  We classify Brauer graph algebras up to derived equivalence by showing that
the set of derived invariants introduced by Antipov is complete. These algebras
first appeared in representation theory of finite groups and can be defined for
any suitably decorated graph on an oriented surface. Motivated by the
connection between Brauer graph algebras and gentle algebras we consider
$A_{\infty}$-trivial extensions of partially wrapped Fukaya categories
associated to surfaces with boundary. This construction naturally enlarges the
class of Brauer graph algebras and provides a way to construct derived
equivalences between Brauer graph algebras with the same derived invariants. As
part of the proof we provide an interpretation of derived invariants of Brauer
graph algebras as orbit invariants of line fields under the action of the
mapping class group.
