Estimating Nonlinear Dynamics with the ConvNet Smoother
This addresses the limitation of traditional Kalman smoothers in real-world applications where linearity and Gaussianity assumptions fail, offering a flexible solution for fields like neuroscience and engineering.
The paper tackled the problem of estimating the state of nonlinear, non-Gaussian dynamical systems from noisy observations by introducing a ConvNet smoother that uses convolutional neural networks trained on simulator-generated data. The method achieved near-optimal performance in Gaussian cases and successfully handled highly nonlinear and non-Gaussian systems, with empirical validation on brain signal data.
Estimating 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 (and the related Kalman filter), relies on assumptions of linearity and Gaussianity that are rarely met in practice. In this paper, we introduced a new dynamical smoothing method that exploits the remarkable capabilities of convolutional neural networks to approximate complex non-linear functions. The main idea is to generate a training set composed of both latent states and observations from an ensemble of simulators and to train the deep network to recover the former from the latter. Importantly, this method only requires the availability of the simulators and can therefore be applied in situations in which either the latent dynamical model or the observation model cannot be easily expressed in closed form. In our simulation studies, we show that the resulting ConvNet smoother has almost optimal performance in the Gaussian case even when the parameters are unknown. Furthermore, the method can be successfully applied to extremely non-linear and non-Gaussian systems. Finally, we empirically validate our approach via the analysis of measured brain signals.