Despite their remarkable performance, deep neural networks remain mostly ``black boxes'', suggesting inexplicability and hindering their wide applications in fields requiring making rational decisions. Here we introduce HOPE (High-order Polynomial Expansion), a method for expanding a network into a high-order Taylor polynomial on a reference input. Specifically, we derive the high-order derivative rule for composite functions and extend the rule to neural networks to obtain their high-order derivatives quickly and accurately. From these derivatives, we can then derive the Taylor polynomial of the neural network, which provides an explicit expression of the network's local interpretations. Numerical analysis confirms the high accuracy, low computational complexity, and good convergence of the proposed method. Moreover, we demonstrate HOPE's wide applications built on deep learning, including function discovery, fast inference, and feature selection. The code is available at https://github.com/HarryPotterXTX/HOPE.git.

翻译：尽管深度神经网络性能卓越，其本质上仍是“黑箱”，这种不可解释性阻碍了其在需要理性决策的领域的广泛应用。本文提出HOPE（高阶多项式展开）方法，该方法可将网络在参考输入上展开为高阶泰勒多项式。具体而言，我们推导了复合函数的高阶导数规则，并将其推广至神经网络，从而快速准确地获得网络的高阶导数。基于这些导数，可进一步得到神经网络的泰勒多项式——该多项式提供了网络局部解释的显式表达式。数值分析证实了所提方法的高精度、低计算复杂度及良好的收敛性。此外，我们展示了HOPE在深度学习领域的广泛应用，包括函数发现、快速推理与特征选择。代码已开源至https://github.com/HarryPotterXTX/HOPE.git。