Physics-Informed Feature Maps for Kernel-Based Learning: A Reinterpretation of Symbolic Regression

Supervised machine learning involves approximating an unknown functional relationship from a limited dataset of features and corresponding labels. Within the framework of regularized approximation in reproducing kernel Hilbert spaces (RKHSs), this task reduces to the minimization of a regularized empirical risk functional, where the choice of the kernel encodes prior assumptions on the solution space. In conventional applications, features are standardized and treated as abstract coordinates, disregarding their physical meaning and structural relationships. This abstraction may hinder both interpretability and stability in scientific problems, where dimensional consistency, conservation laws, or symmetries play a crucial role. This study proposes a physics-informed approach to feature-based machine learning that constructs non-linear feature maps informed by physical laws and dimensional analysis. These maps induce physics-informed kernels that embed domain knowledge directly into the geometry of the RKHS, thereby enhancing the interpretability and, when physical laws are unknown, allow for the identification of relevant mechanisms through feature ranking. The method aims to improve both predictive performance in regression tasks and classification skill scores by integrating domain knowledge into the learning process, while also enabling the potential discovery of new physical equations within the context of explainable machine learning.