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Adaptive Neural Network Subspace Method for Solving Partial Differential Equations with High Accuracy

Zhongshuo Lin, Yifan Wang, Hehu Xie

TL;DR

A new machine learning method for solving partial differential equations based on neural network and adaptive subspace approximation method, which can act as the loss function for adaptively refining the parameters of neural network.

Abstract

Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopted to build the basis of the finite dimensional subspace. Then the discrete solution is obtained by using the subspace approximation. Especially, based on the subspace approximation, a posteriori error estimator can be derivated by the hypercircle technique. This a posteriori error estimator can act as the loss function for adaptively refining the parameters of neural network.

Adaptive Neural Network Subspace Method for Solving Partial Differential Equations with High Accuracy

TL;DR

A new machine learning method for solving partial differential equations based on neural network and adaptive subspace approximation method, which can act as the loss function for adaptively refining the parameters of neural network.

Abstract

Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopted to build the basis of the finite dimensional subspace. Then the discrete solution is obtained by using the subspace approximation. Especially, based on the subspace approximation, a posteriori error estimator can be derivated by the hypercircle technique. This a posteriori error estimator can act as the loss function for adaptively refining the parameters of neural network.

Paper Structure

This paper contains 13 sections, 6 theorems, 102 equations, 14 figures, 8 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Machine learning method for solving PDEs
  3. Neural network subspace for solving PDEs
  4. Elliptic problems with singularity or discontinuous coefficient
  5. Typical two-material diffusion problems
  6. Testing the loss functions
  7. Tests with large coefficient differences
  8. Tests with high-frequency exact solution
  9. Multi-material diffusion problems
  10. Test for continuous interface flux
  11. Test for jump in interface flux
  12. A heterogeneous material problem within the domain
  13. Conclusion

Key Result

Theorem 2.1

The function $u_{\rm NN}^{(1)}$ defined by (Ritz_ML) and the exact solution (Ritz_Form) satisfy the following error estimate

Figures (14)

  • Figure 1: Errors of machine learning method for partial differential equations.
  • Figure 2: Numerical results for solving problem (\ref{['Laplace_Singular']}): the left panel shows the approximate solution, the middle panel shows the exact solution, and the right panel shows the error distribution.
  • Figure 3: The relative error on test points during training for TNNs with different sub-network structures. The left panel shows FNN, and the right panel shows ResNet.
  • Figure 4: The boundary $\partial\Omega$, subdomains $\Omega_1$, $\Omega_2$ and interface $\Gamma$
  • Figure 5: Test (1.2): Numerical results of solving (\ref{['interface_example_1']}) using the posterior error estimator-based loss function: Left: Approximate solution image, Middle: Exact solution image, Right: Error distribution plot
  • ...and 9 more figures

Theorems & Definitions (13)

  • Theorem 2.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.2
  • ...and 3 more