  Atmospheric modeling has recently experienced a surge with the advent of deep
learning. Most of these models, however, predict concentrations of pollutants
following a data-driven approach in which the physical laws that govern their
behaviors and relationships remain hidden. With the aid of real-world air
quality data collected hourly in different stations throughout Madrid, we
present an empirical approach using data-driven techniques with the following
goals: (1) Find parsimonious systems of ordinary differential equations via
sparse identification of nonlinear dynamics (SINDy) that model the
concentration of pollutants and their changes over time; (2) assess the
performance and limitations of our models using stability analysis; (3)
reconstruct the time series of chemical pollutants not measured in certain
stations using delay coordinate embedding results. Our results show that
Akaike's Information Criterion can work well in conjunction with best subset
regression as to find an equilibrium between sparsity and goodness of fit. We
also find that, due to the complexity of the chemical system under study,
identifying the dynamics of this system over longer periods of time require
higher levels of data filtering and smoothing. Stability analysis for the
reconstructed ordinary differential equations (ODEs) reveals that more than
half of the physically relevant critical points are saddle points, suggesting
that the system is unstable even under the idealized assumption that all
environmental conditions are constant over time.
