Programming with equadratures: an open-source package for uncertainty quantification, dimension reduction, and much more

This paper presents an overview of the open-source code equadratures. While originally developed to replicate polynomial chaos results seen in literature, it has since evolved to touch upon multiple aspects of computational engineering and machine learning. Today, the code uses orthogonal polynomial approximations to facilitate various parameter-based studies including uncertainty quantification, sensitivity analysis, dimension reduction, and classification. Additionally, it can address well-known limitations of polynomial approximations. These include the ability to fit to high-dimensional problems without requiring large input-output data pairs, and the ability to negotiate discontinues in any provided data. For the former, subspace-based polynomial approximations are employed, while for the latter, a tree-based piecewise polynomial hierarchy is adopted. Beyond this, ancillary topics such as coefficient computation, dealing with correlated inputs, moment computation, and gradient enhancement are also discussed. Following a deep-dive of the underpinning methods in the code, this paper details numerous case studies—with a slant towards computational aerodynamic problems.