Deep learning is a powerful tool for solving data driven differential problems and has come out to have successful applications in solving direct and inverse problems described by PDEs, even in presence of integral terms. In this paper, we propose to apply radial basis functions (RBFs) as activation functions in suitably designed Physics Informed Neural Networks (PINNs) to solve the inverse problem of computing the peridynamic kernel in the nonlocal formulation of classical wave equation, resulting in what we call RBF-iPINN. We show that the selection of an RBF is necessary to achieve meaningful solutions, that agree with the physical expectations carried by the data. We support our results with numerical examples and experiments, comparing the solution obtained with the proposed RBF-iPINN to the exact solutions.

翻译：深度学习是解决数据驱动微分问题的有力工具，在求解含积分项的偏微分方程正反问题方面已取得成功应用。本文提出在合理设计的物理信息神经网络（PINNs）中采用径向基函数（RBFs）作为激活函数，以求解经典波动方程非局部形式下近场动力学核函数的反问题，由此提出RBF-iPINN方法。我们证明，必须选择径向基函数才能获得与数据所蕴含物理预期一致的具有实际意义的解。通过数值算例与实验，将所提出的RBF-iPINN方法所得解与精确解进行对比，验证了结果的可靠性。