通用可迁移神经网络用于稳态偏微分方程
Generalized Transferable Neural Networks for Steady-State Partial Differential Equations
AI总结：
本文提出通用可迁移神经网络GTransNet，通过增加隐藏层并保留可解释特征生成机制，提升解决稳态偏微分方程的准确性和稳定性。
摘要：
深度学习已成为科学和工程计算中的有力框架，推动了基于神经网络的偏微分方程求解器的发展。在这一领域，具有确定性特征构建的网络架构变得引人注目，提供了高精度和计算效率。其中，可迁移神经网络（TransNet）是一种特殊的浅层神经网络（即单隐藏层架构），其隐藏层参数根据均匀分布超平面的原则预先确定。尽管TransNet在解决具有相对平滑解的偏微分方程方面表现出色，但其精度和稳定性在存在高度振荡解结构时可能下降，其中激活饱和和系统条件问题成为限制因素。本文提出了一种通用可迁移神经网络（GTransNet）用于解决稳态偏微分方程，该网络在原有TransNet设计的基础上增加了额外的隐藏层，同时保留了其可解释的特征生成机制。特别是，GTransNet的第一隐藏层保留了TransNet的参数采样策略，但加入了对神经元偏置的对称约束，而后续的隐藏层则省略了偏置项，并采用方差控制的采样策略来选择神经元权重。
英文摘要：
Deep learning has emerged as a compelling framework for scientific and engineering computing, motivating growing interest in neural network-based solvers for partial differential equations (PDEs). Within this landscape, network architectures with deterministic feature construction have become an appealing approach, offering both high accuracy and computational efficiency in practice. Among them, the transferable neural network (TransNet) is a special class of shallow neural networks (i.e., single-hidden-layer architectures), whose hidden-layer parameters are predetermined according to the principle of uniformly distributed partition hyperplanes. Although TransNet has demonstrated strong performance in solving PDEs with relatively smooth solutions, its accuracy and stability may deteriorate in the presence of highly oscillatory solution structures, where activation saturation and system conditioning issues become limiting factors. In this paper, we propose a generalized transferable neural network (GTransNet) for solving steady-state PDEs, which augments the original TransNet design with additional hidden layers while preserving its interpretable feature-generation mechanism. In particular, the first hidden layer of GTransNet retains TransNet's parameter sampling strategy but incorporates an additional symmetry constraint on the neuron biases, while the subsequent hidden layers omit bias terms and employ a variance-controlled sampling strategy for selecting neuron weights.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
