Abstract
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.
Highlights
K NOWLEDGE of the dynamics generating a signal enables numerous state-space based algorithms to provide both better estimates of the signal of interest in fields such as dynamic tomography [1], as well as predictions of the future which have critical implications such as weather patterns [2], [3]
One successful method is Dynamic Mode Decomposition (DMD), which is inspired by the observation that a Linear Dynamical System (LDS) inherently constructs a Krylov Subspace, a space frequently exploited in efficient numerical algorithms for linear algebra [4]–[6]
DMD inherently searches for low-dimensional approximations of the collective system dynamics, rather than the fundamental governing equations
Summary
K NOWLEDGE of the dynamics generating a signal enables numerous state-space based algorithms to provide both better estimates of the signal of interest in fields such as dynamic tomography [1], as well as predictions of the future which have critical implications such as weather patterns [2], [3]. While understanding global behavior is very important for engineering design and interpreting the impact of boundary conditions, the universe is inherently governed by local interactions which together form these sophisticated behaviors For this reason, it is sometimes preferable in basic scientific research to identify the fundamental governing equations and associated parameters. Proposed alternative formulations use approximations of derivatives to fit nonlinear functions to identify a governing partial differential equation (PDE) [7], [8], use neural networks to identify parameters [9], [10], or build particle simulation models using so-called “interaction networks” [11], [12], oftentimes using hundreds of timesteps While these methods work incredibly well in practice, they often lack rigorous performance bounds due to the inclusion of necessary, but statistically opaque, steps, such as the sophisticated numerical differentiation procedure found in [13]. We derive the bias and MSE of the estimator when applied to a system in steady state, and conclude with simulation results of a diffusion system
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