Finite-element Gaussian processes for the machine learning of steady-state linear partial differential equations

We introduce finite-element Gaussian processes (FEGPs), a novel physics-informed machine learning approach for solving inverse problems involving steady-state, linear partial differential equations (PDEs). Our framework combines a Gaussian process prior for the unknown solution function with a likelihood that incorporates the PDE in its weak form, using a finite-element approximation. This approach offers significantly better scalability than physics-informed Gaussian processes (PIGPs), which rely on the strong form of the PDE. Through numerical experiments on a range of synthetic benchmark problems, we show that FEGPs offer results which outperform PIGPs, and are competitive with physics-informed neural networks (PINNs) with improved uncertainty quantification.