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

05/07/2022
by   Victor Boussange, et al.
0

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. In order to properly capture these phenomena non-local nonlinear PDE models are frequently employed in the literature. 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
06/01/2018

Solving stochastic differential equations and Kolmogorov equations by means of deep learning

Stochastic differential equations (SDEs) and the Kolmogorov partial diff...
research
08/31/2020

Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning

In recent years, tremendous progress has been made on numerical algorith...
research
07/08/2019

Deep splitting method for parabolic PDEs

In this paper we introduce a numerical method for parabolic PDEs that co...
research
07/12/2022

MultiShape: A Spectral Element Method, with Applications to Dynamic Density Functional Theory and PDE-Constrained Optimization

A numerical framework is developed to solve various types of PDEs on com...
research
04/27/2023

A particle method for non-local advection-selection-mutation equations

The well-posedness of a non-local advection-selection-mutation problem d...
research
04/07/2021

Rademacher Complexity and Numerical Quadrature Analysis of Stable Neural Networks with Applications to Numerical PDEs

Methods for solving PDEs using neural networks have recently become a ve...

Please sign up or login with your details

Forgot password? Click here to reset