The relationship between simulated ion cyclotron emission (ICE) signals s and the corresponding 1D velocity distribution function f (v⊥) of the fast ions triggering the ICE is modeled using a two-layer deep neural network. The network architecture (number of layers and number of computational nodes in each layer) and hyperparameters (learning rate and number of learning iterations) are fine-tuned using a bottom-up approach based on cross-validation. Thus, the optimal mapping g (s;θ) of the neural network in terms of the number of nodes, the number of layers, and the values of the hyperparameters, where θ is the learned model parameters, is determined by comparing many different configurations of the network on the same training and test set and choosing the best one based on its average test error. The training and test sets are generated by computing random ICE velocity distribution functions f and their corresponding ICE signals s by modeling the relationship as the linear matrix equation Wf = s. The simulated ICE signals are modeled as edge ICE signals at LHD. The network predictions for f based on ICE signals s are on many simulated ICE signal examples closer to the true velocity distribution function than that obtained by 0th-order Tikhonov regularization, although there might be qualitative differences in which features one technique is better at predicting than the other. Additionally, the network computations are much faster. Adapted versions of the network can be applied to future experimental ICE data to infer fast-ion velocity distribution functions.
Bibliographical noteFunding Information:
This work was carried out within the framework of the EURO-fusion Consortium and has received funding from the Euratom research and training program 2014–2018 and 2019–2020 under Grant Agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. In addition, the work received support from the RCUK Energy Programme [Grant No. EP/T012250/1].