To address the sensitivity of parameters and limited precision for physics-informed extreme learning machines (PIELM) with common activation functions, such as sigmoid, tangent, and Gaussian, in solving high-order partial differential equations (PDEs) relevant to scientific computation and engineering applications, this work develops a Fourier-induced PIELM (FPIELM) method. This approach aims to approximate solutions for a class of fourth-order biharmonic equations with two boundary conditions on both unitized and non-unitized domains. By carefully calculating the differential and boundary operators of the biharmonic equation on discretized collections, the solution for this high-order equation is reformulated as a linear least squares minimization problem. We further evaluate the FPIELM with varying hidden nodes and scaling factors for uniform distribution initialization, and then determine the optimal range for these two hyperparameters. Numerical experiments and comparative analyses demonstrate that the proposed FPIELM method is more stable, robust, precise, and efficient than other PIELM approaches in solving biharmonic equations across both regular and irregular domains.

翻译：针对采用常见激活函数（如 sigmoid、tanh 和高斯函数）的物理信息极限学习机（PIELM）在求解与科学计算和工程应用相关的高阶偏微分方程（PDE）时存在的参数敏感性和精度有限的问题，本研究发展了一种傅里叶诱导的 PIELM（FPIELM）方法。该方法旨在逼近在单位化及非单位化域上具有两类边界条件的一类四阶双调和方程的解。通过在离散点集上仔细计算双调和方程的微分算子与边界算子，将该高阶方程的解重新表述为一个线性最小二乘最小化问题。我们进一步评估了 FPIELM 在不同隐藏层节点数和均匀分布初始化的缩放因子下的性能，从而确定了这两个超参数的最优取值范围。数值实验与对比分析表明，在求解规则和不规则域上的双调和方程时，所提出的 FPIELM 方法比其他 PIELM 方法更稳定、鲁棒、精确和高效。