Automatic selection of inducing points in sparse Gaussian process for approximations of finite element analyses

Gaussian process regression (GPR) is a widely used regression model, but it has poor scalability. Sparse approximation methods improve scalability by using inducing points to approximate the GPR, but determining the optimal number and placement of these points is challenging. Increasing the number of inducing points generally improves the predictive accuracy, but it comes at a computational cost. This article presents a method to estimate the necessary number of inducing points for accurate predictions of finite element method (FEM) analyses using approximate GPR. The approach leverages the proper orthogonal decomposition (POD) technique, using its modes to determine the inducing points. Results demonstrate that the proposed method identifies a sufficient number of inducing points for approximate GPR to achieve predictive accuracy comparable to full GPR, but with half the training time. This approach ensures computational efficiency without significant loss in accuracy, making it a valuable tool for scalable regression in engineering applications. POD has previously been combined with GPR to provide computationally efficient predictions for the full solution field across unseen variable combinations, treating spatial components separately via reduced basis functions. However, this work treats the spatial component as a variable within the GPR approximation, allowing continuous spatial predictions. This ensures that the covariance in the spatial dimension is captured by a single GPR. The method is applied to simulations of a three-span, post-tensioned concrete girder bridge.