  It is well known that a $Q$-manifold gives rise to an $L_\infty$-algebra
structure on the tangent space at a fixed point of the homological vector
field. From the field theory perspective this implies that the expansion of a
classical Batalin-Vilkovisky (BV) formulation around a vacuum solution can be
equivalently cast into the form of an $L_\infty$-algebra. In so doing, the BV
symplectic structure determines a compatible cyclic structure on the
$L_\infty$-algebra. Moreover, $L_\infty$ quasi-isomorphisms correspond to
so-called equivalent reductions (also known as the elimination of generalized
auxiliary fields) of the respective BV systems. Although at the formal level
the relation is straightforward, its implementation in field theory requires
some care because the underlying spaces become infinite-dimensional. In the
case of local gauge theories, the relevant spaces can be approximated by nearly
finite-dimensional ones through employing the jet-bundle technique. In this
work we study the $L_\infty$-counterpart of the jet-bundle BV approach and
generalise it to a more flexible setup of so-called gauge PDEs. It turns out
that in the latter case the underlying $L_\infty$-structure is analogous to
that of Chern-Simons theory. In particular, the underlying linear space is a
module over the space-time exterior algebra and the higher $L_\infty$-maps are
multilinear. Furthermore, a counterpart of the cyclic structure turns out to be
degenerate and possibly nonlinear, and corresponds to a compatible
presymplectic structure which is known to encode the BV symplectic structure
and hence the full-scale Lagrangian BV formulation. Moreover, given a
degenerate cyclic structure one can consistently relax the $L_\infty$-axioms in
such a way that the formalism still describes non-topological models but
involves only finitely-generated modules, as we illustrate in the example of
Yang-Mills theory.
