More Like this

1. A TRAJECTORIAL APPROACH TO THE GRADIENT FLOW PROPERTIES OF LANGEVIN–SMOLUCHOWSKI DIFFUSIONS

   https://doi.org/10.4213/tvp5505  

   KARATZAS, I.; SCHACHERMAYER, W.; TSCHIDERER, B. (January 2022, Theory of probability and its applications)

   We revisit the variational characterization of conservative di↵usion as entropic gra- dient flow and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto that, for diffusions of Langevin–Smoluchowski type, the Fokker–Planck probability density flow maximizes the rate of relative entropy dissipation, as mea- sured by the distance traveled in the ambient space of probability measures with finite second moments, in terms of the quadratic Wasserstein metric. We obtain novel, stochastic-process ver- sions of these features, valid along almost every trajectory of the dffusive motion in the backwards direction of time, using a very direct perturbation analysis. By averaging our trajectorial results with respect to the underlying measure on path space, we establish the maximal rate of entropy dissipation along the Fokker–Planck flow and measure exactly the deviation from this maximum that corresponds to any given perturbation. A bonus of our trajectorial approach is that it derives the HWI inequality relating relative entropy (H), Wasserstein distance (W), and relative Fisher information (I). 

   more » « less
2. Large-Scale Wasserstein Gradient Flows

   Mokrov, Petr; Korotin, Alexander; Li, Lingxiao; Genevay, Aude; Solomon, Justin; Burnaev, Evgeny (December 2021, NeurIPS)

   Wasserstein gradient flows provide a powerful means of understanding and solving many diffusion equations. Specifically, Fokker-Planck equations, which model the diffusion of probability measures, can be understood as gradient descent over entropy functionals in Wasserstein space. This equivalence, introduced by Jordan, Kinderlehrer and Otto, inspired the so-called JKO scheme to approximate these diffusion processes via an implicit discretization of the gradient flow in Wasserstein space. Solving the optimization problem associated with each JKO step, however, presents serious computational challenges. We introduce a scalable method to approximate Wasserstein gradient flows, targeted to machine learning applications. Our approach relies on input-convex neural networks (ICNNs) to discretize the JKO steps, which can be optimized by stochastic gradient descent. Contrarily to previous work, our method does not require domain discretization or particle simulation. As a result, we can sample from the measure at each time step of the diffusion and compute its probability density. We demonstrate the performance of our algorithm by computing diffusions following the Fokker-Planck equation and apply it to unnormalized density sampling as well as nonlinear filtering. 

   more » « less
3. Hyperbolic Problems: Theory, Numerics and Applications (2018): On structure-preserving high order methods for conservation laws

   Liu, Hailiang (January 2020, AIMS on Applied Mathematics)

   Bressan, A; Lewicka, M; Wang, D.; Zheng, Y.X. (Ed.)

   In this paper we review the algorithm development in high order methods for some conservation laws. The emphasis is on our recent contribution in the study of two model classes: Fokker-Planck-type equations and hyperbolic conservation law systems. For the former we will review free-energy-satisfying and positivity-preserving schemes. For the later we will review the general invariant-region-preserving (IRP) limiter, and its application to high order methods for multi-dimensional hyperbolic systems of conservation laws. 

   more » « less
4. Local well-posedness of a nonlinear Fokker–Planck model

   https://doi.org/10.1088/1361-6544/acb7c2  

   Epshteyn, Yekaterina; Liu, Chang; Liu, Chun; Mizuno, Masashi (February 2023, Nonlinearity)

   Abstract Noise or fluctuations play an important role in the modeling and understanding of the behavior of various complex systems in nature. Fokker–Planck equations are powerful mathematical tools to study behavior of such systems subjected to fluctuations. In this paper we establish local well-posedness result of a new nonlinear Fokker–Planck equation. Such equations appear in the modeling of the grain boundary dynamics during microstructure evolution in the polycrystalline materials and obey special energy laws. 

   more » « less
5. A deep learning method for solving Fokker-Planck equations

   Jiayu Zhai, Matthew Dobson (January 2022, Proceedings of Machine Learning Research)

   Joan Bruna, Jan S (Ed.)

   The time evolution of the probability distribution of a stochastic differential equation follows the Fokker-Planck equation, which usually has an unbounded, high-dimensional domain. Inspired by Li (2019), we propose a mesh-free Fokker-Planck solver, in which the solution to the Fokker-Planck equation is now represented by a neural network. The presence of the differential operator in the loss function improves the accuracy of the neural network representation and reduces the demand of data in the training process. Several high dimensional numerical examples are demonstrated. 

   more » « less