  Precision radial velocity (RV) measurements continue to be a key tool to
detect and characterise extrasolar planets. While instrumental precision keeps
improving, stellar activity remains a barrier to obtain reliable measurements
below 1-2 m/s accuracy. Using simulations and real data, we investigate the
capabilities of a Deep Neural Network approach to produce activity free Doppler
measurements of stars. As case studies we use observations of two known stars
(Eps Eridani and AUMicroscopii), both with clear signals of activity induced RV
variability. Synthetic data using the starsim code are generated for the
observables (inputs) and the resulting RV signal (labels), and used to train a
Deep Neural Network algorithm. We identify an architecture consisting of
convolutional and fully connected layers that is adequate to the task. The
indices investigated are mean line-profile parameters (width, bisector,
contrast) and multi-band photometry. We demonstrate that the RV-independent
approach can drastically reduce spurious Doppler variability from known
physical effects such as spots, rotation and convective blueshift. We identify
the combinations of activity indices with most predictive power. When applied
to real observations, we observe a good match of the correction with the
observed variability, but we also find that the noise reduction is not as good
as in the simulations, probably due to the lack of detail in the simulated
physics. We demonstrate that a model-driven machine learning approach is
sufficient to clean Doppler signals from activity induced variability for well
known physical effects. There are dozens of known activity related observables
whose inversion power remains unexplored indicating that the use of additional
indicators, more complete models, and more observations with optimised sampling
strategies can lead to significant improvements in our detrending capabilities.
