One of the obstacles hindering the scaling-up of the initial successes of machine learning in practical engineering applications is the dependence of the accuracy on the size of the database that "drives" the algorithms. Incorporating the already-known physical laws into the training process can significantly reduce the size of the required database. In this study, we establish a neural network-based computational framework to characterize the finite deformation of elastic plates, which in classic theories is described by the F\"oppl--von K\'arm\'an (FvK) equations with a set of boundary conditions (BCs). A neural network is constructed by taking the spatial coordinates as the input and the displacement field as the output to approximate the exact solution of the FvK equations. The physical information (PDEs, BCs, and potential energies) is then incorporated into the loss function, and a pseudo dataset is sampled without knowing the exact solution to finally train the neural network. The prediction accuracy of the modeling framework is carefully examined by applying it to four different loading cases: in-plane tension with non-uniformly distributed stretching forces, in-plane central-hole tension, out-of-plane deflection, and buckling under compression. Two ways of formulating the loss function are compared, one based on the PDEs and BCs, and the other based on the total potential energy of the plate. Through the comparison with the finite element simulation results, it is found that our computational framework is capable of characterizing the elastic deformation of plates with a satisfactory accuracy. Compared with incorporating the PDEs and BCs in the loss, using the total potential energy is a better way in terms of training accuracy and efficiency.

中文翻译：

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弹性板的物理引导神经网络框架：基于控制方程和基于能量的方法的比较

阻碍机器学习在实际工程应用中取得初步成功的障碍之一是准确性对“驱动”算法的数据库大小的依赖性。将已知的物理定律纳入训练过程可以显着减少所需数据库的大小。在这项研究中，我们建立了一个基于神经网络的计算框架来表征弹性板的有限变形，在经典理论中，它由 F\"oppl--von K\'arm\'an (FvK) 方程描述，其中一组边界条件（BCs），以空间坐标为输入，以位移场为输出，逼近FvK方程的精确解，构建神经网络。物理信息（PDEs, BCs, 和势能）然后被合并到损失函数中，并且在不知道最终训练神经网络的确切解的情况下对伪数据集进行采样。通过将其应用于四种不同的载荷情况，仔细检查了建模框架的预测精度：具有非均匀分布的拉伸力的面内张力、面内中心孔张力、面外挠曲和压缩下的屈曲. 比较了两种制定损失函数的方法，一种基于 PDE 和 BC，另一种基于板的总势能。通过与有限元模拟结果的比较，发现我们的计算框架能够以令人满意的精度表征板的弹性变形。