A Neural JKO Scheme for Hellinger-Kantorovich Gradient Flows via Monge-Growth Pairs
This research develops a mesh-free neural scheme to solve advection-reaction-diffusion equations using a gradient flow structure within Hellinger-Kantorovich geometry. The method unifies spatial transport and mass modification into a single neural network step. It provides rigorous guarantees on minimizer existence, density bounds, and error control via primal-dual gaps, linking discrete optimization to physical energy dissipation.