Learning-Based Strong Solutions to Forward and Inverse Problems in PDEs

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 strong 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 that unlike previous methods promote a strong solution. Robust boundary conditions constraints and additional regularizers are included as well. This setting is flexible in the sense that regularizers can be tailored to specific
problems. We demonstrate our method on several free shape 2D second order systems with application to Electrical Impedance Tomography (EIT).