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

TL;DR

This paper introduces an unsupervised deep learning method using neural networks to solve forward and inverse PDE problems, including applications like Electrical Impedance Tomography, without requiring mesh-based discretization.

Contribution

It presents a mesh-free neural network approach that enforces PDEs and boundary conditions through a cost function, applicable to arbitrary domains and complex PDE systems.

Findings

Effective in solving 2D elliptical PDEs with non-constant coefficients

Applicable to inverse problems like Electrical Impedance Tomography

Mesh-free approach simplifies complex domain handling

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.