用改进的分段激活函数提升物理信息神经网络求解精度
About rectified sigmoid function for enhancing the accuracy of Physics-Informed Neural Networks
- 提出分段修正的Sigmoid函数，增强网络拟合能力
- 在谐振子等3类物理问题上，准确率显著优于传统Sigmoid
- 适合需要高精度求解微分方程的科学计算场景
本文研究单隐层神经网络结合改进激活函数求解物理问题。提出一种分段修正Sigmoid激活函数，用于求解常微分方程（ODE）描述的物理系统。设计了基于物理信息的数据驱动初始化算法，以及逐神经元的无梯度优化方法。数值实验表明，在谐振子、相对论弹弓效应和Lorentz系统三类问题中，采用该激活函数的神经网络求解精度显著优于传统Sigmoid网络。
原文摘要 · Abstract (English)
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).
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