Phase field models provide a mathematical framework for describing interface evolution in multi-phase systems by replacing sharp boundaries with smooth transition regions characterized by order parameters. The classical Ginzburg-Landau functional serves as the foundation, incorporating bulk energy and interfacial penalties through gradient terms, naturally handling complex topologies without explicit interface tracking. This makes them also interesting as tools in data analysis, e.g., by providing means to compute minimal cuts on weighted graphs.

 

A central theoretical question concerns the sharp interface limit: how do phase field models behave as interface thickness approaches zero? We briefly examine rigorous convergence analysis showing that phase field equations recover classical sharp interface models.

 

From a computational perspective, convex-concave splitting gradient descent algorithms are interesting due to their unconditional energy stability. We show that they, however, effectively correspond to a limiting of the time step size. More recent advances introduce momentum-based minimization schemes for the Ginzburg-Landau functional. We discuss FISTA-type discretization schemes and their application to phase field problems, including semi-supervised machine learning tasks where these methods provide natural frameworks for graph-based classification.