论文标题
部分可观测时空混沌系统的无模型预测
Entropy-Maximising Diffusions Satisfy a Parallel Transport Law
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
We show that the principle of maximum entropy, a variational method appearing in statistical inference, statistical physics, and the analysis of stochastic dynamical systems, admits a geometric description from gauge theory. Using the connection on a principal $G$-bundle, the gradient flow maximising entropy is written in terms of constraint functions which interact with the dynamics of the probabilistic degrees of freedom of a diffusion process. This allows us to describe the point of maximum entropy as parallel transport over the state space. In particular, it is proven that the solubility of the stationary Fokker--Planck equation corresponds to the existence of parallel transport in a particular associated vector bundle, extending classic results due to Jordan--Kinderlehrer--Otto and Markowich--Villani. A reinterpretation of splitting results in stochastic dynamical systems is also suggested. Beyond stochastic analysis, we are able to indicate a collection of geometric structures surrounding energy-based inference in statistics.
