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A Multiple Transferable Neural Network Method with Domain Decomposition for Elliptic Interface Problems

Tianzheng Lu, Lili Ju, Liyong Zhu

TL;DR

This paper addresses elliptic interface problems with jumps across interfaces and proposes Multi-TransNet, a framework that couples per-subdomain TransNets through interface conditions using nonoverlapping domain decomposition.A core advancement is the empirical formula $\gamma \approx C \frac{M^{1/d}}{R}$ for shaping hidden-layer neurons, extended to multiple subdomains to yield per-subdomain $\gamma_k$ and globally uniform neuron distribution, along with a normalization-based scheme for loss weights.The method is validated through extensive 2D and 3D experiments across low-to-high contrast coefficients, showing superior accuracy, efficiency, and robustness compared with traditional solvers and recent neural methods.Overall, Multi-TransNet provides a scalable, mesh-free approach for elliptic-interface problems with sharp jumps, offering practical impact for simulations in fluids, composite materials, and elasticity.

Abstract

The transferable neural network (TransNet) is a two-layer shallow neural network with pre-determined and uniformly distributed neurons in the hidden layer, and the least-squares solvers can be particularly used to compute the parameters of its output layer when applied to the solution of partial differential equations. In this paper, we integrate the TransNet technique with the nonoverlapping domain decomposition and the interface conditions to develop a novel multiple transferable neural network (Multi-TransNet) method for solving elliptic interface problems, which typically contain discontinuities in both solutions and their derivatives across interfaces. We first propose an empirical formula for the TransNet to characterize the relationship between the radius of the domain-covering ball, the number of hidden-layer neurons, and the optimal neuron shape. In the Multi-TransNet method, we assign each subdomain one distinct TransNet with an adaptively determined number of hidden-layer neurons to maintain the globally uniform neuron distribution across the entire computational domain, and then unite all the subdomain TransNets together by incorporating the interface condition terms into the loss function. The empirical formula is also extended to the Multi-TransNet and further employed to estimate appropriate neuron shapes for the subdomain TransNets, greatly reducing the parameter tuning cost. Additionally, we propose a normalization approach to adaptively select the weighting parameters for the terms in the loss function. Ablation studies and extensive experiments with comparison tests on different types of elliptic interface problems with low to high contrast diffusion coefficients in two and three dimensions are carried out to numerically demonstrate the superior accuracy, efficiency, and robustness of the proposed Multi-TransNet method.

A Multiple Transferable Neural Network Method with Domain Decomposition for Elliptic Interface Problems

TL;DR

This paper addresses elliptic interface problems with jumps across interfaces and proposes Multi-TransNet, a framework that couples per-subdomain TransNets through interface conditions using nonoverlapping domain decomposition.A core advancement is the empirical formula $\gamma \approx C \frac{M^{1/d}}{R}$ for shaping hidden-layer neurons, extended to multiple subdomains to yield per-subdomain $\gamma_k$ and globally uniform neuron distribution, along with a normalization-based scheme for loss weights.The method is validated through extensive 2D and 3D experiments across low-to-high contrast coefficients, showing superior accuracy, efficiency, and robustness compared with traditional solvers and recent neural methods.Overall, Multi-TransNet provides a scalable, mesh-free approach for elliptic-interface problems with sharp jumps, offering practical impact for simulations in fluids, composite materials, and elasticity.

Abstract

The transferable neural network (TransNet) is a two-layer shallow neural network with pre-determined and uniformly distributed neurons in the hidden layer, and the least-squares solvers can be particularly used to compute the parameters of its output layer when applied to the solution of partial differential equations. In this paper, we integrate the TransNet technique with the nonoverlapping domain decomposition and the interface conditions to develop a novel multiple transferable neural network (Multi-TransNet) method for solving elliptic interface problems, which typically contain discontinuities in both solutions and their derivatives across interfaces. We first propose an empirical formula for the TransNet to characterize the relationship between the radius of the domain-covering ball, the number of hidden-layer neurons, and the optimal neuron shape. In the Multi-TransNet method, we assign each subdomain one distinct TransNet with an adaptively determined number of hidden-layer neurons to maintain the globally uniform neuron distribution across the entire computational domain, and then unite all the subdomain TransNets together by incorporating the interface condition terms into the loss function. The empirical formula is also extended to the Multi-TransNet and further employed to estimate appropriate neuron shapes for the subdomain TransNets, greatly reducing the parameter tuning cost. Additionally, we propose a normalization approach to adaptively select the weighting parameters for the terms in the loss function. Ablation studies and extensive experiments with comparison tests on different types of elliptic interface problems with low to high contrast diffusion coefficients in two and three dimensions are carried out to numerically demonstrate the superior accuracy, efficiency, and robustness of the proposed Multi-TransNet method.

