Deep Neural Input-Filtered Gaussian Process Model

This paper presents a new model for addressing the error-in-variables problem in Gaussian process autoregressive models by combining a Gaussian process with a deep neural network. The Gaussian process autoregressive model is a simple and effective method for modeling dynamical systems due to its nonlinear, nonparametric, and Bayesian nature. The analytic solution for the marginal log-likelihood is obtained by considering the training input-output relationship to be static, with dynamics modeled through the inclusion of lagged observations in the input regressor. The limitation of this autoregressive method is that both the outputs and inputs are affected by noise. The simulation is obtained iteratively by propagating the Gaussian distribution through a nonlinear function. This results in a costly estimation of the simulated response with Monte Carlo integration. We propose an alternative approach in which a pre-filtering step is performed using a deep neural network to approximate the intractable recurrent filtering of the latent states. The proposed model improves the autoregressive approach while reducing the computational time of the simulation. The proposed model is validated on two case studies: a synthetic example and a real-world problem.