Mesh-Free Unsupervised Learning-Based PDE Solver of Forward and Inverse problems

Leah Bar, Nir Sochen

Keywords: PDEs, forward problems, inverse problems, unsupervised learning, deep networks, EIT

TL;DR: Solving PDEs with deep learning techniques in an unsupervised fashion with regularizers for forward and inverse problems.

Abstract:

We introduce a novel neural network-based partial differential equations solver for forward and inverse problems. The solver is grid free, mesh free and shape free, and the solution is approximated by a neural network.
We employ an unsupervised approach such that the input to the network is a points set in an arbitrary domain, and the output is the
set of the corresponding function values. The network is trained to minimize deviations of the learned function from the PDE solution and
satisfy the boundary conditions.
The resulting solution in turn is an explicit smooth differentiable function with a known analytical form.

Unlike other numerical methods such as finite differences and finite elements, the derivatives of the desired function can be analytically calculated to any order. This framework therefore, enables the solution of high order non-linear PDEs. The proposed algorithm is a unified formulation of both forward and inverse problems
where the optimized loss function consists of few elements: fidelity terms of L2 and L infinity norms, boundary conditions constraints and additional regularizers. This setting is flexible in the sense that regularizers can be tailored to specific
problems. We demonstrate our method on a free shape 2D second order elliptical system with application to Electrical Impedance Tomography (EIT).