Speaker
Description
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 physical systems. Parametric matrix models work by replacing operators in the known or supposed governing equations with trainable, parametrized ones. Similar to how physics problems are usually solved, parametric matrix models learn the governing equations that lead to the desired outputs. Parametric matrix models take the additional step of applying the principles of model order reduction and reduced basis methods to find efficient approximate matrix equations with finite dimensions and can be efficiently trained from empirical data. Such equations are guaranteed to exist and can be constructed, in theory, via methods such as the proper orthogonal decomposition.
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. We first demonstrate the superior performance of PMMs for three scientific computing examples: multivariable regression, quantum computing, and quantum many-body systems. We then show the broad versatility and efficiency of PMMs on several supervised image classification benchmarks as well as hybrid machine learning when paired together with both trainable and pre-trained convolutional neural networks. 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
