AI-Lorenz: A physics-data-driven framework for black-box and gray-box
  identification of chaotic systems with symbolic regression
Mario De Florio Ioannis G. Kevrekidis George Em Karniadakis
Discovering mathematical models that characterize the observed behavior of
dynamical systems remains a major challenge, especially for systems in a
chaotic regime. The challenge is even greater when the physics underlying such
systems is not yet understood, and scientific inquiry must solely rely on
empirical data. Driven by the need to fill this gap, we develop a framework
that learns mathematical expressions modeling complex dynamical behaviors by
identifying differential equations from noisy and sparse observable data. We
train a small neural network to learn the dynamics of a system, its rate of
change in time, and missing model terms, which are used as input for a symbolic
regression algorithm to autonomously distill the explicit mathematical terms.
This, in turn, enables us to predict the future evolution of the dynamical
behavior. The performance of this framework is validated by recovering the
right-hand sides and unknown terms of certain complex, chaotic systems such as
the well-known Lorenz system, a six-dimensional hyperchaotic system, and the
non-autonomous Sprott chaotic system, and comparing them with their known
analytical expressions.