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

Inverse problems in partial differential equations are fundamental in science and mathematics with wide applications in medical imaging, signal processing, computer vision, remote sensing, electromagnetism and more. Classical methods such as finite differences, finite volume and finite elements are numerical discretization-based methods where the domain is divided into a uniform grid or polygon mesh. The differential equation is then reduced to a system of algebraic equations. These methods may have some limitations: the solution is numeric and may suffer from high condition number, highly dependent on the discretization and even the second derivative is sensitive to noise.