Physics-informed Deep Neural Networks for High Dimensional Integration Público

Abstract

Monte Carlo methods are one of the most prevailing computational algorithms that use random sampling process to generate numerical estimations for problems in various fields, including mathematics, physical sciences, finance, etc. One of the applications of Monte-Carlo methods in mathematics is numerical integration, in which the speed of convergence is still slow. The Quasi- Monte Carlo method can improve the performance of the Monte Carlo method, especially in high dimensions, by generating low-discrepancy sequences. Existing Quasi-Monte Carlo methods use sequences such as Halton sequence, Sobol sequence, or Faure sequence that may fail to guarantee the low discrepancy due to improper predefined parameters. In this paper, we propose a different approach for generating low-discrepancy sequences through deep neural networks. By modeling the sequences as dynamic molecules and minimizing the total energy of the datasets in the deep neural networks, we are able to guarantee the low-discrepancy of the sequences that are independent of the initial distributions. We demonstrate the effectiveness of our methods using various numerical experiments for problems ranging from low to high dimensions. 

Table of Contents

1 Introduction 1

1.1 Backgrounds ........................................ 1

1.2 Outline of the Thesis.................................... 2

2 Monte Carlo Methods 3

2.1 Monte Carlo Methods for Numerical Integration .................... 3

2.2 Quasi-MonteCarloMethods................................ 4

3 Motivation from Physics 7

3.1 The Notion of Equilibrium................................. 7

3.2 Boltzmann Distribution .................................. 8

4 Basics of Deep Learning 10

4.1 Multi-layer Perceptron................................... 10

4.2 Automatic Differentiation ................................. 11

5 Dynamics, System Energy and Deep Neural Networks 13

5.1 Deep Learning Approach.................................. 14

5.2 Adaptive Training Approach ............................... 16

6 Numerical Experiments 19

6.1 Energy Functions...................................... 19

6.2 Low Dimensional Examples ................................ 21

6.3 Results Evaluation ..................................... 26

7 Discussion and Future Work 31

Bibliography 34 

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Statistics Mathematics
Palabra Clave	Deep Learning Monte Carlo Methods
Committee Chair / Thesis Advisor	Yuanzhe Xi, Emory University
Committee Members	Manuela Manetta, Emory University Weihua An, Emory University

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