Nonlinear Diffusion Equations: Full characterization of Entropies

What the math gives to ML

The paper characterizes Lyapunov entropies for nonlinear Fokker–Planck flows through the chemical potential \(\phi(u)+V\), where the diffusion nonlinearity determines the entropy geometry and uniform convexity of \(V\) drives exponential relaxation. This suggests replacing the usual quadratic or KL objective in neural density samplers with a mobility-matched Bregman entropy whose gradient flow has an explicit equilibrium and dissipation identity. The most practical transfer is a nonlinear diffusion sampler or density-model regularizer: choose \(P\), construct \(\phi\), and train a neural velocity or score field to follow the corresponding entropy flow, testing whether it mixes faster or remains stable on concentrated target distributions.