Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamics

What the math gives to ML

The paper develops a constructive modulation analysis for a nonlinear parabolic flow near a family of shrinking desingularizations. Its transferable asset is the explicit separation of a rapidly evolving scale parameter from a perturbation, together with weighted norms that prevent the perturbation from overwhelming the shrinking core and spectral-rate conditions that determine admissible damping. A neural analogue is to give a scale-like network degree of freedom its own coordinate and update rule, while controlling the orthogonal residual in a geometry-aware weighted norm rather than letting ordinary Euclidean optimization mix the two dynamics. This is most plausibly useful for scale-sensitive residual networks, neural ODEs, or low-rank models where parameter norm and function scale evolve on very different time scales.