物理信息神经网络的两层隐藏层是否仍然足够？

本文讨论了各种方法和技术，用于初始化和训练具有单隐藏层的神经网络，以及使用由单隐藏层神经网络组成的可分离物理信息神经网络来求解由常微分方程（ODE）和偏微分方程（PDE）描述的物理问题。提出了一种用于求解由ODE描述的物理问题的单隐藏层神经网络严格确定性初始化方法。对现有损失函数加权方法的修改，以及开发的新方法用于训练严格确定性初始化的神经网络以求解ODE（分离、基于二阶导数的额外加权、基于预测解的加权、相对残差）。提出了一种物理信息数据驱动初始化单隐藏层神经网络的算法。 presented a neural network with pronounced generalizing properties, whose generalizing abilities of which can be precisely controlled by adjusting network parameters. A metric for measuring the generalization of such neural network has been introduced. A gradient-free neuron-by-neuron fitting method has been developed for adjusting the parameters of a single-hidden-layer neural network, which does not require the use of an optimizer or solver for its implementation. The proposed methods have been extended to 2D problems using the separable physics-informed neural networks approach. Numerous experiments have been carried out to develop the above methods and approaches. Experiments on physical problems, such as solving various ODEs and PDEs, have demonstrated that these methods for initializing and training neural networks with one or two hidden layers (SPINN) achieve competitive accuracy and, in some cases, state-of-the-art results.