The field of numerical methods for partial differential equations (PDEs) is moving towards the development of more accurate and stable methods for solving inverse problems and PDEs with complex boundary conditions. Researchers are exploring new approaches, such as probabilistic numerical methods and shifted boundary methods, to improve the convergence and stability of existing methods. Additionally, there is a growing interest in using neural networks to solve PDEs, with a focus on improving gradient fidelity and overall solution accuracy. Notable papers in this area include: A probabilistic approach to spectral analysis of Cauchy-type inverse problems, which provides a comprehensive convergence and stability analysis of probabilistic numerical methods. SSBE-PINN, which proposes a novel method for improving the stability and accuracy of physics-informed neural networks for solving PDEs.