Parametric Matrix Models

We present a general class of machine learning algorithms called parametric matrix models. Parametric matrix models are based on matrix equations, and the design is motivated by the efficiency of reduced basis methods for approximating solutions of parametric equations. The dependent variables can be defined implicitly or explicitly, and the equations may use algebraic, differential, or integral relations. Parametric matrix models can be trained with empirical data only, and no high-fidelity model calculations are needed. While originally designed for scientific computing, parametric matrix models are universal function approximators that can be applied to general machine learning problems. After introducing the underlying theory, we apply parametric matrix models to a series of different challenges that show their performance for a wide range of problems. For all the challenges tested here, parametric matrix models produce accurate results within a computational framework that allows for parameter extrapolation and interpretability.