Learning Inverse Problem from Sparse Noisy Data with Operator Split and Physics-Constraints Machine Learning

Abstract Inverse problems (IPs) begin with measured data and try to estimate the model parameters. In science and engineering, many problems are seen as inverse problems. Partial differential equations (PDEs) or variational problems are also used to characterize similar issues (VPs). A VP is usually an energy functional that is solved by lowering the energy function. Since curvature-driven regularities have been proven to need considerable prior understanding of physics, they have gotten a lot of attention. Unfortunately, the curvaturedriven regularities correlate to the higher-order EulerLagrangian equations. Furthermore, they frequently have non-smooth and non-convex features, making numerical solutions a difficult challenge in a variety of applications. In this paper (AD), we introduced a method based on physics-constrained deep learning (PCL) and automatic differentiation to handle inverse issues from noisy data. In addition, to address this challenge, we combine standard variational approaches (VMs) with DL-based algorithms. The operator split technique may successfully break non-convex variational models into multiple simple sub-problems to solve. Each sub-problem corresponds to an Euler-Lagrangian PDE, which is effectively solved using deep neural networks (DNNs) via the AD process.