Gaussian Processes simplify differential equations

Abstract

In this paper we use Gaussian processes (kernel methods) to learn mappings between trajectories of distinct differential equations. Our goal is to simplify both the representation and the solution of these equations. We begin by examining the Cole-Hopf transformation, a classical result that converts the nonlinear, viscous Burgers' equation into the linear heat equation. We demonstrate that this transformation can be effectively learned using Gaussian process regression, either from single or from multiple initial conditions of the Burgers equation. We then extend our methodology to discover mappings between initial conditions of a nonlinear partial differential equation (PDE) and a linear PDE, where the exact form of the linear PDE remains unknown and is inferred through Computational Graph Completion (CGC), a generalization of Gaussian Process Regression from approximating single input/output functions to approximating multiple input/output functions that interact within a computational graph. Further, we employ CGC to identify a local transformation from the nonlinear ordinary differential equation (ODE) of the Brusselator to its Poincar\'{e} normal form, capturing the dynamics around a Hopf bifurcation. We conclude by addressing the broader question of whether systematic transformations between nonlinear and linear PDEs can generally exist, suggesting avenues for future research.