Partial differential equations and stochastic methods in molecular dynamics

Acta Numerica ◽  
2016 ◽  
Vol 25 ◽  
pp. 681-880 ◽  
Author(s):  
Tony Lelièvre ◽  
Gabriel Stoltz
Keyword(s):  
Molecular Dynamics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Statistical Physics ◽  
Stochastic Dynamics ◽  
Planck Equation ◽  
Stochastic Methods ◽  
High Dimensions ◽  
Partial Differential ◽  
Long Time

The objective of molecular dynamics computations is to infer macroscopic properties of matter from atomistic models via averages with respect to probability measures dictated by the principles of statistical physics. Obtaining accurate results requires efficient sampling of atomistic configurations, which are typically generated using very long trajectories of stochastic differential equations in high dimensions, such as Langevin dynamics and its overdamped limit. Depending on the quantities of interest at the macroscopic level, one may also be interested in dynamical properties computed from averages over paths of these dynamics.This review describes how techniques from the analysis of partial differential equations can be used to devise good algorithms and to quantify their efficiency and accuracy. In particular, a crucial role is played by the study of the long-time behaviour of the solution to the Fokker–Planck equation associated with the stochastic dynamics.

scholarly journals The asymptotic solution of a singularly perturbed Cauchy problem for Fokker-Planck equation

2021 ◽  
Vol 29 (2) ◽  
pp. 126-145
Author(s):  
Mohamed A. Bouatta ◽  
Sergey A. Vasilyev ◽  
Sergey I. Vinitsky
Keyword(s):  
Cauchy Problem ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Small Parameter ◽  
Asymptotic Method ◽  
Planck Equation ◽  
Singularly Perturbed ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Fokker Planck

The asymptotic method is a very attractive area of applied mathematics. There are many modern research directions which use a small parameter such as statistical mechanics, chemical reaction theory and so on. The application of the Fokker-Planck equation (FPE) with a small parameter is the most popular because this equation is the parabolic partial differential equations and the solutions of FPE give the probability density function. In this paper we investigate the singularly perturbed Cauchy problem for symmetric linear system of parabolic partial differential equations with a small parameter. We assume that this system is the Tikhonov non-homogeneous system with constant coefficients. The paper aims to consider this Cauchy problem, apply the asymptotic method and construct expansions of the solutions in the form of two-type decomposition. This decomposition has regular and border-layer parts. The main result of this paper is a justification of an asymptotic expansion for the solutions of this Cauchy problem. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations.

scholarly journals Solving high-dimensional partial differential equations using deep learning

2018 ◽  
Vol 115 (34) ◽  
pp. 8505-8510 ◽  
Author(s):  
Jiequn Han ◽  
Arnulf Jentzen ◽  
Weinan E
Keyword(s):  
Deep Learning ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ad Hoc ◽  
Difficult Problem ◽  
High Dimensional ◽  
Parabolic Pdes ◽  
Partial Differential ◽  
Long Time ◽  
Hamilton Jacobi Bellman

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the “curse of dimensionality.” This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black–Scholes equation, the Hamilton–Jacobi–Bellman equation, and the Allen–Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. This opens up possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their interrelationships.

scholarly journals Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

2020 ◽  
Vol 476 (2244) ◽  
pp. 20190630
Author(s):  
Martin Hutzenthaler ◽  
Arnulf Jentzen ◽  
Thomas Kruse ◽  
Tuan Anh Nguyen ◽  
Philippe von Wurstemberger
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Curse Of Dimensionality ◽  
Computational Effort ◽  
Parabolic Partial Differential Equations ◽  
Heat Equations ◽  
Partial Differential ◽  
Lipschitz Continuous ◽  
Semilinear Pdes ◽  
Long Time

For a long time it has been well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows at most polynomially both in the dimension and in the reciprocal of the required accuracy.

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