This study investigates different Scientific Machine Learning (SciML) approaches for the analysis of functionally graded (FG) porous beams and compares them under a new framework. The beam material properties are assumed to vary as an arbitrary continuous function. The methods consider the output of a neural network/operator as an approximation to the displacement fields and derive the equations governing beam behavior based on the continuum formulation. The methods are implemented in the framework and formulated by three approaches: (a) the vector approach leads to a Physics-Informed Neural Network (PINN), (b) the energy approach brings about the Deep Energy Method (DEM), and (c) the data-driven approach, which results in a class of Neural Operator methods. Finally, a neural operator has been trained to predict the response of the porous beam with functionally graded material under any porosity distribution pattern and any arbitrary traction condition. The results are validated with analytical and numerical reference solutions. The data and code accompanying this manuscript will be publicly available at https://github.com/eshaghi-ms/DeepNetBeam.

翻译：本研究探讨了用于功能梯度多孔梁分析的不同科学机器学习方法，并在一个新框架下对它们进行了比较。假设梁的材料属性随任意连续函数变化。这些方法将神经网络/算子的输出视为位移场的近似，并基于连续介质公式推导出控制梁行为的方程。这些方法在框架中通过三种途径实现：(a) 向量法导出了物理信息神经网络，(b) 能量法引出了深度能量法，(c) 数据驱动法产生了一类神经算子方法。最后，训练了一个神经算子来预测具有功能梯度材料的多孔梁在任何孔隙分布模式和任意牵引条件下的响应。结果通过解析和数值参考解进行了验证。本文附带的数据和代码将在 https://github.com/eshaghi-ms/DeepNetBeam 公开提供。