  Emerging sampling algorithms based on normalizing flows have the potential to
solve ergodicity problems in lattice calculations. Furthermore, it has been
noted that flows can be used to compute thermodynamic quantities which are
difficult to access with traditional methods. This suggests that they are also
applicable to the density-of-states approach to complex action problems. In
particular, flow-based sampling may be used to compute the density directly, in
contradistinction to the conventional strategy of reconstructing it via
measuring and integrating the derivative of its logarithm. By circumventing
this procedure, the accumulation of errors from the numerical integration is
avoided completely and the overall normalization factor can be determined
explicitly. In this proof-of-principle study, we demonstrate our method in the
context of two-component scalar field theory where the $O(2)$ symmetry is
explicitly broken by an imaginary external field. First, we concentrate on the
zero-dimensional case which can be solved exactly. We show that with our
method, the Lee-Yang zeroes of the associated partition function can be
successfully located. Subsequently, we confirm that the flow-based approach
correctly reproduces the density computed with conventional methods in one- and
two-dimensional models.
