Learning-Based Strong Solutions to Forward and Inverse Problems in Partial Differential Equations
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 and initial 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 several free shape 2D second order systems with application to Electrical Impedance Tomography (EIT), diffusion and wave equations. Bio: Leah Bar holds B.Sc. in Physics, M.Sc. in Bio-Medical Engineering and PhD in Electrical Engineering from Tel-Aviv University. She worked as a post-doctoral fellow in the Department of Electrical Engineering at the University of Minnesota. She is currently a senior researcher at MaxQ-AI, a medical AI start-up, and in addition a researcher at the Mathematics Department in Tel-Aviv University. Her research interest are: machine learning, image processing, computer vision and variational methods.
Date: Tue 18 Feb 2020
Start Time: 11:30
End Time: 12:30
1061 | Electrical Eng. Building