The field of numerical methods for partial differential equations is witnessing significant developments, with a focus on improving the accuracy, stability, and efficiency of existing methods. Researchers are exploring new techniques, such as energy-stable schemes and weak random feature methods, to tackle complex problems like mean curvature flow and surface diffusion. These innovations have the potential to advance our understanding of various physical phenomena and improve the simulation of real-world systems. Notably, the introduction of patch bubbles for advection-dominated problems and the enhancement of the random feature method for solving partial differential equations are noteworthy contributions. Specifically, the proposal of the BGN-MDR method for simulating mean curvature flow and surface diffusion, and the development of the Weak RFM for finding weak solutions to partial differential equations, are particularly noteworthy for their potential to improve the accuracy and efficiency of numerical simulations.