Given a smooth complex algebraic variety, the Riemann-Hilbert-correspondence tells us, that the category of perverse sheaves is equivalent to the category of regular, holonomic D-modules.

However not every interesting holonomic D-module is regular. For example the solution sheaves of all the $D_{\mathbb A^1}$-modules $\mathbb C[x]e^{\chi x}$ are isomorphic to the constant sheaf and only for $\chi=0$ our module $\mathbb C[x]e^{\chi x}$ is regular.

So my question is, is there an analogue of the Riemann-Hilbert correspondence if we replace regular by something else (and perhaps also perverse sheaves by something else)?

For example in the above example one could do the following: One could fix a $\chi$ and tensor first with $\mathbb C[x] e^{-\chi x} $, before applying the deRham functor. This gives an equivalence between perverse sheaves and holonomic modules with "$e^{-\chi x}$-like " singularities.