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Neural Parametric Fokker-Planck Equations [article]

Shu Liu, Wuchen Li, Hongyuan Zha, Haomin Zhou
2021 arXiv   pre-print
We call such ODEs neural parametric Fokker-Planck equations.  ...  Moreover, we also establish bounds for the asymptotic convergence analysis of the neural parametric Fokker-Planck equation as well as the error analysis for both the continuous and discrete versions.  ...  We also believe that the approaches in algorithm design and error analysis developed in this study can be extended to other equations, such as porous media equation, Schrödinger equation, and Schrödinger  ... 
arXiv:2002.11309v3 fatcat:b72ly75bizcjxfe2g676h3t3pa

Parametric Fokker-Planck equation [article]

Wuchen Li, Shu Liu, Hongyuan Zha, Haomin Zhou
2019 arXiv   pre-print
We derive the Fokker-Planck equation on the parametric space. It is the Wasserstein gradient flow of relative entropy on the statistical manifold.  ...  Parametric Fokker-Planck equation In this section, we briefly review the fact that Fokker-Planck equation is a Wasserstein gradient flow of relative entropy.  ...  Fokker-Planck equation.  ... 
arXiv:1903.10076v1 fatcat:pg2mvg46cjfr5lugh5utvwbkji

Preface to Vol 2, No 3 of the Journal of Intelligent Industrial Systems

G. Rigatos, P. Siano
2016 Intelligent Industrial Systems  
Alternatively it is proposed to apply non-parametric estimators of the neural network type so as to extract the model of industrial systems out of data sets.  ...  Moreover, it is proposed to exploit neural networks for accomplishing fault diagnosis tasks, In such an approach the neural network is used for assigning the industrial systems' output to fault classes  ...  It is shown that the procedure for numerical solution of the Fokker-Planck PDE results into a set of nonlinear ordinary differential equations (ODEs) and an associated state equations model.  ... 
doi:10.1007/s40903-016-0058-5 fatcat:y6ivw4qcj5hpdcz3zyhqswpy7a

Neural Galerkin Scheme with Active Learning for High-Dimensional Evolution Equations [article]

Joan Bruna and Benjamin Peherstorfer and Eric Vanden-Eijnden
2022 arXiv   pre-print
fail, especially when features of the solutions evolve locally such as in high-dimensional wave propagation problems and interacting particle systems described by Fokker-Planck and kinetic equations.  ...  This work proposes Neural Galerkin schemes based on deep learning that generate training data samples with active learning for numerically solving high-dimensional partial differential equations.  ...  Fokker-Planck equations, Boltzmann equations, etc.  ... 
arXiv:2203.01360v3 fatcat:xhmep6kt5ndctmyf2gdl47dtcy

Stochastic Resonance, Brownian Ratchets and the Fokker-Planck Equation

A. Allison
2003 AIP Conference Proceedings  
The evidence suggests that some forms of stochastic resonance can be formulated in terms of a Fokker-Planck equation and some cannot.  ...  We pose a general philosophical question: "Is it is possible to relate stochastic resonance and Brownian ratchets in a formal way, through a consideration of the Fokker-Planck equation?"  ...  It should be admitted that not all forms of SR can be formulated in terms of a Fokker-Planck equation.  ... 
doi:10.1063/1.1584877 fatcat:dhl4xxsghzdrxedrbufumeawry

Fokker-Planck perspective on stochastic delay systems: Exact solutions and data analysis of biological systems

T. D. Frank, P. J. Beek, R. Friedrich
2003 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
Stochastic delay systems with additive noise are examined from the perspective of Fokker-Planck equations.  ...  For a linear system, the exact stationary probability density is derived by means of a delay Fokker-Planck equation.  ...  That is, the delay Fokker-Planck equation ͑7͒ does not provide us with sufficient information to determine K() as a function of .  ... 
doi:10.1103/physreve.68.021912 pmid:14525011 fatcat:z4phmxvfzrdlflhvgt35zs6vf4

Computational Nonlinear Stochastic Control

Mrinal Kumar, Suman Chakravorty, John L. Junkins
2009 Journal of Guidance Control and Dynamics  
Forward and Backward Kolmogorov Equations In this section, we shall introduce the forward Kolmogorov or the FokkerPlanck equation and the backward Kolmogorov equation (BKE).  ...  L., “The Partition of Unity Finite Element Approach to the Stationary Fokker- Planck Equation,” AIAA Paper 2006-6285, Aug. 2006.  ... 
doi:10.2514/1.37128 fatcat:v7pabaf6uzfmnchmzhuxvx7c5y

Nonparametric inference of stochastic differential equations based on the relative entropy rate [article]

Min Dai, Jinqiao Duan, Jianyu Hu, Xiangjun Wang
2021 arXiv   pre-print
Discovering governing equations and quantifying dynamical properties of complex systems are among central challenges.  ...  Meanwhile, this approach enables to extract the governing equations. We illustrate our approach in several examples.  ...  Karniadakis, “Solving Inverse Stochastic Problems from Dis- crete Particle Observations Using the FokkerPlanck Equation and Physics-Informed Neural Net- works,” SIAM Journal on Scientific Computing  ... 
arXiv:2112.04692v1 fatcat:hnb25znupvbsjbesvmslryexuu

