We propose a computationally efficient estimator for multi-dimensional linear space-invariant system dynamics with periodic boundary conditions that attains low mean squared error from very few temporal steps. By exploiting the inherent redundancy found in many real-world spatiotemporal systems, the estimator performance improves with the dimensionality of the system. This paper provides a detailed analysis of maximum likelihood estimation of the state transition operator in linear space-invariant systems driven by Gaussian noise. The key result of this work is that, by incorporating the space-invariance prior, the mean squared error of a estimator normalized to the number of parameters is upper bounded by $N^{-1}M^{-1} + O(N^{-1} M^{-2})$, where $N$ is the number of spatial points, and $M$ is the number of observed timesteps after the initial value.

中文翻译：

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线性空间不变动力学的估计

我们为具有周期性边界条件的多维线性空间不变系统动力学提出了一种计算效率高的估计器，该估计器从很少的时间步骤中获得低均方误差。通过利用在许多现实世界时空系统中发现的固有冗余，估计器性能随着系统的维数而提高。本文详细分析了由高斯噪声驱动的线性空间不变系统中状态转移算子的最大似然估计。这项工作的主要结果是，通过结合空间不变性先验，标准化为参数数量的估计器的均方误差上限为$N^{-1}M^{-1} + O(N^{-1} M^{-2})$， 在哪里 $N$ 是空间点的数量，并且 百万美元 是初始值后观察到的时间步数。