The BdryMatérn GP: A New Gaussian Process Model for Incorporating Boundary Information

Abstract. Gaussian processes (GPs) are broadly used as surrogate models for expensive computer simulators of complex phenomena. However, a key bottleneck is that its training data are generated from this expensive simulator and thus can be highly limited. A promising solution is to supplement the surrogate with boundary information from scientific knowledge. However, despite recent work, existing boundary-integrated GPs face key limitations for incorporating broad boundary information on irregular (i.e., nonhypercube) domains: they do not provide approximation error analysis nor a means for enforcing sample path smoothness, both of which are important for reliable surrogates. We thus propose a novel BdryMatérn GP framework, which can reliably integrate Dirichlet, Neumann, and Robin boundary information on an irregular domain with mild smoothness conditions on its boundaries. Our model leverages a new BdryMatérn covariance kernel derived in path integral form via a stochastic partial differential equation formulation. Similar to the GP with a Matérn kernel, we prove that sample paths from the BdryMatérn GP satisfy the desired boundaries with smoothness control on its derivatives. We further present an efficient approximation procedure for the BdryMatérn kernel using finite element modeling with rigorous error analysis. Finally, we demonstrate the effectiveness of the BdryMatérn GP in a suite of simulation experiments and PDE applications that provide boundary information on irregular domains.