Fig. 1: Dynamics reconstruction from random and sparse data.
a The textbook case of a random time series sampled at a frequency higher than the Nyquist frequency. b Training data from the target system (left) and a segment of time series of six data points in a time interval containing approximately two cycles of oscillation. According to the Nyquist criterion, the signal can be faithfully reconstructed with more than 12 uniformly sampled data points (see text). When the data points are far fewer than 12 and are randomly sampled, reconstruction becomes challenging. However, if training data from the same target system are available, existing machine-learning methods can be used to reconstruct the dynamics from the sparse data31. c If no training data from the target system are available, hybrid machine learning proposed here provides a viable solution to reconstructing the dynamics from sparse data. d Problem statement. Given random and sparse data, the goal is to reconstruct the dynamics of the target system governed by dx/dt = f(x, t). A hurdle that needs to be overcome is that, for any given three points, there exist infinitely many ways to fit the data, as illustrated on the right side. e Training of the machine-learning framework using complete data from a large number of synthetic dynamical systems [h1, h2, ⋯ , hk]. The framework is then adapted to reconstruct and predict the dynamics of the target systems [f1, ⋯  , fm]. f An example: in the testing (deployment) phase, sparse observations are provided to the trained neural network for dynamics reconstruction.
