Gradient flows I

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Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusion (2023, methodology): “We can similarly derive Thus, Proposition 8.1.8 of (Ambrosio et al., 2005) implies that for p T -a.e. x , the following reverse time ODE (which is the Eq.”
On a class of first order Hamilton–Jacobi equations in metric spaces (2014, methodology): “Application to a compressible Euler equation Let X := P2(R) be the space of probability measures on R with finite second moment, and let d(ρ, γ) denote the 2-Wasserstein so that, (X, d) is a complete and geodesic metric space (chapter 7 of [2]).”
Action Matching: A Variational Method for Learning Stochastic Dynamics from Samples (2022, methodology): “Kinetic Energy From Ambrosio et al. (2008), we know that ∇ s ∗ t ( x ) , deﬁned in Theorem 1, minimizes the kinetic energy functional K ( v t ) , deﬁned below, along where the optimization is over all v t satisfying the continuity equation with q t .”
Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss (2020, positive): “• (Divergence form) It can be shown that a Wasserstein gradient ﬂow as deﬁned in Deﬁnition 2.1 satisﬁes, in the sense of distributions, the following partial diﬀerential equation (Ambrosio et al., 2008) ∂ t µ t = − div( ∇ F µ t µ t ) .”