The recent developments in the research area of numerical methods for partial differential equations (PDEs) have shown a significant shift towards integrating deep learning techniques with traditional numerical methods. This integration aims to leverage the strengths of both approaches, enhancing the accuracy and efficiency of solving complex PDEs. One notable trend is the development of intrinsic methods for enforcing boundary conditions in deep neural networks, which addresses the limitations of traditional penalty-type methods. These new approaches promise to simplify the optimization process and improve robustness. Additionally, there is a growing interest in geometry-aware solvers and preconditioners, which adapt to different problem geometries without requiring extensive retraining, demonstrating a robust and versatile solution for various PDEs. Furthermore, the field is witnessing advancements in numerical homogenization, where deep learning is used to correct coarse-scale approximations, particularly for problems with highly oscillatory coefficients. These innovations collectively push the boundaries of what is achievable in computational mathematics, offering new tools for tackling the complexities inherent in PDEs.

Noteworthy papers include one that introduces a novel intrinsic approach to impose essential boundary conditions in deep neural networks, and another that presents a geometry-aware preconditioner for linear PDEs, which remains robust across different geometries without additional fine-tuning.