Neural network is a dynamical system described by two different types of degrees of freedom: fast-changing non-trainable variables (e.g. state of neurons) and slow-changing trainable variables (e.g. weights and biases). We show that the non-equilibrium dynamics of trainable variables can be described by the Madelung equations, if the number of neurons is fixed, and by the Schrodinger equation, if the learning system is capable of adjusting its own parameters such as the number of neurons, step size and mini-batch size. We argue that the Lorentz symmetries and curved space-time can emerge from the interplay between stochastic entropy production and entropy destruction due to learning. We show that the non-equilibrium dynamics of non-trainable variables can be described by the geodesic equation (in the emergent space-time) for localized states of neurons, and by the Einstein equations (with cosmological constant) for the entire network. We conclude that the quantum description of trainable variables and the gravitational description of non-trainable variables are dual in the sense that they provide alternative macroscopic descriptions of the same learning system, defined microscopically as a neural network.
翻译:神经网络是一个动态系统,由两种不同类型的自由度描述为动态系统:快速变化的不可训练变量(例如神经元状况)和缓慢变化的可训练变量(例如重量和偏向)。我们显示,可训练变量的非平衡动态可以用Madelung方程式描述,如果神经元数量是固定的,也可以用Schrodinger方程式描述,如果学习系统能够调整其本身的参数,例如神经数量、步数和小批量大小。我们认为,Lorentz对称和弯曲的空间时段可以从Stochetic entropy生产与因学习而导致的酶破坏之间的相互作用中产生。我们显示,不可训练变量的非平衡动态可以用大地测量方程式(在新兴空间时段)描述,而爱因斯坦方方方程式(有宇宙常数)能够调整其参数。我们的结论是,可训练变量的量描述和曲线时空位空间时空位时间可以产生双重的系统变数,我们的结论是,它们作为感化的宏观网络定义的系统变数,提供了一种非感官的宏观变式的系统描述。