Obtaining the solutions of partial differential equations based on various machine learning methods has drawn more and more attention in the fields of scientific computation and engineering applications. In this work, we first propose a coupled Extreme Learning Machine (called CELM) method incorporated with the physical laws to solve a class of fourth-order biharmonic equations by reformulating it into two well-posed Poisson problems. In addition, some activation functions including tangent, gauss, sine, and trigonometric (sin+cos) functions are introduced to assess our CELM method. Notably, the sine and trigonometric functions demonstrate a remarkable ability to effectively minimize the approximation error of the CELM model. In the end, several numerical experiments are performed to study the initializing approaches for both the weights and biases of the hidden units in our CELM model and explore the required number of hidden units. Numerical results show the proposed CELM algorithm is high-precision and efficient to address the biharmonic equation in both regular and irregular domains.

翻译：基于各种机器学习方法获取偏微分方程的解，在科学计算和工程应用领域日益受到关注。本文首次提出一种融入物理规律的耦合极端学习机（CELM）方法，通过将一类四阶双调和方程重新表述为两个适定的泊松问题来求解。此外，引入正切、高斯、正弦和三角（sin+cos）函数等激活函数评估我们的CELM方法。值得注意的是，正弦和三角函数在有效最小化CELM模型的逼近误差方面展现出卓越能力。最后，通过多个数值实验研究了我们CELM模型中隐含单元权重和偏置的初始化方法，并探索了所需的隐含单元数量。数值结果表明，所提出的CELM算法在求解规则和不规则区域上的双调和方程时具有高精度和高效性。