Nonlinear gradient denoising: Finding accurate extrema from inaccurate functional derivatives

A method for nonlinear optimization with machine learning (ML) models, called nonlinear gradient denoising (NLGD), is developed, and applied with ML approximations to the kinetic energy density functional in an orbital-free density functional theory. Due to systematically inaccurate gradients of ML models, in particular when the data is very high-dimensional, the optimization must be constrained to the data manifold. We use nonlinear kernel principal component analysis (PCA) to locally reconstruct the manifold, enabling a projected gradient descent along it. A thorough analysis of the method is given via a simple model, designed to clarify the concepts presented. Additionally, NLGD is compared with the local PCA method used in previous work. Our method is shown to be superior in cases when the data manifold is highly nonlinear and high dimensional. Further applications of the method in both density functional theory and ML are discussed. (c) 2015 Wiley Periodicals, Inc.