Parametric Matrix Models: Applications in Physics*

We present a general class of machine learning algorithms called parametric matrix models. In contrast with most existing

machine learning models that imitate the biology of neurons, parametric matrix models use matrix equations that emulate

the physics of quantum systems. Similar to how physics problems are usually solved, parametric matrix models learn the

governing equations that lead to the desired outputs. Parametric matrix models can be efficiently trained from empirical data,

and the equations may use algebraic, differential, or integral relations. While originally designed for scientific computing, we

prove that 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 an efficient and interpretable computational framework that allows for input feature extrapolation.