Solving inverse problems with sparse noisy data, operator splitting and physics-constraints machine learning

Abstract Using measurable data as a starting point, inverse problems can be used to estimate model parameters. In science and engineering, several problems can be viewed as inverse problems. Partial differential equations (PDEs) or variational problems are also used to characterize similar issues. Typically, an energy function is used to address a variational problem. Because curvature-driven regularities have been shown to require a lot of prior understanding of physics concepts, they have received a lot of attention. Unfortunately, curvature-driven regularities correlate with the higher-order Euler-Lagrangian equations and frequently have nonsmooth and nonconvex features, making numerical solutions a difficult challenge in a variety of applications. Thus, we introduce a method based on physics-constrained deep learning based on automatic differentiation (AD) to manage inverse problems from noisy data. Additionally, to address this challenge, we combine standard variational approaches with deep learning-based algorithms. The operator split technique can successfully break nonconvex variational models into multiple simple subproblems to solve. Each subproblem corresponds to an Euler-Lagrangian PDE, which is effectively solved using deep neural networks via the AD process.