The article is devoted to the study of neural networks with one hidden layer and a modified activation function for solving physical problems. A rectified sigmoid activation function has been proposed to solve physical problems described by the ODE with neural networks. Algorithms for physics-informed data-driven initialization of a neural network and a neuron-by-neuron gradient-free fitting method have been presented for the neural network with this activation function. Numerical experiments demonstrate the superiority of neural networks with a rectified sigmoid function over neural networks with a sigmoid function in the accuracy of solving physical problems (harmonic oscillator, relativistic slingshot, and Lorentz system).

翻译：本文致力于研究具有单隐藏层及改进激活函数的神经网络在物理问题求解中的应用。针对常微分方程描述的物理问题，提出了一种修正S型激活函数用于神经网络求解。针对采用该激活函数的神经网络，提出了基于物理信息数据驱动的初始化算法以及逐神经元无梯度拟合方法。数值实验证明，在求解物理问题（谐振子、相对论弹弓效应及洛伦兹系统）的精度方面，采用修正S型函数的神经网络优于采用标准S型函数的神经网络。