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Unsupervised Deep Learning Algorithm for PDE-based Forward and Inverse Problems

Leah Bar, Nir Sochen

TL;DR

The paper presents an unsupervised, neural-network-based framework for solving both forward and inverse PDE problems by directly approximating the solution and coefficients with neural nets and training them to satisfy the PDE and boundary conditions. It introduces a mixed fidelity loss that combines $L_2$ residuals with an $L_\infty$-like term, along with regularizers, enabling accurate, mesh-free solutions on arbitrary domains. The approach is demonstrated on a 2D divergence-form elliptic PDE, with Electrical Impedance Tomography as a concrete application, showing improved edge fidelity and robust conductivity reconstruction. This framework provides a flexible, unsupervised tool for PDE-based imaging and inverse problems, with broad potential across scientific computing and engineering.

Abstract

We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost function, and satisfies the PDE, boundary conditions, and additional regularizations. The method is mesh free and can be easily applied to an arbitrary regular domain. We focus on 2D second order elliptical system with non-constant coefficients, with application to Electrical Impedance Tomography.

Unsupervised Deep Learning Algorithm for PDE-based Forward and Inverse Problems

TL;DR

The paper presents an unsupervised, neural-network-based framework for solving both forward and inverse PDE problems by directly approximating the solution and coefficients with neural nets and training them to satisfy the PDE and boundary conditions. It introduces a mixed fidelity loss that combines residuals with an -like term, along with regularizers, enabling accurate, mesh-free solutions on arbitrary domains. The approach is demonstrated on a 2D divergence-form elliptic PDE, with Electrical Impedance Tomography as a concrete application, showing improved edge fidelity and robust conductivity reconstruction. This framework provides a flexible, unsupervised tool for PDE-based imaging and inverse problems, with broad potential across scientific computing and engineering.

Abstract

We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost function, and satisfies the PDE, boundary conditions, and additional regularizations. The method is mesh free and can be easily applied to an arbitrary regular domain. We focus on 2D second order elliptical system with non-constant coefficients, with application to Electrical Impedance Tomography.

Paper Structure

This paper contains 6 sections, 11 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Mathematical Formulation
  3. Application to Electrical Impedance Tomography
  4. Forward Problem
  5. Inverse Problem
  6. Discussion

Figures (11)

  • Figure 1: Network architecture: the point $(x,y)\in \mathbb{R}^2$ serves as an input and $u$ as the output.
  • Figure 2: Electrical current $\psi_n$ for $n=1,2,3$
  • Figure 3: The conductivity $\sigma$ of phantom $1$
  • Figure 4: Top: ground truth (FEM) of $u$ for $n=1,2,3$ given phantom $1$. Middle: reconstruction by the proposed method. MSE = $(3.15e-3,1.33e-3,6.93e-4)$, PSNR=$(37.26,36.12,35.76)$. Bottom: relative error
  • Figure 5: Top: ground truth (FEM) of $\partial u/\partial x$ for $n=1,2,3$ given phantom $1$. Middle: $\partial u/\partial x$ reconstruction by the proposed method. MSE =$(3.77e-8,3.20e-8,2.84e-8)$ PSNR = $(37.03,31.22,34.02)$. Bottom: relative error.
  • ...and 6 more figures