用机器学习方法解分数阶偏微分方程和不连续问题

目录

第一个书签之前

用机器学习方法解分数阶偏微分方程和不连续问题

Solving Fractional Partial Differential Equations and Discontinuous Problems via Deep Learning

摘 要

ABSTRACT

Contents

第1章 Introduction

1.1 Research Background and Significance

1.1.1 Source

1.1.2 Background and Significance

1.2 Research Status

1.2.1 Numerical Methods of Fractional PDEs

1.2.2 Deep Learning Methods on PDEs

1.2.3 Deep Learning Methods on Fractional PDEs

1.3 Thesis Outline

第2章 Basic Model

2.1 Fractional Derivative and Differential Equations

2.1.1 Fractional Derivative

2.1.2 Fractional Differential Equations and Numerical Methods

2.1.3 Spectral Methods

2.1.4 Numerical Examples

2.2 Nonlocal Models and Singularity

2.2.1 PDEs with Singularity

2.2.2 Nonlocal Model

2.3 Machine Learning Methods

2.3.1 Deep Learning

2.3.2 Active Function

2.3.3 Numerical Example

2.4 Summary

第3章 Fractional PDEs and Deep Galerkin Method

3.1 Deep Galerkin Method

3.1.1 Problem Statement

3.1.2 Deep Learning Models

3.2 Deep Learning Based Methods on Fractional PDEs

3.2.1 Numerical Integral

3.2.2 Theoritic Basis

3.2.3 Algorithm

3.3 Numerical Examples

3.3.1 Accuracy Test

3.3.2 Fractional Allen-Cahn Equation

3.3.3 3D Fractioanl Heat Equation

3.4 Summary

第4章 Deep Learning Based Methods for Discontinuity Problems

4.1 Partial Differential Equations with Discontinuity

4.2 Deep Learning Network Frame with Discontinuity

4.3 Numercal Results

4.3.1 Viscous Burgers' Equation

4.3.2 Transport Equation

4.3.3 Inviscid Burgers' Equation

4.4 Summary

Conclusions

结 论

哈尔滨工业大学与南方科技大学联合培养研究生学位论文原创性声明和使用权限

Acknowledgements