Locomotion in viscous-dominated environments, where inertial effects are negligible and the Reynolds number is small, is a central topic in engineering mathematics. In this regime, propulsion emerges from geometric and kinematic interactions between deforming bodies and the surrounding Stokes flow. Classical theoretical developments—including kinematic reversibility, reciprocal-theorem formulations, geometric descriptions of swimming strokes, and optimal control of body deformation—have established a rigorous framework for understanding microswimmer propulsion and fluid–structure interactions in highly viscous fluids. These studies have revealed how hydrodynamic forces, elasticity, and internal actuation combine to generate motion through complex yet mathematically tractable mechanisms. Many modern problems in microscale locomotion, however, involve high-dimensional control spaces and strongly coupled physical processes. Systems with flexible filaments, distributed actuation, or chemically active interfaces introduce parameter regimes that are difficult to explore systematically using purely analytical or traditional numerical approaches. Machine learning offers complementary tools for organizing and reducing such complexity. When combined with the physical structure of Stokesian hydrodynamics—such as linear force balance, geometric constraints, and viscous dissipation—data-driven methods can help identify effective propulsion strategies, reduced dynamical descriptions, and efficient computational representations. This collection invites original research and review articles that integrate modern machine learning methodologies with the mathematical principles governing low-Reynolds-number locomotion. Topics of interest include physics-informed neural networks for viscous flow problems, operator learning for Stokes solvers, sparse model discovery for swimmer dynamics, reinforcement learning for stroke optimization and control, and reduced-order representations of fluid–structure interactions. By emphasizing mathematical formulation, physical consistency, and reproducibility, the collection aims to highlight emerging frameworks that connect machine learning with the analytical structure of viscous flows and the mechanics of microscale propulsion.

Hydrogels are soft, fluid-filled solids used in diverse applications such as regenerative medicine, soft robotics, and agriculture. The rich behaviours of hydrogels are driven by a complex interplay between fluid and solid mechanics, thermodynamics, and environmental interactions. This special issue aims to showcase the wide variety of mathematical challenges that are encountered when studying hydrogels and highlight the pivotal role that mathematics can play in unlocking the full potential of these important materials.