The recent developments in the research area have significantly advanced numerical methods for complex models, particularly in the context of wave turbulence theory and phase field equations. There is a notable trend towards the development of more robust and adaptive numerical schemes that ensure energy conservation and stability, even in the absence of global Lipschitz continuity. These advancements are crucial for accurately capturing long-term asymptotic behavior and for optimizing energy dissipation rates. Additionally, hybrid methods that combine different numerical techniques are being explored to address the limitations of individual methods, offering a more versatile approach to solving a wide range of partial differential equations. Overall, the field is moving towards more unified and efficient frameworks that enhance the accuracy and adaptability of numerical solutions.