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A reduced order Schwarz method for nonlinear multiscale elliptic equations based on two-layer neural networks

Shi Chen, Zhiyan Ding, Qin Li, Stephen J. Wright

TL;DR

The paper addresses solving multiscale fully nonlinear elliptic PDEs with high-contrast oscillatory coefficients by integrating domain decomposition Schwarz methods with offline-trained two-layer neural networks that approximate the boundary-to-boundary maps between neighboring subdomains. It exploits homogenization to justify a low-dimensional boundary operator and leverages Barron-type results to motivate a dimension-robust two-layer NN surrogate, trained on data from enlarged patches and initialized via a linearized operator. In the online phase, these NN surrogates replace expensive local solves within the Schwarz iterations, delivering substantial speedups while preserving accuracy for both a semilinear equation and a multiscale $p$-Laplace equation. The results demonstrate significant efficiency gains and good generalization, suggesting scalability of the approach for nonlinear multiscale PDE solvers.

Abstract

Neural networks are powerful tools for approximating high dimensional data that have been used in many contexts, including solution of partial differential equations (PDEs). We describe a solver for multiscale fully nonlinear elliptic equations that makes use of domain decomposition, an accelerated Schwarz framework, and two-layer neural networks to approximate the boundary-to-boundary map for the subdomains, which is the key step in the Schwarz procedure. Conventionally, the boundary-to-boundary map requires solution of boundary-value elliptic problems on each subdomain. By leveraging the compressibility of multiscale problems, our approach trains the neural network offline to serve as a surrogate for the usual implementation of the boundary-to-boundary map. Our method is applied to a multiscale semilinear elliptic equation and a multiscale $p$-Laplace equation. In both cases we demonstrate significant improvement in efficiency as well as good accuracy and generalization performance.

A reduced order Schwarz method for nonlinear multiscale elliptic equations based on two-layer neural networks

TL;DR

The paper addresses solving multiscale fully nonlinear elliptic PDEs with high-contrast oscillatory coefficients by integrating domain decomposition Schwarz methods with offline-trained two-layer neural networks that approximate the boundary-to-boundary maps between neighboring subdomains. It exploits homogenization to justify a low-dimensional boundary operator and leverages Barron-type results to motivate a dimension-robust two-layer NN surrogate, trained on data from enlarged patches and initialized via a linearized operator. In the online phase, these NN surrogates replace expensive local solves within the Schwarz iterations, delivering substantial speedups while preserving accuracy for both a semilinear equation and a multiscale -Laplace equation. The results demonstrate significant efficiency gains and good generalization, suggesting scalability of the approach for nonlinear multiscale PDE solvers.

Abstract

Neural networks are powerful tools for approximating high dimensional data that have been used in many contexts, including solution of partial differential equations (PDEs). We describe a solver for multiscale fully nonlinear elliptic equations that makes use of domain decomposition, an accelerated Schwarz framework, and two-layer neural networks to approximate the boundary-to-boundary map for the subdomains, which is the key step in the Schwarz procedure. Conventionally, the boundary-to-boundary map requires solution of boundary-value elliptic problems on each subdomain. By leveraging the compressibility of multiscale problems, our approach trains the neural network offline to serve as a surrogate for the usual implementation of the boundary-to-boundary map. Our method is applied to a multiscale semilinear elliptic equation and a multiscale -Laplace equation. In both cases we demonstrate significant improvement in efficiency as well as good accuracy and generalization performance.

Paper Structure

This paper contains 17 sections, 3 theorems, 33 equations, 16 figures, 8 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Domain Decomposition and the Schwarz method for multiscale elliptic PDEs
  3. Nonlinear elliptic equation with multiscale medium
  4. Domain decomposition and Schwarz iteration
  5. Reduced order Schwarz method based on neural networks
  6. Two observations
  7. Offline training and the full algorithm
  8. Generating training data
  9. Initialization
  10. Numerical results
  11. Semilinear elliptic equations
  12. Offline training
  13. Online phase: Schwarz iteration
  14. $p$-Laplace equations
  15. Offline training
  16. ...and 2 more sections

Key Result

Theorem 2.1

Suppose that the nonlinear function $F^\epsilon$ is uniform elliptic and $u\mapsto F^\epsilon(\cdot,\cdot,u,\cdot)$ is nondecreasing. Let $F^\epsilon$ be pseudo-periodic as defined in eqn:pseudo_periodic. The solution $u^\epsilon$ to eqn:elliptic converges uniformly as $\epsilon \to 0$ to the unique where the homogenized nonlinear function $\bar{F}(R,p,u,x)$ is defined as follows: For a fixed set

Figures (16)

  • Figure 2.1: Domain decomposition for a square 2D geometry. Each patch is labeled by a multi-index $m = (m_1,m_2)$. The patches adjacent to $\Omega_m$ are those on its north/south/west/east sides.
  • Figure 3.1: Singular values of the boundary-to-boundary operator $\mathcal{Q}_m^\epsilon$ for the linear elliptic equation \ref{['eqn:linear_initial_semi']} with medium $\kappa^\epsilon$ defined in \ref{['eqn:medium']} for different values of $\epsilon$ and $\Delta x$ on a local patch. Left plot: $\Delta x = 2^{-8}$. Right plot: $\epsilon = 2^{-4}$. To ensure the regularity of the test function space, the discrete version of the boundary-to-boundary map is represented on basis functions composed of piecewise linear function with fixed step size $2^{-8}$.
  • Figure 3.2: Local enlargement of patches is used to damp boundary effects.
  • Figure 4.1: Medium $\kappa$ for semilinear elliptic equation.
  • Figure 4.2: Training loss for loss function $\mathcal{L}$\ref{['eqn:loss_detail']} for patch (2,2). For the variants that use random initializations, we use the PyTorch default, which generate the weights and biases in each layer uniformly from $(-\sqrt{d_\mathrm{input}},\sqrt{d_\mathrm{input}})$, where $d_\mathrm{input}$ is the input dimension of the layer.
  • ...and 11 more figures

Theorems & Definitions (4)

  • Theorem 2.1: Ev:1992periodic, Theorem 3.3
  • Remark 1
  • Theorem 3.1: Barron's Theorem
  • Corollary 3.1