The field of inverse problems and numerical methods is witnessing significant developments, with a focus on innovative techniques for solving complex equations and improving the accuracy of solutions. Researchers are exploring new approaches, such as the use of Carleman convexification and phase retrieval via the Wentzel-Kramers-Brillouin approximation, to address challenging inverse problems. Additionally, semi-analytical methods and perturbation solutions are being developed to tackle nonlinear boundary value problems. The Lightning Method, a recent development in numerical solutions, is being applied to solve linear PDEs with spectral accuracy. Data-driven self-supervised learning is also being used to detect singularities in solutions of partial differential equations. Noteworthy papers include: The 'Inverse scattering without phase' paper, which presents a robust numerical framework for reconstructing spatially varying dielectric constants from phaseless backscattering measurements. The 'Data-Driven Self-Supervised Learning for the Discovery of Solution Singularity for Partial Differential Equations' paper, which proposes a self-supervised learning framework for estimating the location of singularities in PDE solutions.