Echo state network ensembles for surrogate models with an application to urban mobility

In this paper we investigate the use of nonlinear embeddings to represent dynamic inputs in surrogate models, used to estimate the outcome of time-consuming simulation tools in feasible time. By encoding the temporal sequences, together with non-dynamic inputs, into a space of static features, we are able to formalize the surrogate modeling problem as a standard supervised learning one. To this purpose, we propose to use the echo state network (ESN) paradigm to generate the embeddings of the dynamic inputs. The main advantage of this approach is that the embedding does not require training, since it is provided by a reservoir of neurons with randomly generated parameters. In order to enhance the robustness of this method based on randomization, we propose to use an ensemble of different ESN embeddings. Within this scheme, we consider a procedure aimed at controlling diversity among the ensemble elements before the simulations are run. Furthermore, in order to improve accuracy, we investigate also nonlinear mappings for the generation of the ESN outputs. The proposed approach is applied to an urban traffic network example, a typical case in which simulation models are generally complex and very time-consuming, and the scenarios to be simulated involve both static inputs and timevarying quantities.