This paper presents a concise mathematical framework for investigating both feed-forward and backward process, during the training to learn model weights, of an artificial neural network (ANN). Inspired from the idea of the two-step rule for backpropagation, we define a notion of F-adjoint which is aimed at a better description of the backpropagation algorithm. In particular, by introducing the notions of F-propagation and F-adjoint through a deep neural network architecture, the backpropagation associated to a cost/loss function is proven to be completely characterized by the F-adjoint of the corresponding F-propagation relatively to the partial derivative, with respect to the inputs, of the cost function.

翻译：本文提出了一个简洁的数学框架，用于研究人工神经网络（ANN）在训练过程中学习模型权重的前馈和反向过程。受反向传播两步法则思想的启发，我们定义了“F-伴随”这一概念，旨在更清晰地描述反向传播算法。具体而言，通过引入深度神经网络架构中的“F-传播”和“F-伴随”概念，我们证明了与代价/损失函数相关的反向传播完全可由相应F-传播的F-伴随（相对于代价函数关于输入的部分导数）来刻画。