  In this work we present a data-driven method for the discovery of parametric
partial differential equations (PDEs), thus allowing one to disambiguate
between the underlying evolution equations and their parametric dependencies.
Group sparsity is used to ensure parsimonious representations of observed
dynamics in the form of a parametric PDE, while also allowing the coefficients
to have arbitrary time series, or spatial dependence. This work builds on
previous methods for the identification of constant coefficient PDEs, expanding
the field to include a new class of equations which until now have eluded
machine learning based identification methods. We show that group sequentially
thresholded ridge regression outperforms group LASSO in identifying the fewest
terms in the PDE along with their parametric dependency. The method is
demonstrated on four canonical models with and without the introduction of
noise.
