THE VARIATIONAL FORMULATION OF THE FOKKER-PLANCK EQUATION

By R. Jordan, D. Kinderlehrer, Felix Otto. 1996

Subfields: Probabilistic & Bayesian Methods

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Citations in the corpus

Denoising Diffusion Implicit Models (2020, methodology): “Since Langevin dynamics is a discretization of a gradient ﬂow (Jordan et al., 1998), both DDPM and NCSN require many steps to achieve good sample quality.”
Wasserstein Adversarial Examples via Projected Sinkhorn Iterations (2019, methodology): “Proximal operators with respect to Wasserstein instead of Euclidean distances are used in gradient flow problems (Jordan et al., 1998; Peyré, 2015), but are not relevant to our setting.”
Improving Task-free Continual Learning by Distributionally Robust Memory Evolution (2022, background): “(9) in the space of probability measures (Jordan et al., 1998).”
A mean field view of the landscape of two-layer neural networks (2018, neutral): “sup k∈[0,T/ε]∩N ∣∣RN (θ )−R(ρkε)∣∣≤CeCT errN ,D(z ), [19]”
Mean Field Analysis of Neural Networks: A Law of Large Numbers (2018, background): “As it is well known in the literature (see, for example, [2], [10], [28]), the PDE (1.”
Computational Optimal Transport (2018, background): “Interest in Wasserstein gradient flows was sparked by the seminal paper of Jordan, Kinderlehrer and Otto [Jordan et al., 1998], and these evolutions are often called “JKO flows” following their work.”
Stochastic Gradient Descent Performs Variational Inference, Converges to Limit Cycles for Deep Networks (2017, background): “As ﬁrst discovered in the works of Jordan, Kinderleherer and Otto (Jordan et al., 1998; Otto, 2001), certain partial differential equations can be seen as coming from a variational principle, i.e., they perform steepest descent with respect to functionals of their state distribution.”
Long-Time Behaviour and Phase Transitions for the Mckean–Vlasov Equation on the Torus (2018, background): “It is well known, starting from the seminal work in [43,59], that this equation belongs to a larger class of dissipative PDEs including the heat equation, the porous medium equation, and the aggregation equation, which can be written in the form”
A New Class of Efficient and Robust Energy Stable Schemes for Gradient Flows (2017, unknown): “where G= ( ) (0 <<1) (cf. [1]). For more complicated dissipation mechanisms, G may be nonlinear and may depend on ˚. An example is the Wasserstein gradient ow for ˚>0, where G= r(˚r) (cf. [20,40]). As long as Gis non-positive, the free energy is non-increasing, (1.2) dE[˚] dt = E ˚ @˚ @t = ”
Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems (2008, unknown): “[2], [4], [12], [22], [37].”
A Family of Nonlinear Fourth Order Equations of Gradient Flow Type (2009, methodology): “Although the link of (1.1) to (1.22) is less evident than to (1.20), the weaker form (1.22) is wellsuited to our analysis for several reasons: first, (1.22) is the natural variational formulation of the gradient flow equation that is obtained by the method of Jordan, Kinderlehrer and Otto [27], see ”
Convergence of pointed non‐compact metric measure spaces and stability of Ricci curvature bounds and heat flows (2013, unknown): “Adopting the point of view of probability measures and following the approach introduced in [31] and further developed by [6], one can consider the metric gradient flow à la De Giorgi of the relative entropy functional Entm in (P2(X),W2).”
Optimal transportation with infinitely many marginals (2012, background): “This type of convexity is well known, and has been exploited in, for example, [16].”
Gradient flows of the entropy for jump processes (2012, background): “One of the most striking examples is Otto’s discovery [20,26] that many diffusion equations can be interpreted as gradient flows of a suitable free energy functional with respect to the L2-Wasserstein distance on the space of probability measures.”
The Splitting Theorem in Non-Smooth Context (2013, unknown): “– In accordance with the smooth case ([42]), the gradient flow in L2(X,m) of the generically non-quadratic - natural Dirichlet energy on (X, d,m) and the gradient flow in (P2(X),W2) of the relative entropy functional coincide ([32], [37], [5]).”
Gradient Flows: In Metric Spaces and in the Space of Probability Measures (2005, methodology): “In this way we recover in a general framework the Riemannian interpretation of the Wasserstein distance developed by Otto in [106] (see also [105], [82] and also [38]): indeed, the right hand side in (25) is nothing but the minimal length, computed with respect to the metric tensor, of all absolutel”
Scaling Algorithms for Unbalanced Transport Problems (2016, background): “Initiated by [39], the study of such flows when T is the space of probability measures endowed with the Wasserstein metric d = W (see Section 2.”
On the geometry of metric measure spaces. II (2006, methodology): “Following [27], the heat semigroup on a metric measure space (M, d,m) should be obtained as the gradient flow on P2(M) for the relative entropy Ent( · |m).”