The recent developments in the research area of computational methods for partial differential equations (PDEs) and machine learning applications have shown a significant shift towards integrating advanced learning techniques with traditional numerical methods. A common theme across several papers is the use of neural networks and operator learning to enhance the efficiency and accuracy of solving complex PDEs, particularly those with discontinuous solutions or heterogeneous coefficients. These approaches often leverage physics-informed constraints and multi-task learning to improve generalization and reduce overfitting. Notably, there is a growing interest in developing frameworks that can handle geometrical variability and real-time simulations, which are crucial for engineering applications. Additionally, the incorporation of thermodynamic principles and iterative methods within neural operators is emerging as a promising direction for improving the robustness and reliability of predictions in heterogeneous material structures. The field is also witnessing advancements in the systematic design of morphing structures and the enforcement of boundary conditions through innovative deep learning algorithms. Overall, the integration of deep learning with traditional computational methods is paving the way for more efficient, accurate, and versatile solutions to complex PDE problems.