We develop a new formulation of deep learning based on the Mori-Zwanzig (MZ) formalism of irreversible statistical mechanics. The new formulation is built upon the well-known duality between deep neural networks and discrete dynamical systems, and it allows us to directly propagate quantities of interest (conditional expectations and probability density functions) forward and backward through the network by means of exact linear operator equations. Such new equations can be used as a starting point to develop new effective parameterizations of deep neural networks, and provide a new framework to study deep-learning via operator theoretic methods. The proposed MZ formulation of deep learning naturally introduces a new concept, i.e., the memory of the neural network, which plays a fundamental role in low-dimensional modeling and parameterization. By using the theory of contraction mappings, we develop sufficient conditions for the memory of the neural network to decay with the number of layers. This allows us to rigorously transform deep networks into shallow ones, e.g., by reducing the number of neurons per layer (using projection operators), or by reducing the total number of layers (using the decay property of the memory operator).

翻译：我们基于不可逆统计力学的Mori-Zwanzig (MZ)形式体系，提出了一种新的深度学习表述方法。该表述建立在深度神经网络与离散动力系统已知的对偶性基础之上，通过精确线性算子方程，能够直接沿网络前向和后向传播感兴趣的量（条件期望与概率密度函数）。这些新方程可作为开发深度神经网络新型有效参数化的起点，并提供一套通过算子理论方法研究深度学习的新框架。所提出的MZ深度学习形式体系自然引入了新概念——神经网络的记忆性，该概念在低维建模与参数化中具有基础性作用。利用压缩映射理论，我们给出了神经网络记忆性随层数衰减的充分条件。由此可通过减少每层神经元数量（使用投影算子），或利用记忆算子的衰减特性减少总层数，将深度网络严格转化为浅层网络。