Weak gauge PDEs and AKSZ-like constructions

AKSZ construction is a powerful tool for studying and classifying topological field theories which involves the language of graded geometry. The generalisation of the non-Lagrangian version of AKSZ sigma-models to generic local gauge theories is known under the name of gauge PDEs. Gauge PDEs are supergeometrical objects that encode the BV description of local non-Lagrangian gauge theories. Just like AKSZ sigma-models, gPDEs behave well when restricted to submanifolds, which makes them useful in the study of gauge theories on manifolds with boundaries. However, gPDEs are infinite-dimensional unless the underlying system is a PDE of finite type or a topological theory. Weak gauge PDE is a finite-dimensional analogue of a gPDE in which the BRST-differential $Q$ is nilpotent modulo certain integrable $Q$-invariant distribution. These finite-dimensional objects can be considered as a far-reaching generalisation of the AKSZ construction.