  A Deep Learning approach is devised to estimate the elastic energy density
$\rho$ at the free surface of an undulated stressed film. About 190000
arbitrary surface profiles h(x) are randomly generated by Perlin noise and
paired with the corresponding elastic energy density profiles $\rho(x)$,
computed by a semi-analytical Green's function approximation, suitable for
small-slope morphologies. The resulting dataset and smaller subsets of it are
used for the training of a Fully Convolutional Neural Network. The trained
models are shown to return quantitative predictions of $\rho$, not only in
terms of convergence of the loss function during training, but also in
validation and testing, with better results in the case of the larger dataset.
Extensive tests are performed to assess the generalization capability of the
Neural Network model when applied to profiles with localized features or
assigned geometries not included in the original dataset. Moreover, its
possible exploitation on domain sizes beyond the one used in the training is
also analyzed in-depth. The conditions providing a one-to-one reproduction of
the ground-truth $\rho(x)$ profiles computed by the Green's approximation are
highlighted along with critical cases. The accuracy and robustness of the
deep-learned $\rho(x)$ are further demonstrated in the time-integration of
surface evolution problems described by simple partial differential equations
of evaporation/condensation and surface diffusion.
