The recent developments in the research area of numerical methods and computational techniques have shown a significant shift towards integrating machine learning with traditional computational methods. This trend is evident in the use of Physics-Informed Neural Networks (PINNs) for solving inverse problems and complex PDEs, which demonstrates a novel approach to incorporating physical laws into neural network training. Additionally, there is a growing focus on developing adaptive and robust numerical schemes that can handle multiscale and nonlinear problems efficiently. For instance, the introduction of entropy-stable neural networks and adaptive neural network basis methods highlights advancements in ensuring stability and accuracy in long-term simulations. Furthermore, the field is witnessing innovations in mesh-free and mesh-based methods, with notable improvements in computational efficiency and conservation properties. The integration of optimal transport techniques and the development of new finite element methods tailored for specific problems, such as those with variable viscosity, underscore the continued evolution and specialization of numerical methods. Notably, the use of generative models in Bayesian inference for data assimilation in geophysical fluid dynamics represents a cutting-edge application of machine learning in enhancing the predictive capabilities of numerical simulations.

Noteworthy Papers:

• The use of Physics-Informed Neural Networks for frictionless contact problems under large deformation showcases a robust framework for complex engineering applications.
• The introduction of an Asymptotic-Preserving Random Feature Method for multiscale radiative transfer equations demonstrates significant computational advantages and robustness across different scales.