自动微分在聚合物黏度拟合中的应用

摘要

在聚合物七参数Cross-WLF黏度模型的非线性拟合中,需要解决的一个关键问题是如何精确计算梯度矢量Δf2（x）及汉森矩阵Δf2（x）。因7个黏度参数的数量级差别巨大,传统的数值微分算法因为容易产生歧义而不再适用,本文采用自动微分方法计算Δf2（x）及Δf2（x）,在忽略截断误差的情况下,计算结果即为解析解,该方法不仅有效地保障了Δf2（x）及Δf2（x）的计算精度,而且提高了拟合计算的稳定性,本文算例结果表明拟合误差（RMS误差）可达1×10-5量级。%The key problem in evaluating the parameters of the 7-parameter Cross-WLF viscosity model of polymer materials with nonlinear fitting method is how exactly to calculate the gradient vector f2（x） and the Hessian matrix f2（x）.Numerical method loses its efficiency here and singularity problem arises due to the significant difference of the parameters in magnitude.In order to overcome this problem,automatic differentiation method was adopted in this paper to calculate？ f2（x） and？ f2（x）.Its differentiation accuracy is up to that of analytical method when neglecting the truncation error.Further more,this method is of high stability and benefits much to the final result of the fitting program.Test case shows the RMS error between the fitting viscosity and the experimental viscosity is less than 1×10-5.