Long-time behavior of a class of nonlocal partial differential equations

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.

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Partial differential equations and stochastic methods in molecular dynamics

Acta Numerica ◽

10.1017/s0962492916000039 ◽

2016 ◽

Vol 25 ◽

pp. 681-880 ◽

Cited By ~ 38

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.

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On a class of stochastic partial differential equations with multiple invariant measures

Nonlinear Differential Equations and Applications NoDEA ◽

10.1007/s00030-021-00691-x ◽

2021 ◽

Vol 28 (3) ◽

Author(s):

Bálint Farkas ◽

Martin Friesen ◽

Barbara Rüdiger ◽

Dennis Schroers

Keyword(s):

Hilbert Space ◽

Partial Differential Equations ◽

Differential Equations ◽

Stochastic Partial Differential Equations ◽

Invariant Measures ◽

Transition Probabilities ◽

Time Behavior ◽

Initial State ◽

Partial Differential ◽

Unique Mild Solution

AbstractIn this work we investigate the long-time behavior for Markov processes obtained as the unique mild solution to stochastic partial differential equations in a Hilbert space. We analyze the existence and characterization of invariant measures as well as convergence of transition probabilities. While in the existing literature typically uniqueness of invariant measures is studied, we focus on the case where the uniqueness of invariant measures fails to hold. Namely, introducing a generalized dissipativity condition combined with a decomposition of the Hilbert space, we prove the existence of multiple limiting distributions in dependence of the initial state of the process and study the convergence of transition probabilities in the Wasserstein 2-distance. Finally, we apply our results to Lévy driven Ornstein–Uhlenbeck processes, the Heath–Jarrow–Morton–Musiela equation as well as to stochastic partial differential equations with delay.

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

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2019.0630 ◽

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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