Physics-informed multi-output Gaussian process for dynamical system modeling.

Abstract

Learning accurate dynamics models is crucial for model-based reinforcement learning. Gaussian processes (GPs), as a probabilistic modeling approach, have been widely used for dynamical system modeling. However, standard GPs are designed for single-output scenarios, modeling each dimension of a dynamical system independently and failing to capture dependencies among them. Multi-output GPs address this limitation by jointly modeling multiple outputs, capturing their correlations, and enabling information transfer across dimensions. As a result, each output can leverage information from others to improve modeling accuracy. Existing multi-output GPs typically capture the correlation between multiple outputs using covariance functions, but this approach often lacks interpretability in the context of dynamical system modeling. This paper proposes a physics-informed multi-output Gaussian process (P-MO-GP) for dynamical system modeling. P-MO-GP incorporates a physical model derived from the Lagrangian method as prior knowledge and models each dimension of the dynamical system as a GP with the mean function defined by the discretized physical model. The mean functions of all dimensions share the same parameter, namely the physical parameter. P-MO-GP treats all hyperparameters (the physical parameter, kernel parameters, and noise variances) as random variables and adopts a fully Bayesian framework. With a probabilistic graphical model, we prove that when the physical parameter is unknown, all dimensions of the dynamical system become dependent. We propose an interpretable and principled approach to correlating all dimensions of a dynamical system by leveraging the natural assumption that all dimensions share a common physical parameter. Through extensive simulations, we demonstrate that the proposed model, P-MO-GP, learns more accurate dynamics models, achieves better control performance, and is more robust to observation noise than both single-output GPs and existing multi-output GPs.