End-to-end Artificial Intelligence to analyze dynamical processes: A linear benchmark test

We envisage AI architectures to analyze complex time series in an end-to-end fashion, meaning that the quantitative metrics of the time series are learned directly from data, without the use of specific human-thought algorithms. That is, we challenge AI to learn those specific algorithms. We present a first step in this direction, a benchmark test on linear dynamical processes. We tackle the archetypical task of learning the eigenvalues of the state-transition matrix of a linear (discrete-time, stable) dynamical system, from output data. We train a scalable LSTM neural network with artificially generated data from random matrices of dimension 2-to-5. With noise-free data, the performance of the trained network is very good (average $R^2=0.955$), especially in estimating the dominant eigenvalues, whereas there is space for improvements on non-dominant real eigenvalues and on the dimension of the generating matrix. Remarkably, the performance is robust to measurement noise and the network outperforms the mean-square identification of the corresponding AR process (the latter giving exact eigenvalues on noise-free data) at noise standard deviation starting from $10^{-6}$.