Harnessing fluctuations to discover dissipative evolution equations

Li, Xiaoguai; Dirr, Nicolas; Embacher, Peter; Zimmer, Johannes; Reina, Celia

doi:10.1016/j.jmps.2019.05.017

Cited by 11 publications

(6 citation statements)

References 36 publications

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“…Assume the validity of Assumption FD1 (discretised differential operators), Assumption FD2 (Brownian particle system), Assumption FD3 (scaling assumptions), and Assumption FD4 (discretised mean-field limit), all given below. In particular, assume that the mean-field limit ρ h in (20) satisfies ρ min ≤ ρ h ≤ ρ max for some positive ρ min , ρ max on [0, T ]. Let ρ h be the solution of the discretised Dean-Kawasaki model given in Definition FD-DK on a time interval [0, T ].…”

Section: Main Results and Summarymentioning

confidence: 99%

“…The fluctuation-dissipation relation -implicitly contained for instance in the Dean-Kawasaki equation -may be used to recover macroscopic diffusion properties from fluctuations in finite particle number simulations, see for instance [10,20]. Outside of the realm of physics, the concept of fluctuating hydrodynamics has also been applied to systems of interacting agents, see e. g. [16].…”

Section: Introductionmentioning

confidence: 99%

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The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles

Cornalba¹,

Fischer²

2021

Preprint

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The Dean-Kawasaki equation -a strongly singular SPDE -is a basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N independent diffusing particles in the regime of large particle numbers N1. The singular nature of the Dean-Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification. Besides being non-renormalisable by the theory of regularity structures by Hairer et al., it has recently been shown to not even admit nontrivial martingale solutions.In the present work, we give a rigorous and fully quantitative justification of the Dean-Kawasaki equation by considering the natural regularisation provided by standard numerical discretisations: We show that structure-preserving discretisations of the Dean-Kawasaki equation may approximate the density fluctuations of N noninteracting diffusing particles to arbitrary order in N −1 (in suitable weak metrics). In other words, the Dean-Kawasaki equation may be interpreted as a "recipe" for accurate and efficient numerical simulations of the density fluctuations of independent diffusing particles.

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Section: Main Results and Summarymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles

Cornalba¹,

Fischer²

2021

Preprint

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show abstract

“…Similar ideas have been used for model inference in References [30, 31] with the latter employing a “weak” form of an evolution equation similar to Equation (23) to learn an unknown discrete operator.…”

Section: A Data‐driven Model For Delay Photocurrentsmentioning

confidence: 99%

“…that is, the learning of the spatial ETI operator employs a "loss" function given by the Euclidean distance in R 3 . Similar ideas have been used for model inference in References [30,31] with the latter employing a "weak" form of an evolution equation similar to Equation ( 23) to learn an unknown discrete operator.…”

Section: Learning the Spatial Eti Operatormentioning

confidence: 99%

Development of data‐driven exponential integrators with application to modeling of delay photocurrents

Bochev

Paskaleva

2021

Numerical Methods Partial

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Radiation-induced photocurrent effects represent a threat to microelectronic components operating in space, manmade and terrestrial radiation environments. Analysis of these threats by circuit simulations requires accurate and computationally efficient compact models. Most existing compact models are based on closed form analytic solutions of the governing equations and require empirical assumptions and idealizations that can limit their validity. In this paper we formulate an alternative numerical, data-driven approach that learns a compact model from data representative of the type of measurements one can obtain in an experimental facility. To develop the model we start from a generic discrete-time dynamical system and then use physics knowledge to refine its structure. Numerical studies demonstrate the potential of the model and establish some empirical guidelines for its training.

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“…For recent numerical approaches to fluctuating hydrodynamics, we refer the reader e. g. to [ 1 – 3 , 9 , 13 , 14 , 24 , 33 , 36 ] (in particular, [ 2 ] contains the extension of the current work to the case of weakly interacting particles). Note that the small prefactor of the noise term in the Dean–Kawasaki equation ( 1 ) enables the use of certain higher-order timestepping schemes [ 22 ], a fact that we also use in our numerical simulations.…”

Section: Introductionmentioning

confidence: 99%

The Dean–Kawasaki Equation and the Structure of Density Fluctuations in Systems of Diffusing Particles

Cornalba

Fischer

2023

Arch Rational Mech Anal

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The Dean–Kawasaki equation—a strongly singular SPDE—is a basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N independent diffusing particles in the regime of large particle numbers $$N\gg 1$$ N ≫ 1 . The singular nature of the Dean–Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification. Besides being non-renormalisable by the theory of regularity structures by Hairer et al., it has recently been shown to not even admit nontrivial martingale solutions. In the present work, we give a rigorous and fully quantitative justification of the Dean–Kawasaki equation by considering the natural regularisation provided by standard numerical discretisations: We show that structure-preserving discretisations of the Dean–Kawasaki equation may approximate the density fluctuations of N non-interacting diffusing particles to arbitrary order in $$N^{-1}$$ N - 1 (in suitable weak metrics). In other words, the Dean–Kawasaki equation may be interpreted as a “recipe” for accurate and efficient numerical simulations of the density fluctuations of independent diffusing particles.

show abstract

Harnessing fluctuations to discover dissipative evolution equations

Cited by 11 publications

References 36 publications

The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles

The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles

Development of data‐driven exponential integrators with application to modeling of delay photocurrents

The Dean–Kawasaki Equation and the Structure of Density Fluctuations in Systems of Diffusing Particles

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