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Tensor Neural Network Based Machine Learning Method for Elliptic Multiscale Problems

Zhongshuo Lin, Haochen Liu, Hehu Xie

TL;DR

A type of tensor neural network based machine learning method is introduced to solve elliptic multiscale problems with reasonable accuracy and brings a new way to design numerical methods for computing more general multiscale problems with high accuracy.

Abstract

In this paper, we introduce a type of tensor neural network based machine learning method to solve elliptic multiscale problems. Based on the special structure, we can do the direct and highly accurate high dimensional integrations for the tensor neural network functions without Monte Carlo process. Here, with the help of homogenization techniques, the multiscale problem is first transformed to the high dimensional limit problem with reasonable accuracy. Then, based on the tensor neural network, we design a type of machine learning method to solve the derived high dimensional limit problem. The proposed method in this paper brings a new way to design numerical methods for computing more general multiscale problems with high accuracy. Several numerical examples are also provided to validate the accuracy of the proposed numerical methods.

Tensor Neural Network Based Machine Learning Method for Elliptic Multiscale Problems

TL;DR

A type of tensor neural network based machine learning method is introduced to solve elliptic multiscale problems with reasonable accuracy and brings a new way to design numerical methods for computing more general multiscale problems with high accuracy.

Abstract

In this paper, we introduce a type of tensor neural network based machine learning method to solve elliptic multiscale problems. Based on the special structure, we can do the direct and highly accurate high dimensional integrations for the tensor neural network functions without Monte Carlo process. Here, with the help of homogenization techniques, the multiscale problem is first transformed to the high dimensional limit problem with reasonable accuracy. Then, based on the tensor neural network, we design a type of machine learning method to solve the derived high dimensional limit problem. The proposed method in this paper brings a new way to design numerical methods for computing more general multiscale problems with high accuracy. Several numerical examples are also provided to validate the accuracy of the proposed numerical methods.
Paper Structure (11 sections, 2 theorems, 57 equations, 10 figures, 4 tables)

This paper contains 11 sections, 2 theorems, 57 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Solving multiscale partial differential equations
  3. Tensor neural network architecture
  4. Tensor neural network based machine learning method
  5. The solving procedure of TNN-based method
  6. Quadrature scheme for TNN
  7. Numerical results
  8. One-dimensional elliptic two-scale problems
  9. Two-dimensional elliptic two-scale problems
  10. One-dimensional elliptic three-scale problems
  11. Conclusions

Key Result

Theorem 3.1

WangJinXie Assume that each $\Omega_i$ is an interval in $\mathbb R$ for $i=1, \cdots, d$, $\Omega=\Omega_1\times\cdots\times\Omega_d$, and the function $f(x)\in H^m(\Omega)$. Then for any tolerance $\varepsilon>0$, there exist a positive integer $p$ and the corresponding TNN defined by (def_TNN_nor

Figures (10)

  • Figure 1: Architecture of TNN. Black arrows mean linear transformation (or affine transformation). Each ending node of blue arrows is obtained by taking the scalar multiplication of all starting nodes of blue arrows that end in this ending node. The final output of TNN is derived from the summation of all starting nodes of red arrows.
  • Figure 2: The $L^2$ error and $H^1$ semi-norm error of $\chi^{\text{TNN}}(x_i, \cdot)$ at the selected points $\{x_i\}, i=1,\cdots, 20$. On the selected points $x_i$, the maximum $L^2$ norm error is 1.3736e-05, the maximum $H^1$ semi-norm error is 1.3949e-04, the maximum relative $L^2$ norm error is 4.7948e-04, the maximum relative $H^1$ semi-norm error is 7.7027e-04.
  • Figure 3: Relative errors of the homogenized coefficient during the training process for problem (\ref{['cell-prob']}).
  • Figure 4: The figures of $A^{\text{TNN}}(x)$, $\frac{\partial A^{\text{TNN}}(x)}{\partial x}$ and their exact ones for the problem (\ref{['ex_1D']}), where the exact homogenized coefficient $A^*(x)$ is defined by formula (\ref{['homo-coefficient-analytical']}) and $A^{\text{TNN}}(x)$ is the approximate coefficient obtained by the TNN method. Left: the homogenized coefficients $A^*(x)$ and $A^{\text{TNN}}(x)$. Right: the homogenized coefficient $\frac{\partial A^*(x)}{\partial x}$ and $\frac{\partial A^{\text{TNN}}(x)}{\partial x}$.
  • Figure 5: Left: the figures of $u_0^{\text{TNN}}$, $u_0^{\text{TNN}} + \varepsilon u_1^{\text{TNN}}$ and $u_{\varepsilon}^{\text{FEM}}$ of the problem (\ref{['ex_1D']}) for the case of $\varepsilon = 1/10$. Right: the figures of $\frac{\partial u_0^{\text{TNN}}}{\partial x}$, $\frac{\partial u_0^{TNN}}{\partial x} + \frac{\partial u_1^{TNN}}{\partial y}$ and $\frac{\partial u_{\varepsilon}^{\text{FEM}}}{\partial x}$ of the problem (\ref{['ex_1D']}) for the case of $\varepsilon = 1/10$. Here, the final $H^1$ semi-norm absolute error $|u_{0}^{\text{TNN}} + \varepsilon u_1^{\text{TNN}} - u_{\varepsilon}^{\text{FEM}}|_{H^1(\Omega)} = 2.699\text{e{-04}}$.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Theorem 3.1
  • Theorem 3.2