Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

Victor Boussange (), Sebastian Becker (), Arnulf Jentzen (), Benno Kuckuck () and Loïc Pellissier ()
Additional contact information
Victor Boussange: Swiss Federal Research Institute for Forest, Snow and Landscape (WSL)
Sebastian Becker: ETH Zürich
Arnulf Jentzen: The Chinese University of Hong Kong, Shenzhen
Benno Kuckuck: University of Münster
Loïc Pellissier: Swiss Federal Research Institute for Forest, Snow and Landscape (WSL)

Partial Differential Equations and Applications, 2023, vol. 4, issue 6, 1-51

Abstract: Abstract Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately account for certain non-local phenomena such as, e.g., interactions at a distance. Non-local nonlinear PDE models can accurately capture these phenomena, but traditional numerical approximation methods are infeasible when the considered non-local PDE is high-dimensional. In this article we propose two numerical methods based on machine learning and on Picard iterations, respectively, to approximately solve non-local nonlinear PDEs. The proposed machine learning-based method is an extended variant of a deep learning-based splitting-up type approximation method previously introduced in the literature and utilizes neural networks to provide approximate solutions on a subset of the spatial domain of the solution. The Picard iterations-based method is an extended variant of the so-called full history recursive multilevel Picard approximation scheme previously introduced in the literature and provides an approximate solution for a single point of the domain. Both methods are mesh-free and allow non-local nonlinear PDEs with Neumann boundary conditions to be solved in high dimensions. In the two methods, the numerical difficulties arising due to the dimensionality of the PDEs are avoided by (i) using the correspondence between the expected trajectory of reflected stochastic processes and the solution of PDEs (given by the Feynman–Kac formula) and by (ii) using a plain vanilla Monte Carlo integration to handle the non-local term. We evaluate the performance of the two methods on five different PDEs arising in physics and biology. In all cases, the methods yield good results in up to 10 dimensions with short run times. Our work extends recently developed methods to overcome the curse of dimensionality in solving PDEs.

Keywords: Non-local; Partial differential equation; PDE; Deep learning; Neural networks; Neumann boundary condition; Reflected Brownian motion; Primary 35R09; Secondary 65M75; 45K05; 35K20; 65C05; 65M22; 68T07 (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s42985-023-00244-0 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:pardea:v:4:y:2023:i:6:d:10.1007_s42985-023-00244-0

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/42985/

DOI: 10.1007/s42985-023-00244-0

Access Statistics for this article

Partial Differential Equations and Applications is currently edited by Zhitao Zhang

More articles in Partial Differential Equations and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().