边界积分方程的多层随机梯度神经求解器
A multilevel stochastic-gradient neural solver for boundary integral equations
- National Chung Cheng University（国立中正大学）
- The University of Texas at Austin（德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出一种多层随机梯度神经求解器，通过逐步细化求积网格训练多层感知机，利用神经正切核谱分析和多网格思想克服频谱偏差，实现高效求解第二类边界积分方程。
AI中文摘要：
我们为第二类边界积分方程开发了一种多层随机梯度神经求解器。未知密度由多层感知机表示，通过在一系列逐步细化的求积网格上最小化Nyström离散残差来训练，每一层从前一层的参数热启动。每一步仅需在随机选取的配置行小批量上进行稠密矩阵-向量乘积和网络前向传播，这些操作可直接映射到GPU硬件。残差收缩由经验神经正切核（NTK）控制，该核是单个连续核的离散样本。在固定网格上，一旦残差集中在网络收缩缓慢的模式上，训练就会停滞，这一平台期由频率原理描述；谱分析解释并通过实验证实，细化求积如何解析更多连续核的谱并将这些模式重新纳入优化器的可达范围。频谱偏差在其他地方是神经网络求解器的障碍，在此处却充当了多重网格型迭代的光滑器，求积细化替代了粗网格校正。在网络均匀正则性假设下，总工作量是最终网格工作量的常数倍，离散第二类算子的均匀条件使得NTK成为唯一的速率决定谱，同时将训练残差转化为后验误差界。在内Dirichlet拉普拉斯/泊松问题和外Neumann亥姆霍兹问题上的实验，使用参数化和符号距离曲面表示，证明了所提方法在可比容差下与GMRES相比的有效性和效率。
英文摘要：
We propose a multilevel stochastic-gradient neural solver (MLSG) for second-kind boundary integral equations. MLSG represents the unknown boundary density using a neural network optimized by stochastic residual minimization over a hierarchy of successively refined Nyström discretizations. Upon transitioning from one level to the next, the network parameters obtained on the previous level initialize training on the current one. This coarse-to-fine strategy retains a continuous, grid-independent density representation and is designed to reduce the total computational effort required to reach a prescribed residual tolerance at the target resolution. The algorithm avoids grid-transfer operators and hierarchical fast-summation machinery, relying instead on batched kernel evaluations and standard network forward and backward passes that map efficiently onto modern GPU architectures. For uniformly stable second-kind discretizations, so strongly nonuniform contraction rates originate in the empirical neural tangent kernel (NTK) rather than in the discretized operator. Within each level, parameter updates can reshape the NTK, while refinement re-samples the tangent kernel on a richer discrete space and reveals directions that were not adequately resolved on coarser levels. A cross-level estimate bounds the warm-start loss in terms of the preceding training tolerance and quadrature error, motivating a tolerance schedule that balances optimization and discretization errors. Experiments on Laplace/Poisson and Helmholtz problems in two and three dimensions, together with an exterior Robin problem on a hypersurface in $\R^4$, demonstrate the method under both parametric and signed-distance surface representations at up to million-scale discretizations.
