figure 4: The reconstructions for Example \ref{['exam:neu2d3']} with exact data in (b) and noisy data $(\delta=10\%)$ in (d). From the top to bottom, the results are for $A_{11}$, $A_{12}$ and $A_{22}$, respectively.
  • Figure 5: The reconstructions for Example \ref{['exam:neu2d3']} with exact data in (b) and noisy data $(\delta=10\%)$ in (d) using the FEM. From the top to bottom, the results are for $A_{11}$, $A_{12}$ and $A_{22}$, respectively.
  • ...and 13 more figures

Theorems & Definitions (21)

  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Lemma 3.2
  • Theorem 3.2
  • proof
  • Example 4.1
  • ...and 11 more