Fighting seizures with seizures: diffusion and stability in neural systems [article]

Erik D. Fagerholm, Chayanin Tangwiriyasakul, Karl J. Friston, Inês R. Violante, Steven Williams, David W. Carmichael, Suejen Perani, Federico E. Turkheimer, Rosalyn J. Moran, Robert Leech, Mark P. Richardson
2020 arXiv   pre-print
Here, we demonstrate that the (neuronal) state equation used in Dynamic Causal Modelling generalises to a Fokker-Planck formalism when propagation of neuronal activity along structural connections is considered  ...  We show, for neural systems prone to epileptic seizures, that such a reduction can be achieved via external stimulation.  ...  Therefore, [10] takes the form of the discretized Fokker-Planck equation, with a drift term: ( , , ) and a diffusion term: ∑ #$ $ -$%& Linear stability analysis: In order to model neural time series,  ... 
arXiv:2005.00805v1 fatcat:mta6kz6lmreh5gug7llu7uhu2i

Quantum Statistical Learning via Quantum Wasserstein Natural Gradient

Simon Becker, Wuchen Li
2021 Journal of statistical physics  
For continuous parametric models of density operators, we pull back the quantum Wasserstein metric such that the parameter space becomes a Riemannian manifold with quantum Wasserstein information matrix  ...  (\theta ) \in {\mathcal {P}}_{\theta }} W_{Q}^2 (\rho _{\star },\rho (\theta ))$$ argmin ρ ( θ ) ∈ P θ W Q 2 ( ρ ⋆ , ρ ( θ ) ) to approximate a target quantum state $$\rho _{\star }$$ ρ ⋆ by a set of parametrized  ...  Fokker-Planck equation.  ... 
doi:10.1007/s10955-020-02682-1 fatcat:pwm7nbng2jan7ngpf6simnne3u

Scalable Inference in SDEs by Direct Matching of the Fokker-Planck-Kolmogorov Equation [article]

Arno Solin, Ella Tamir, Prakhar Verma
2021 arXiv   pre-print
We address this issue by revisiting the classical SDE literature and derive direct approximations to the (typically intractable) Fokker-Planck-Kolmogorov equation by matching moments.  ...  These methods are general-purpose and used with parametric and non-parametric models, and neural SDEs.  ...  This PDE is also known as the Fokker-Planck equation (in physics) and the forward Kolmogorov equation (in stochastics).  ... 
arXiv:2110.15739v1 fatcat:fafoab423fhpzi4u4u7r5oxs64

Page 5918 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews  
(NL-VUAMH; Amsterdam) Nonlinear Fokker-Planck equations whose stationary solutions make entropy-like functionals stationary. (English summary) Phys. A 272 (1999), no. 3-4, 497-508.  ...  Summary: “The present study extends the correspondence prin- ciple of Martinez et al. that establishes a link between nonlinear Fokker-Planck equations (NLFPEs) and the variational principle approach of  ... 

Synthesis and Computer Study of Population Dynamics Controlled Models Using Methods of Numerical Optimization, Stochastization and Machine Learning

Anastasia V. Demidova, Olga V. Druzhinina, Olga N. Masina, Alexey A. Petrov
2021 Mathematics  
A methods for the synthesis of controllers based on the use of artificial neural networks and machine learning are developed.  ...  The transition is made from the model specified by the vector ordinary differential equation to the corresponding stochastic model.  ...  The Fokker-Planck equation describes the time evolution of the probability density function.  ... 
doi:10.3390/math9243303 fatcat:plrjdwzkqvartgelb4kbdxzk5i

Dynamics of the Desai-Zwanzig model in multiwell and random energy landscapes

Susana N. Gomes, Serafim Kalliadasis, Grigorios A. Pavliotis, Petr Yatsyshin
2019 Physical review. E  
Our numerical experiments are based on Monte Carlo simulations, the numerical solution of the time-dependent nonlinear Fokker-Planck (McKean-Vlasov equation), the minimization of the free energy functional  ...  The McKean-Vlasov equation is a nonlinear, nonlocal equation, sometimes referred to as the McKean-Vlasov-Fokker-Planck equation.  ...  The invariant measure(s) of the system (1) satisfy the stationary Fokker-Planck equation (7) .  ... 
doi:10.1103/physreve.99.032109 fatcat:mnsxsmvd6fddrewf4fnl6bgrga

Discrete time symmetry breaking in quantum circuits: exact solutions and tunneling

Feng-Xiao Sun, Qiongyi He, Qihuang Gong, Run Yan Teh, Margaret D Reid, Peter D Drummond
2019 New Journal of Physics  
We discuss general properties of discrete time quantum symmetry breaking in degenerate parametric oscillators.  ...  Recent experiments in superconducting quantum circuit with Josephson junction nonlinearities give rise to new properties of strong parametric coupling and nonlinearities.  ...  From the dimensionless Fokker-Planck equation, equation (2.32), the mean-field limit corresponds to  ¥ n .  ... 
doi:10.1088/1367-2630/ab3f5e fatcat:gmoyt5cr4jdp5bp7yyolqfxrdq
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