Estimating Nonlinear Dynamics with the ConvNet Smoother

The estimation of the state of a dynamical system from a series of noise-corrupted observations is fundamental in many areas of science and engineering. The most well-known method, the Kalman smoother, relies on assumptions of linearity and Gaussianity that are rarely met in practice. In this paper, we introduced a new smoothing method that exploits the remarkable capabilities of constitutional neural networks to approximate complex non-linear functions. The main idea is to generate a training set composed by both latent states and observations from a simulator and to train the deep network to recover the former from the latter. Importantly, this method only requires the availability of a simulator and can therefore be applied in situations in which either the latent dynamical model or the observation model cannot be expressed in closed form. In our simulation studies, we showed that the resulting ConvNet smoother has almost optimal performance in the Gaussian case and can be successfully applied to extremely non-linear and non-Gaussian systems. Finally, we showed an example of analysis on real brain signals.