Interpretability of neural networks and their underlying theoretical behavior remain an open field of study even after the great success of their practical applications, particularly with the emergence of deep learning. In this work, NN2Poly is proposed: a theoretical approach to obtain an explicit polynomial model that provides an accurate representation of an already trained fully-connected feed-forward artificial neural network (a multilayer perceptron or MLP). This approach extends a previous idea proposed in the literature, which was limited to single hidden layer networks, to work with arbitrarily deep MLPs in both regression and classification tasks. The objective of this paper is to achieve this by using a Taylor expansion on the activation function, at each layer, and then using several combinatorial properties to calculate the coefficients of the desired polynomials. Discussion is presented on the main computational challenges of this method, and the way to overcome them by imposing certain constraints during the training phase. Finally, simulation experiments as well as an application to a real data set are presented to demonstrate the effectiveness of the proposed method.

翻译：在深度学习取得巨大实践成功，特别是随着其实际应用的涌现，神经网络的可解释性及其潜在的理论行为仍是一个开放的研究领域。本文提出NN2Poly方法：这是一种理论方法，旨在获得一个显式多项式模型，该模型能够精确表征已训练的全连接前馈人工神经网络（即多层感知机，MLP）。该方法将文献中先前仅适用于单隐层网络的思路，扩展至可处理任意深度的MLP，适用于回归与分类任务。本文的目标是通过对各层激活函数进行泰勒展开，并利用若干组合数学性质计算所需多项式系数来实现这一扩展。文中讨论了该方法面临的主要计算挑战，以及通过在训练阶段施加特定约束来克服这些挑战的途径。最后，通过仿真实验及实际数据集的应用验证了该方法的有效性。