  Nonlinear dynamics is a pervasive phenomenon observed in scientific and
engineering disciplines. However, the task of deriving analytical expressions
to describe nonlinear dynamics from limited data remains challenging. In this
paper, we shall present a novel deep symbolic learning method called the
"finite expression method" (FEX) to discover governing equations within a
function space containing a finite set of analytic expressions, based on
observed dynamic data. The key concept is to employ FEX to generate analytical
expressions of the governing equations by learning the derivatives of partial
differential equation (PDE) solutions through convolutions. Our numerical
results demonstrate that our FEX surpasses other existing methods (such as
PDE-Net, SINDy, GP, and SPL) in terms of numerical performance across a range
of problems, including time-dependent PDE problems and nonlinear dynamical
systems with time-varying coefficients. Moreover, the results highlight FEX's
flexibility and expressive power in accurately approximating symbolic governing
equations.
