Time discretization in the solution of parabolic PDEs with ANNs

We investigate the resolution of parabolic PDEs via Extreme Learning Machine (ELMs) Neural Networks. An Artificial Neural Network (ANN) is an interconnected group of nodes - where activation functions act- organized in numbered layers. From a mathematical point of view, an ANN represents a combination of activation functions that is linear if only one hidden layer is admitted. In the present paper, the ELMs setting is considered, and this gives that a single hidden layer is admitted and that the ANN can be trained at a modest computational cost as compared to 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 θmethod and Backward Differentiation Formulae (BDF) techniques are investigated on some linear parabolic PDEs that are challenging 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.