论文标题
作为神经网络的世界
The world as a neural network
论文作者
论文摘要
我们讨论了整个宇宙最基本水平的可能性是神经网络。我们确定了两种不同类型的动态自由度：“可训练”变量（例如偏置向量或权重矩阵）和“隐藏”变量（例如，神经元的状态向量）。我们首先考虑可训练的变量的随机演变，以争辩说，Madelung方程（具有代表相位的自由能）接近平衡的动态，并远离Hamilton-Jacobi方程的平衡（具有代表汉密尔顿主要功能的自由能）。这表明可训练的变量确实可以通过代表隐藏变量的神经元的状态向量表现出经典和量子行为。然后，我们通过考虑具有平均状态矢量的$ d $非交互子系，$ \ bar {\ bf x}^{1} $，...，$ \ bar {\ bf x}^{d}^{d} $和总体平均状态矢量$ \ bar {$ bf bf x}^0来研究隐藏变量的随机演变。在重量矩阵是置换矩阵时的限制中，可以用相对论字符串描述$ \ bar {\ bf x}^μ$的动力学。如果子系统与度量张量描述的相互作用最小相互作用，则出现的时空弯曲。我们认为，这种系统中的熵产生是度量张量的局部函数，应由Onsager Tensor的对称性确定。事实证明，非常简单且高度对称的Onsager张量会导致爱因斯坦 - 希尔伯特（Einstein-Hilbert）术语描述的熵产生。这表明神经网络的学习动力学确实可以表现出量子力学和一般相对论所描述的近似行为。我们还讨论了这两个描述是彼此全息双重的可能性。
We discuss a possibility that the entire universe on its most fundamental level is a neural network. We identify two different types of dynamical degrees of freedom: "trainable" variables (e.g. bias vector or weight matrix) and "hidden" variables (e.g. state vector of neurons). We first consider stochastic evolution of the trainable variables to argue that near equilibrium their dynamics is well approximated by Madelung equations (with free energy representing the phase) and further away from the equilibrium by Hamilton-Jacobi equations (with free energy representing the Hamilton's principal function). This shows that the trainable variables can indeed exhibit classical and quantum behaviors with the state vector of neurons representing the hidden variables. We then study stochastic evolution of the hidden variables by considering $D$ non-interacting subsystems with average state vectors, $\bar{\bf x}^{1}$, ..., $\bar{\bf x}^{D}$ and an overall average state vector $\bar{\bf x}^{0}$. In the limit when the weight matrix is a permutation matrix, the dynamics of $\bar{\bf x}^μ$ can be described in terms of relativistic strings in an emergent $D+1$ dimensional Minkowski space-time. If the subsystems are minimally interacting, with interactions described by a metric tensor, then the emergent space-time becomes curved. We argue that the entropy production in such a system is a local function of the metric tensor which should be determined by the symmetries of the Onsager tensor. It turns out that a very simple and highly symmetric Onsager tensor leads to the entropy production described by the Einstein-Hilbert term. This shows that the learning dynamics of a neural network can indeed exhibit approximate behaviors described by both quantum mechanics and general relativity. We also discuss a possibility that the two descriptions are holographic duals of each other.
