Use of neural networks to solve the integral equation of the Laser Intensity Modulation Method (LIMM)

Determination of the polarization distribution in ferroelectrics using LIMM requires the solution of a Fredholm integral equation of the 1st kind. The constrained regularization method which has been used successfully to solve this problem for single-layer ferroelectrics fails when applied to thin films on a multi-layer substrate. A new approach based on neural networks has been developed to alleviate these problems. In this method, a functional form containing a small number of undetermined parameters is selected to represent the unknown polarization. The LIMM equation is used to synthesize a very large number of data sets (current vs frequency), each one corresponding to a polarization distribution with a different set of parameters. These data are used to 'train' the neural network. Then the neural network is used to predict parameters for data sets on which it was not 'trained.'