  The 'Cavity-Mean-Field' approximation developed for the Random Transverse
Field Ising Model on the Cayley tree [L. Ioffe and M. M\'ezard, PRL 105, 037001
(2010)] has been found to reproduce the known exact result for the surface
magnetization in $d=1$ [O. Dimitrova and M. M\'ezard, J. Stat. Mech. (2011)
P01020]. In the present paper, we propose to extend these ideas in finite
dimensions $d>1$ via a non-linear transfer approach for the surface
magnetization. In the disordered phase, the linearization of the transfer
equations correspond to the transfer matrix for a Directed Polymer in a random
medium of transverse dimension $D=d-1$, in agreement with the leading order
perturbative scaling analysis [C. Monthus and T. Garel, arxiv:1110.3145]. We
present numerical results of the non-linear transfer approach in dimensions
$d=2$ and $d=3$. In both cases, we find that the critical point is governed by
Infinite Disorder scaling. In particular exactly at criticality, the one-point
surface magnetization scales as $\ln m_L^{surf} \simeq - L^{\omega_c} v$, where
$\omega_c(d)$ coincides with the droplet exponent $\omega_{DP}(D=d-1)$ of the
corresponding Directed Polymer model, with $\omega_c(d=2)=1/3$ and
$\omega_c(d=3) \simeq 0.24$. The distribution $P(v)$ of the positive random
variable $v$ of order O(1) presents a power-law singularity near the origin
$P(v) \propto v^a$ with $a(d=2,3)>0$ so that all moments of the surface
magnetization are governed by the same power-law decay $\bar{(m_L^{surf})^k}
\propto L^{- x_S}$ with $x_S=\omega_c (1+a)$ independently of the order $k$.
