We present an Artificial Neural Network (ANN) model of photospheric lithium depletion in cool stars (3000 < Teff / K < 6500), producing estimates and probability distributions of age from Li I 6708A equivalent width (LiEW) and effective temperature data inputs. The model is trained on the same sample of 6200 stars from 52 open clusters, observed in the Gaia-ESO spectroscopic survey, and used to calibrate the previously published analytical EAGLES model, with ages 2 - 6000 Myr and -0.3 < [Fe/H] < 0.2. The additional flexibility of the ANN provides some improvements, including better modelling of the "lithium dip" at ages < 50 Myr and Teff ~ 3500K, and of the intrinsic dispersion in LiEW at all ages. Poor age discrimination is still an issue at ages > 1 Gyr, confirming that additional modelling flexibility is not sufficient to fully represent the LiEW - age - Teff relationship, and suggesting the involvement of further astrophysical parameters. Expansion to include such parameters - rotation, accretion, and surface gravity - is discussed, and the use of an ANN means these can be more easily included in future iterations, alongside more flexible functional forms for the LiEW dispersion. Our methods and ANN model are provided in an updated version 2.0 of the EAGLES software.