Encoder-X: Solving Unknown Coefficients Automatically in Polynomial Fitting by Using an Autoencoder.

Modeling, prediction, and recognition tasks depend on the proper representation of the objective curves and surfaces. Polynomial functions have been proved to be a powerful tool for representing curves and surfaces. Until now, various methods have been used for polynomial fitting. With a recent boom in neural networks, researchers have attempted to solve polynomial fitting by using this end-to-end model, which has a powerful fitting ability. However, the current neural network-based methods are poor in stability and slow in convergence speed. In this article, we develop a novel neural network-based method, called Encoder-X, for polynomial fitting, which can solve not only the explicit polynomial fitting but also the implicit polynomial fitting. The method regards polynomial coefficients as the feature value of raw data in a polynomial space expression and therefore polynomial fitting can be achieved by a special autoencoder. The entire model consists of an encoder defined by a neural network and a decoder defined by a polynomial mathematical expression. We input sampling points into an encoder to obtain polynomial coefficients and then input them into a decoder to output the predicted function value. The error between the predicted function value and the true function value can update parameters in the encoder. The results prove that this method is better than the compared methods in terms of stability, convergence, and accuracy. In addition, Encoder-X can be used for solving other mathematical modeling tasks.