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The discrete inverse conductivity problem solved by the weights of an interpretable neural network

Elena Beretta, Maolin Deng, Alberto Gandolfi, Bangti Jin

TL;DR

This work tackles the discrete inverse conductivity problem on a square lattice by designing a three-layer neural network whose second-layer weights directly encode edge conductivities, while the first layer approximates the discrete Green's kernel. The model is trained using Dirichlet-to-Neumann data and imposes symmetry and conservation constraints to render the learned conductivities interpretable. For noiseless data, the framework guarantees that optimal minima yield the true conductivities, and with noise or partial DtN data the method remains robust, often outperforming the Curtis–Morrow algorithm. A sensitivity analysis based on the Jacobian of the DtN map supports stability insights and clarifies the impact of partial data on reconstruction accuracy, suggesting practical applicability to nonlinear inverse problems with interpretable ML components.

Abstract

In this work, we develop a novel neural network (NN) approach to solve the discrete inverse conductivity problem of recovering the conductivity profile on network edges from the discrete Dirichlet-to-Neumann map on a square lattice. The novelty of the approach lies in the fact that the sought-after conductivity is not provided directly as the output of the NN but is instead encoded in the weights of the post-trainig NN in the second layer. Hence the weights of the trained NN acquire a clear physical meaning, which contrasts with most existing neural network approaches, where the weights are typically not interpretable. This work represents a step toward designing NNs with interpretable post-training weights. Numerically, we observe that the method outperforms the conventional Curtis-Morrow algorithm for both noisy full and partial data.

The discrete inverse conductivity problem solved by the weights of an interpretable neural network

TL;DR

This work tackles the discrete inverse conductivity problem on a square lattice by designing a three-layer neural network whose second-layer weights directly encode edge conductivities, while the first layer approximates the discrete Green's kernel. The model is trained using Dirichlet-to-Neumann data and imposes symmetry and conservation constraints to render the learned conductivities interpretable. For noiseless data, the framework guarantees that optimal minima yield the true conductivities, and with noise or partial DtN data the method remains robust, often outperforming the Curtis–Morrow algorithm. A sensitivity analysis based on the Jacobian of the DtN map supports stability insights and clarifies the impact of partial data on reconstruction accuracy, suggesting practical applicability to nonlinear inverse problems with interpretable ML components.

Abstract

In this work, we develop a novel neural network (NN) approach to solve the discrete inverse conductivity problem of recovering the conductivity profile on network edges from the discrete Dirichlet-to-Neumann map on a square lattice. The novelty of the approach lies in the fact that the sought-after conductivity is not provided directly as the output of the NN but is instead encoded in the weights of the post-trainig NN in the second layer. Hence the weights of the trained NN acquire a clear physical meaning, which contrasts with most existing neural network approaches, where the weights are typically not interpretable. This work represents a step toward designing NNs with interpretable post-training weights. Numerically, we observe that the method outperforms the conventional Curtis-Morrow algorithm for both noisy full and partial data.
Paper Structure (12 sections, 5 theorems, 46 equations, 12 figures, 1 table)

This paper contains 12 sections, 5 theorems, 46 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. The discrete conductivity problem
  3. The neural network architecture
  4. Optimal weights for noiseless data
  5. Numerical experiments and discussions
  6. Data generation and training
  7. Numerical results and discussions
  8. Neural network approach
  9. Comparison with Curtis-Morrow algorithm
  10. Sensitivity analysis
  11. Conclusions
  12. Proof of Proposition \ref{['prop:Jacobian']}

Key Result

Theorem 4.1

Fix $\alpha>0$. Given a conductivity profile $\bm{\gamma}=[\gamma_{\{p,q\}}]_{\{p,q\}\in B}$, consider $m \geq 3n$ Cauchy data pairs solving problem EQ1 for the given ${\bm\gamma}$ such that $\{\overline{\bf u}(k)\}_{k=1, \dots, m}$ contain a basis of the vector space of the subspace of $\mathbb R^{4n}$ generated by the last $3n$ coordinates. Next, consider the FNN in Eq2 and Eq3. Then for the in

Figures (12)

  • Figure 1: Schematic illustrations of (a) the network and (b) partial DtN data.
  • Figure 2: The architecture of the proposed FNN.
  • Figure 3: The training dynamics of the algorithm for (a) exact data with $n=6,8,10,12$, $\alpha=1$, random initial conductivity; and (b) noisy data at different noise level with $n=10$, $\alpha=1$. It takes 7.29s, 13.5s, 18.8s and 27.6s for per $10^4$ iterations when $n=6,8,10$ and $12$ for full data, respectively.
  • Figure 4: The recovered conductivity for exact data in the case $n=6,8,10,12$ (top), and pointwise error $\log_{10}|\bm e_{\bm\gamma}|$ (bottom)
  • Figure 5: The recovered conductivity $\widehat{\bm\gamma}$ (top), and the log error $\log_{10} |e_{\bm\gamma}|$ (bottom) at three noise levels. The maximum error $\|\bm e_{\bm\gamma}\|_\infty$ is 0.077, 0.12 and 0.47 for $\epsilon=0.001\%$, $0.01\%$ and $0.1\%$, respectively.
  • ...and 7 more figures

Theorems & Definitions (13)

  • Remark 3.1
  • Remark 3.2
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • Remark 4.1
  • Remark 5.1
  • Proposition 6.1
  • Lemma 6.1
  • proof
  • ...and 3 more