By modifying the tracial techniques of Guionnet and Shlyakhtenko in [GS14] we produce free monotone transport in the context of a finitely generated free Araki-Woods factor, which can be considered a non-tracial analogue of the free group factors. We solve a free analogue of the Monge-Ampère equation to produce a criterion for when an N-tuple of non-commutative random variables generate a free Araki-Woods factor. The criterion, that the joint law satisfies a certain non-commutative differential equation involving a canonical potential, is precisely the tracial criterion established in [GS14] modulo modifications to the differential operators and potential that are completely natural in light of the structure of the free Araki-Woods factor. We provide two applications of this result. The first is that for small |q|, the q-deformed free Araki-Woods algebras are isomorphic to the free Araki-Woods factor with the same number of generators and orthogonal representation of R. This is obtained using similar estimates to some found in [Dab14], which were used to prove the tracial analogue in [GS14] that the q-deformed free group factors are isomorphic to the free group factor for small |q|. The second application is to finite depth subfactor planar algebras, where it is shown that the transport machinery can be expressed diagrammatically via planar tangles. From this one obtains a criterion for when towers of von Neumann algebras are isomorphic.