Frontiers in multiscale computations

Multiscale problems, characterized by phenomena spanning multiple spatial and temporal scales, are central to many areas of science and engineering. Typically modeled by systems of partial differential equations (PDEs), these problems challenge conventional numerical methods, which must capture fine- and coarse-scale interactions simultaneously at high computational cost. Recent advances in multiscale techniques have greatly improved efficiency and robustness, especially in settings without clear scale separation or obvious structural invariants, expanding their range of applications.

Key open challenges remain. Integrating multiscale methods with machine learning and quantum computing offers exciting opportunities: operator compression could accelerate simulations by approximating models across scales, while deep learning may uncover hidden coarse variables and latent dynamics. Adapting multiscale techniques to quantum architectures represents an emerging frontier. Another crucial direction is extending these methods from well-structured, linear problems to highly nonlinear, coupled PDE systems such as those in quantum and plasma physics, where resolving interactions across fundamentally different scales is essential.