发表机构
School of Mathematics and Statistics, Wuhan University; Department of Mathematics, University of South Carolina（武汉大学数学与统计学院; 南卡罗来纳大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出LD-GTransNet方法，通过升阶-解耦策略统一逼近分段光滑解，简化为线性最小二乘问题，高效精确求解椭圆和移动界面问题，优于现有方法。
AI 中文摘要
本文提出了一种高效且精确的方法，称为“升阶-解耦广义可迁移神经网络”（LD-GTransNet），用于求解椭圆和移动界面问题的数值解。我们的方法利用了多层GTransNet方法，并引入了一种新颖的升阶-解耦策略，其中隐藏层神经元参数被预设，物理坐标和添加的辅助变量由单独的子网络处理。这种设计使得无需显式域分解即可实现对分段光滑解的统一全局逼近，同时保持了网络的表达能力，并避免了传统升阶方法中固有的冗余。由此产生的公式简化为线性最小二乘问题，从而消除了非线性训练的需求，确保了高计算效率和精度。在二维和三维基准界面问题上的大量数值实验表明，与现有的基于神经网络的方法和传统数值方法相比，我们提出的LD-GTransNet始终提供优越的性能，特别是在高对比度系数和复杂界面几何的问题上。
英文摘要
In this paper, we propose a highly efficient and accurate method, called ``lift-and-decoupling generalized transferable neural network" (LD-GTransNet), for numerical solution of elliptic and moving interface problems. Our method makes use of the multi-layer GTransNet approach and incorporates a novel {lift-and-decoupling} strategy, in which the hidden-layer neuron parameters are preset and the physical coordinates and the added auxiliary variable are processed by separate subnetworks. Such design enables a unified global approximation of piecewise smooth solutions without explicit domain decomposition, while preserving network expressivity and avoiding the redundancy inherent in conventional {lift} approaches. The resulting formulation reduces to a linear least-squares problem, thereby eliminating the need of nonlinear training and ensuring the high computational efficiency and accuracy. Extensive numerical experiments on benchmark interface problems in two and three dimensions demonstrate that, compared with existing neural network-based and traditional numerical methods, our proposed LD-GTransNet consistently delivers superior performance, particularly for problems with high-contrast coefficients and complex interface geometries.
