  Discovering dynamical models to describe underlying dynamical behavior is
essential to draw decisive conclusions and engineering studies, e.g.,
optimizing a process. Experimental data availability notwithstanding has
increased significantly, but interpretable and explainable models in science
and engineering yet remain incomprehensible. In this work, we blend machine
learning and dictionary-based learning with numerical analysis tools to
discover governing differential equations from noisy and sparsely-sampled
measurement data. We utilize the fact that given a dictionary containing huge
candidate nonlinear functions, dynamical models can often be described by a few
appropriately chosen candidates. As a result, we obtain interpretable and
parsimonious models which are prone to generalize better beyond the sampling
regime. Additionally, we integrate a numerical integration framework with
dictionary learning that yields differential equations without requiring or
approximating derivative information at any stage. Hence, it is utterly
effective in corrupted and sparsely-sampled data. We discuss its extension to
governing equations, containing rational nonlinearities that typically appear
in biological networks. Moreover, we generalized the method to governing
equations that are subject to parameter variations and externally controlled
inputs. We demonstrate the efficiency of the method to discover a number of
diverse differential equations using noisy measurements, including a model
describing neural dynamics, chaotic Lorenz model, Michaelis-Menten Kinetics,
and a parameterized Hopf normal form.
