The fields of numerical methods, parametric problems, metamaterials, and neural networks are experiencing significant advancements, driven by the development of innovative techniques and technologies. A common theme across these areas is the focus on improving computational efficiency, accuracy, and expressivity.
In numerical methods, high-order techniques such as the third-order finite volume semi-implicit method for the Shallow Water-Exner model are demonstrating improved stability and accuracy. Variational Multiscale methods are being applied to nonlinear problems, including the Navier-Stokes equations, to preserve high-order accuracy and desirable conservation properties. Preconditioners for linear poroelasticity and elasticity are being developed to efficiently handle large-scale saddle-point systems.
Parametric problems are witnessing a shift towards adaptive methods, combining sparse and low-rank approximations, adaptive rank approaches, and dynamical low-rank optimization. Notable papers include the introduction of a new approximation format for parametric elliptic PDEs and a study proposing an adaptive scheme for optimizing damper positions in vibrational systems.
The field of metamaterials is advancing with the development of innovative solutions to complex problems, including the design and optimization of materials with unique properties. Machine learning models are being used to capture the relationships between 3D topology, density condition, and mechanical properties. New methods, such as progressive search algorithms, are being proposed to address the challenges of high-dimensional design spaces.
Neural networks are becoming more efficient and expressive, with the use of tensor decompositions, low-rank adaptations, and subspace-based methods. Novel frameworks such as NeuronSeek and TensorGuide have demonstrated enhanced stability and expressivity, while methods like Subspace Boosting and PLoP have improved model merging and adapter placement.
Overall, these advancements have the potential to significantly impact various fields, including engineering, geophysics, and biology, by providing more accurate and efficient numerical simulations of complex systems, reducing computational costs, and enabling the development of more efficient and effective devices.
