As a method of function approximation, polynomial fitting has always been the main research hotspot in mathematical modeling. In many disciplines such as computer, physics, biology, neural networks have been widely used, and most of the applications have been transformed into fitting problems using neural networks. One of the main reasons that neural networks can be widely used is that it has a certain sense of universal approximation. In order to fit the polynomial, this paper constructs a three-layer feedforward neural network, uses Taylor series as the activation function, and determines the number of hidden layer neurons according to the order of the polynomial and the dimensions of the input variables. For explicit polynomial fitting, this paper uses non-linear functions as the objective function, and compares the fitting effects under different orders of polynomials. For the fitting of implicit polynomial curves, the current popular polynomial fitting algorithms are compared and analyzed. Experiments have proved that the algorithm used in this paper is suitable for both explicit polynomial fitting and implicit polynomial fitting. The algorithm is relatively simple, practical, easy to calculate, and can efficiently achieve the fitting goal. At the same time, the computational complexity is relatively low, which has certain application value.

Different face regions have different contributions to recognition. Especially in the wild environment, the difference of contributions will be further amplified due to a lot of interference. Based on this, this paper proposes an attention-aware face recognition method based on a deep convolutional neural network and reinforcement learning. The proposed method composes of an Attention-Net and a Feature-net. The Attention-Net is used to select patches in the input face image according to the facial landmarks and trained with reinforcement learning to maximize the recognition accuracy. The Feature-net is used for extracting discriminative embedding features. In addition, a regularization method has also been introduced. The mask of the input layer is also applied to the intermediate feature maps, which is an approximation to train a series of models for different face patches and provide a combined model. Our method achieves satisfactory recognition performance on its application to the public prevailing face verification database.

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.