How deep is your network? Deep vs. shallow learning of transfer operators

We introduce RaNNDy, a randomized neural network framework for learning transfer operators and their spectral decompositions. By fixing the weights of the hidden layers and training only the output layer, RaNNDy achieves significant reductions in computational costs---often by two orders of magnitude---with similar accuracy compared to state-of-the-art deep learning methods. The method is less sensitive to hyperparameter tuning and provides a closed-form solution for the output layer weights that directly represent eigenfunctions of the learned operator. Moreover, it is possible to estimate uncertainties associated with the computed spectral properties via ensemble learning. We demonstrate the efficacy of the proposed approach using diverse benchmark problems, including stochastic dynamical systems and protein folding processes, highlighting the strengths but also weaknesses of RaNNDy.