Quasi-Monte Carlo sampling for machine-learning partial differential equations

11/05/2019
by   Jingrun Chen, et al.
0

Solving partial differential equations in high dimensions by deep neural network has brought significant attentions in recent years. In many scenarios, the loss function is defined as an integral over a high-dimensional domain. Monte-Carlo method, together with the deep neural network, is used to overcome the curse of dimensionality, while classical methods fail. Often, a deep neural network outperforms classical numerical methods in terms of both accuracy and efficiency. In this paper, we propose to use quasi-Monte Carlo sampling, instead of Monte-Carlo method to approximate the loss function. To demonstrate the idea, we conduct numerical experiments in the framework of deep Ritz method proposed by Weinan E and Bing Yu. For the same accuracy requirement, it is observed that quasi-Monte Carlo sampling reduces the size of training data set by more than two orders of magnitude compared to that of MC method. Under some assumptions, we prove that quasi-Monte Carlo sampling together with the deep neural network generates a convergent series with rate proportional to the approximation accuracy of quasi-Monte Carlo method for numerical integration. Numerically the fitted convergence rate is a bit smaller, but the proposed approach always outperforms Monte Carlo method. It is worth mentioning that the convergence analysis is generic whenever a loss function is approximated by the quasi-Monte Carlo method, although observations here are based on deep Ritz method.

READ FULL TEXT

page 8

page 14

page 19

research
10/28/2022

Convergence analysis of a quasi-Monte Carlo-based deep learning algorithm for solving partial differential equations

Deep learning methods have achieved great success in solving partial dif...
research
10/30/2021

On quadrature rules for solving Partial Differential Equations using Neural Networks

Neural Networks have been widely used to solve Partial Differential Equa...
research
08/18/2020

On dropping the first Sobol' point

Quasi-Monte Carlo (QMC) points are a substitute for plain Monte Carlo (M...
research
03/07/2019

Deep learning observables in computational fluid dynamics

Many large scale problems in computational fluid dynamics such as uncert...
research
08/01/2023

Learning Green's Function Efficiently Using Low-Rank Approximations

Learning the Green's function using deep learning models enables to solv...
research
11/02/2016

Deep Learning Approximation for Stochastic Control Problems

Many real world stochastic control problems suffer from the "curse of di...
research
10/27/2020

Nonlinear Monte Carlo Method for Imbalanced Data Learning

For basic machine learning problems, expected error is used to evaluate ...

Please sign up or login with your details

Forgot password? Click here to reset