Obtaining the solutions of partial differential equations based on 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, gaussian, sine, and trigonometric functions are introduced to assess our CELM method. Furthermore, we introduce several activation functions, such as tangent, Gaussian, sine, and trigonometric functions, to evaluate the performance of 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 ways 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 equations on both regular and irregular domains.

翻译：基于机器学习方法求解偏微分方程在科学计算和工程应用领域日益受到关注。本文首先提出一种融入物理规律的耦合极限学习机（CELM）方法，通过将四阶双调和方程重构为两个适定的泊松问题来求解该类方程。此外，我们引入了包括正切、高斯、正弦和三角函数在内的多种激活函数来评估CELM方法的性能。值得注意的是，正弦和三角函数在有效最小化CELM模型逼近误差方面展现出显著能力。最后，通过数值实验研究了CELM模型中隐含层权值与偏置的初始化方式，并探索了所需隐含层单元数量。数值结果表明，所提出的CELM算法在规则域与非规则域上求解双调和方程时均具有高精度与高效率。