Projection-based model reduction: Formulations for physics-based machine learning

This paper considers the creation of parametric surrogate models for applications in science and engineering where the goal is to predict high-dimensional output quantities of interest, such as pressure, temperature and strain fields. The proposed methodology develops a low-dimensional parametrization of these quantities of interest using the proper orthogonal decomposition (POD), and combines this parametrization with machine learning methods to learn the map between the input parameters and the POD expansion coefficients. The use of particular solutions in the POD expansion provides a way to embed physical constraints, such as boundary conditions and other features of the solution that must be preserved. The relative costs and effectiveness of four different machine learning techniques—neural networks, multivariate polynomial regression, k-nearest-neighbors and decision trees—are explored through two engineering examples. The first example considers prediction of the pressure field around an airfoil, while the second considers prediction of the strain field over a damaged composite panel. The case studies demonstrate the importance of embedding physical constraints within learned models, and also highlight the important point that the amount of model training data available in an engineering setting is often much less than it is in other machine learning applications, making it essential to incorporate knowledge from physical models.