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A Least-Squares-Based Regularity-Conforming Neural Networks (LS-ReCoNNs) for Solving Parametric Transmission Problems

Shima Baharlouei, Jamie Taylor, David Pardo

TL;DR

Numerical experiments in one and two dimensions demonstrate that LS-ReCoNN effectively captures singularities while maintaining solution accuracy across a wide range of parameter values.

Abstract

This article focuses on solving parametric transmission problems in one and two spatial dimensions. These problems belong to a class of partial differential equations that arise in the modeling of physical systems with heterogeneous materials. They often exhibit discontinuities across interfaces and singularities at points where interfaces intersect. To address these problems, we propose a new deep learning approach named {\it{Least-Squares-Based Regularity-Conforming Neural Network (LS-ReCoNN)}}. This approach proposes a loss function that is shown to be a consistent upper bound for the energy-norm error. The method represents the solution as the sum of a principal component and a singular component. The principal component is decomposed into smooth and gradient-jump parts, which capture both the regular solution behavior and reduced regularity across interfaces in one- and two-dimensional problems. The singular component is introduced to model junction singularities and it is approximated using basis functions computed from a one-dimensional finite element eigenvalue problem. For the principal component, a separated representation is employed, consisting of parameter-dependent coefficients and space-dependent functions. A deep neural network approximates the space-dependent functions, while the parameter-dependent coefficients are determined by a least-squares solver, where the optimal coefficients for each parameter instance are obtained online by solving a low-dimensional least-squares problem. Numerical experiments in one and two dimensions demonstrate that LS-ReCoNN effectively captures singularities while maintaining solution accuracy across a wide range of parameter values.

A Least-Squares-Based Regularity-Conforming Neural Networks (LS-ReCoNNs) for Solving Parametric Transmission Problems

TL;DR

Numerical experiments in one and two dimensions demonstrate that LS-ReCoNN effectively captures singularities while maintaining solution accuracy across a wide range of parameter values.

Abstract

This article focuses on solving parametric transmission problems in one and two spatial dimensions. These problems belong to a class of partial differential equations that arise in the modeling of physical systems with heterogeneous materials. They often exhibit discontinuities across interfaces and singularities at points where interfaces intersect. To address these problems, we propose a new deep learning approach named {\it{Least-Squares-Based Regularity-Conforming Neural Network (LS-ReCoNN)}}. This approach proposes a loss function that is shown to be a consistent upper bound for the energy-norm error. The method represents the solution as the sum of a principal component and a singular component. The principal component is decomposed into smooth and gradient-jump parts, which capture both the regular solution behavior and reduced regularity across interfaces in one- and two-dimensional problems. The singular component is introduced to model junction singularities and it is approximated using basis functions computed from a one-dimensional finite element eigenvalue problem. For the principal component, a separated representation is employed, consisting of parameter-dependent coefficients and space-dependent functions. A deep neural network approximates the space-dependent functions, while the parameter-dependent coefficients are determined by a least-squares solver, where the optimal coefficients for each parameter instance are obtained online by solving a low-dimensional least-squares problem. Numerical experiments in one and two dimensions demonstrate that LS-ReCoNN effectively captures singularities while maintaining solution accuracy across a wide range of parameter values.
Paper Structure (37 sections, 7 theorems, 93 equations, 18 figures, 2 algorithms)

This paper contains 37 sections, 7 theorems, 93 equations, 18 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model Problem: Parametric Elliptic Transmission Problem
  3. Formulation of the Problem
  4. Decomposition of the Solution
  5. Continuum loss function
  6. LS-ReCoNN Framework
  7. Solution Space Discretization
  8. LS-ReCoNN Approximate Solution
  9. LS-ReCoNN Loss Function
  10. Loss Discretization
  11. Analysis of Computational Cost
  12. Numerical Examples
  13. One-Dimensional Problems
  14. Comparison to single-instance problem:
  15. Two-Dimensional Problems
  16. ...and 22 more sections

Key Result

Theorem 2.1

Let $u\in \mathbb{X}^p$ and $\Theta>0$. Then, there exists a constant $C>0$, depending only on $\Theta$ and the set of discontinuities $\Gamma$ of $p$, such that

Figures (18)

  • Figure 1: Illustration of the domain, subdomains, and relative notations in one and two dimensions.
  • Figure 2: Active regions of the decomposed solution components: the smooth part $\mathfrak{w}^p$ is active over the entire domain $\Omega$ except on the material interfaces, while the gradient jump part $\mathfrak{v}^p$ and the singular component $\mathfrak{s}^p$ are localized to the material interfaces $\Gamma$ and the singular vertices, respectively.
  • Figure 3: NN architecture for the parameter-independent trial functions. The network maps spatial coordinates to a vector of raw outputs that represent the smooth and gradient jump components of the solution.
  • Figure 4: Schematic of the LS-ReCoNN architectures for solving one-dimensional parametric transmission problems.
  • Figure 5: Schematic of the LS-ReCoNN architectures for solving two-dimensional parametric transmission problems.
  • ...and 13 more figures

Theorems & Definitions (13)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma E.1
  • proof
  • Lemma E.2
  • proof
  • Theorem E.3
  • ...and 3 more