  We investigate the resolution of parabolic PDEs via Extreme Learning Machine
(ELMs) Neural Networks, which have a single hidden layer and can be trained at
a modest computational cost as compared with Deep Learning Neural Networks. Our
approach addresses the time evolution by applying classical ODEs techniques and
uses ELM-based collocation for solving the resulting stationary elliptic
problems. In this framework, the $\theta$-method and Backward Difference
Formulae (BDF) techniques are investigated on some linear parabolic PDEs that
are challeging problems for the stability and accuracy properties of the
methods. The results of numerical experiments confirm that ELM-based solution
techniques combined with BDF methods can provide high-accuracy solutions of
parabolic PDEs.