Paper Structure

This paper contains 21 sections, 2 theorems, 56 equations, 23 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Review of the transferable neural network method
  3. Geometrization of the neural feature space
  4. Generating the hidden-layer neuron location --- uniform neuron distribution
  5. Tuning the hidden-layer neuron shape and solving the PDE
  6. An efficient empirical formula-based prediction strategy for the shape parameter of TransNet
  7. A multiple transferable neural network method for elliptic interface problems
  8. Nonoverlapping domain decomposition and integrated subdomain TransNets
  9. Globally uniform neuron distribution across subdomains
  10. Extended empirical formula and strategies for the hidden-layer neuron shapes of Multi-TransNet
  11. Adaptively determining the loss weighting parameters through normalization
  12. Numerical experiments
  13. Ablation studies
  14. TransNet
  15. Multi-TransNet
  16. ...and 6 more sections

Key Result

Theorem 1

Given a set of $M$ partition hyperplanes of $\mathbb{R}^d$ defined by If $\left\{\boldsymbol{a}_m\right\}_{m=1}^M$ are i.i.d. and uniformly distributed on the d-dimensional unit sphere, and $\left\{r_m\right\}_{m=1}^M$ are i.i.d. and uniformly distributed in $[0,1]$, then for a fixed $\uptau \in(0,1)$, where $D_M^\uptau(\boldsymbol{x})$ is the density function of the neurons defined by with $\c

Figures (23)

  • Figure 1: Some sample domains with interfaces. Left: $K=2$ with $\Gamma = \overline\Omega_1\cap\overline\Omega_2$; Right: $K=3$ with $\Gamma_1 = \overline\Omega_1\cap\overline\Omega_2$ and $\Gamma_2 = \overline\Omega_2\cap\overline\Omega_3$.
  • Figure 2: Illustration of the density distribution of the partition hyperplanes of hidden-layer neurons from TransNet with different $\boldsymbol{x}_c$ and $R$. Here $M=30,000$ and $\uptau=0.1$. Top-left to top-right: with a scaling of factor of 2.0, top-left to bottom-left: with a scaling of factor of 0.5, and top-left to bottom-right: with a scaling of factor of 0.5 and a translation of the center $(0.5,0.5)$.
  • Figure 3: Illustration of the solution process of the proposed Multi-TransNet method for the elliptic interface problem \ref{['equ:IP']} with $K=2$.
  • Figure 4: Comparisons of the numerical solutions (left) and gradients (right) produced by one TransNet with 100 hidden-layer neurons and the proposed Multi-TransNet with 5 hidden-layer neurons for each of the two subdomain TransNets for the 1D elliptic interface problem \ref{['equ:1D-IP']}.
  • Figure 5: Comparison of the number distribution of hidden-layer neurons of the Multi-TransNet with two subdomain TransNets with the subdomains $\Omega_1 = B_{0.5}\left(0.25, 0.25\right)$ and $\Omega_2 = B_{1}\left(0, 0\right)\setminus B_{0.5}\left(0.25, 0.25\right)$. Here $M_1+M_2$ and $\uptau$ are fixed to 60000 and 0.1, respectively. Top row: $M_1=M_2=30000$; Bottom row: $M_1=20000$ and $M_2=40000$ thus ${M_1}/{R_1} = {M_2}/{R_2}$.
  • ...and 18 more figures

Theorems & Definitions (5)

  • Theorem 1: Uniform neuron distribution in the unit ball $B_{1}(\boldsymbol{0})$
  • Theorem 2: Uniform neuron distribution in the ball $B_{R}(\boldsymbol{x}_c)$
  • Remark 1: The selection of parameters $\boldsymbol{x}_c$ and $R$ for a general domain $\Omega$
  • Remark 2
  • Remark 3