[Paper Review] Conductivity Imaging from Internal Measurements with Mixed Least-Squares Deep Neural Networks

Bangti Jin, Xi‐Yao Li+2 more|arXiv (Cornell University)|Mar 29, 2023

Groundwater flow and contamination studiesEnvironmental Science3 citations

TL;DR

This paper proposes a mixed least-squares deep neural network (DNN) framework for reconstructing conductivity distributions in elliptic PDEs from full-domain internal measurements. By jointly approximating conductivity and flux via DNNs using a least-squares loss with H¹ regularization, the method achieves stable, noise-robust reconstructions with rigorous error bounds explicitly depending on data noise, network depth, width, and sampling resolution.

ABSTRACT

In this work we develop a novel approach using deep neural networks to reconstruct the conductivity distribution in elliptic problems from one measurement of the solution over the whole domain. The approach is based on a mixed reformulation of the governing equation and utilizes the standard least-squares objective, with deep neural networks as ansatz functions to approximate the conductivity and flux simultaneously. We provide a thorough analysis of the deep neural network approximations of the conductivity for both continuous and empirical losses, including rigorous error estimates that are explicit in terms of the noise level, various penalty parameters and neural network architectural parameters (depth, width and parameter bound). We also provide multiple numerical experiments in two- and multi-dimensions to illustrate distinct features of the approach, e.g., excellent stability with respect to data noise and capability of solving high-dimensional problems.

Motivation & Objective

• To develop a stable, data-driven method for reconstructing conductivity distributions from a single internal measurement of the solution in elliptic PDEs.
• To overcome limitations of classical numerical methods in high-dimensional or noisy settings by leveraging the expressivity of deep neural networks.
• To establish rigorous generalization error bounds for the conductivity reconstruction that explicitly depend on noise level, network architecture (depth, width, parameter bounds), and sampling density.
• To provide a theoretical foundation for DNN-based inverse problems by combining mixed least-squares formulation with approximation theory and statistical learning theory.

Proposed method

• Formulate the conductivity imaging problem as a mixed least-squares minimization problem involving both the conductivity $ q $ and the flux $ oldsymbol{ ho} = q\nabla u $, using a variational formulation of the PDE.
• Approximate both $ q $ and $ oldsymbol{ ho} $ using deep neural networks as ansatz functions, enabling high-dimensional and nonlinear representation of the solution manifold.
• Define a least-squares loss functional that penalizes violations of the PDE and boundary conditions, augmented with an $ H^1(\Omega) $-type penalty on $ q $ to enforce regularity.
• Minimize the empirical loss over sampled points in the domain $ \Omega $ and on the boundary $ \partial\Omega $, using Monte Carlo quadrature to approximate the integrals.
• Employ Rademacher complexity and covering number arguments to derive generalization error bounds for the DNN approximations of $ q $, $ \boldsymbol{\rho} $, and related quantities.
• Establish theoretical error estimates that are explicit in terms of data noise $ \delta $, network depth $ L $, width $ W $, parameter bounds $ R $, and sampling counts $ n $.

Experimental results

Research questions

• RQ1 Can deep neural networks effectively reconstruct conductivity distributions from a single full-domain internal measurement of the solution in elliptic PDEs?
• RQ2 How does the generalization error of the DNN-based reconstruction scale with respect to data noise, network architecture (depth, width, parameter bounds), and sampling resolution?
• RQ3 What is the theoretical stability and convergence behavior of the proposed mixed least-squares DNN framework under noisy data and finite sampling?
• RQ4 How do the theoretical error bounds derived via statistical learning and approximation theory compare with empirical performance in numerical experiments?

Key findings

• The proposed method achieves stable conductivity reconstructions even under significant data noise, demonstrating robustness in numerical experiments across 2D and multi-dimensional problems.
• Theoretical error bounds for the conductivity approximation are explicitly derived and depend polynomially on the noise level $ \delta $, network depth $ L $, width $ W $, parameter bounds $ R $, and sampling counts $ n $, with logarithmic dependencies on network size.
• The Rademacher complexity of the DNN function class is bounded by $ \mathcal{O}(n^{-1/2} R_{\sigma}^2 N_{\kappa}^2 (N_{\kappa} + N_{\theta})^{1/2} (\log R_{\sigma} + \log N_{\kappa} + \log n)^{1/2}) $, where $ N_{\kappa} $ and $ N_{\theta} $ denote the number of parameters in the flux and conductivity networks.
• The error bounds for the flux and conductivity approximations scale with $ n^{-1/2} $, indicating convergence with increasing sampling, and are influenced by the network depth and width through logarithmic and polynomial terms.
• The method achieves high accuracy in high-dimensional problems due to the expressive power of DNNs and the stability induced by the mixed least-squares formulation with $ H^1 $ regularization.
• Empirical results confirm that the method outperforms standard approaches in terms of noise resilience and scalability to higher dimensions, with error decreasing as sampling density increases.