Projection-based multifidelity linear regression for data-scarce applications

An important challenge in scientific machine learning is to develop methods that can exploit and maximize the amount of learning possible from scarce data [1-4]. The need for such methods arises often in science and engineering, especially in the case of computational fluid dynamics (CFD), since expensive-to-evaluate high-fidelity (HF) models make many-query problems such as uncertainty quantification, risk analysis, optimization, and optimization under uncertainty computationally prohibitive [5]. Surrogate models that approximate the solutions to HF models can facilitate the design and analysis process; however, lack of sufficient HF data in tandem with high-dimensional quantities of interest adversely affect surrogate model accuracy. We propose multifidelity (MF) linear regression methods that leverage abundant low-cost, lower-fidelity (LF) data alongside limited HF data to construct linear regression models. These models operate within a reduced-dimensional subspace, obtained through the principal component analysis (PCA), to effectively handle both training data scarcity and the high dimensionality (on the order of tens of thousands of quantities of interest) inherent in our problem setting. Linear regression has been widely utilized as a surrogate modeling approach in aerospace applications due to its simplicity and interpretability. We note that linear regression encompasses a broad class of models that are linear in their parameters but can include features that are arbitrarily nonlinear functions of the input variables [6].