arXiv:2502.19893math.NAcs.LG2025-02被引 18

用分域神经网络解决带界面不连续的椭圆型问题,精度效率双提升。

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

  • 分域设计多套可迁移神经网络,每域自适应配置隐层节点数。
  • 在二维和三维低至高对比度系数问题中,误差降低20%以上。
  • 适合需要高鲁棒性求解界面问题的科研与工程场景。

迁移神经网络(TransNet)是一种具有预设均匀分布隐层神经元的两层浅层网络,适用于偏微分方程求解。本文将TransNet与非重叠域分解及界面条件结合,提出一种新型多迁移神经网络(Multi-TransNet)方法,用于求解通常在界面处存在解及其导数不连续的椭圆型界面问题。首先提出一个经验公式,描述域覆盖球半径、隐层神经元数量与最优神经元形状之间的关系。在Multi-TransNet中,为每个子域分配独立的TransNet,自适应确定其隐层神经元数量以保持全域均匀分布,并通过将界面条件项引入损失函数来联合各子域网络。该经验公式也被扩展至Multi-TransNet,用于估算子域TransNet的合适神经元形状,显著降低参数调优成本。此外,提出一种归一化方法,自适应选择损失函数中各项的权重。通过消融实验与大量对比测试,在二维和三维不同类型的椭圆型界面问题上,涵盖低至高对比度扩散系数,数值验证了所提方法在精度、效率和鲁棒性上的优越性。

原文摘要 · Abstract (English)

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.

神经网络界面问题域分解迁移学习

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