  The Wagner function in classical unsteady aerodynamic theory represents the
response in lift on an airfoil that is subject to a sudden change in
conditions. While it plays a fundamental role in the development and
application of unsteady aerodynamic methods, explicit expressions for this
function are difficult to obtain. The Wagner function requires computation of
an inverse Laplace transform, or similar inversion, of a non-rational function
in the Laplace domain, which is closely related to the Theodorsen function.
This has led to numerous proposed approximations to the Wagner function, which
facilitate convenient and rapid computations. While these approximations can be
sufficient for many purposes, their behavior is often noticeably different from
the true Wagner function, especially for long-time asymptotic behavior. In
particular, while many approximations have small maximum absolute error across
all times, the relative error of the asymptotic behavior can be substantial. As
well as documenting this error, we propose an alternative approximation
methodology that is accurate for all times, for a variety of accuracy measures.
This methodology casts the Wagner function as the solution of a nonlinear
scalar ordinary differential equation, which is identified using a variant of
the sparse identification of nonlinear dynamics (SINDy) algorithm. We show that
this approach can give accurate approximations using either first- or
second-order differential equations. We additionally show that this method can
be applied to model the analogous lift response for a more realistic
aerodynamic system, featuring a finite thickness airfoil and a nonplanar wake.
