基于优化Kolmogorov--Arnold神经网络的二维双调和方程无网格数值逼近
Mesh-Free Numerical Approximation of the Biharmonic Equation via Optimized Kolmogorov--Arnold Neural Networks
- Texas A&M University-Corpus Christi（德克萨斯农工大学科珀斯克里斯蒂分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出基于Kolmogorov--Arnold物理信息神经网络的无网格方法求解双调和方程，通过归一化残差和混合优化，在单位方形基准上实现相对L2误差1.593e-5，仅用7801参数，无需网格或辅助变量。
AI中文摘要：
本文开发了一种基于Kolmogorov--Arnold物理信息神经网络（KAN-PINNs）的无网格数值框架，用于逼近四阶椭圆边值问题，并具体应用于控制薄板挠度的双调和方程。与依赖固定节点激活函数的传统多层感知器不同，本文在网络边上部署了通过径向基函数参数化的可学习单变量函数，同时通过双曲正切激活映射保持高阶可微性。针对四阶微分算子和完全固支边界条件固有的严重数值刚性，本文采用直接归一化残差公式，并结合混合多阶段AdamW到L-BFGS优化流程加以解决。实现了一种自动化的24/7爬山搜索协议，以系统校准边界惩罚权重和优化调度。在单位正方形上的光滑制造基准测试中，连续迭代实现了超过150倍的误差降低因子，最终相对L2误差达到1.593×10^{-5}（0.00159%），训练损失为3.065×10^{-6}，仅使用7,801个可训练参数。这些结果表明，使用紧凑的无网格KAN-PINNs架构可以准确求解高阶偏微分方程问题，而无需辅助变量变换或离散网格生成。
英文摘要：
A mesh-free numerical framework based on \textit{Kolmogorov--Arnold Physics-Informed Neural Networks} (KAN-PINNs) is developed for the approximation of fourth-order elliptic boundary value problems, with specific application to the biharmonic equation governing thin plate deflection. Unlike conventional Multi-Layer Perceptrons relying on fixed nodal activations, learnable univariate functions parameterized via radial basis functions are deployed on network edges, while high-order differentiability is preserved through hyperbolic tangent activation mappings. The severe numerical stiffness inherent to fourth-order differential operators and fully clamped boundary conditions is addressed through a direct normalized residual formulation coupled with a hybrid, multi-stage AdamW-to-L-BFGS optimization pipeline. An automated \textit{24/7 hill-climbing search protocol} is implemented to systematically calibrate boundary penalty weights and optimization schedules. When evaluated on a smooth manufactured benchmark on the unit square, an error reduction factor exceeding $150\times$ is attained over successive iterations, culminating in a final relative $L_2$ error of $1.593 \times 10^{-5}$ ($0.00159\%$) and a training loss of $3.065 \times 10^{-6}$ utilizing only $7,801$ trainable parameters. These results demonstrate that high-order PDE problems can be accurately resolved using compact, mesh-free KAN-PINNs architecture without requiring auxiliary variable transformations or discrete mesh generation.
