How deep is your network? Deep vs. shallow learning of transfer operators
This work addresses the challenge of efficiently analyzing complex dynamical systems, such as in physics and biology, by providing a faster and more robust method for operator learning, though it appears incremental in its adaptation of randomized neural networks.
The authors tackled the problem of learning transfer operators and their spectral decompositions from data by proposing RaNNDy, a randomized neural network approach that reduces training time and resources without significant accuracy loss, as demonstrated in numerical examples like stochastic dynamical systems and protein folding.
We propose a randomized neural network approach called RaNNDy for learning transfer operators and their spectral decompositions from data. The weights of the hidden layers of the neural network are randomly selected and only the output layer is trained. The main advantage is that without a noticeable reduction in accuracy, this approach significantly reduces the training time and resources while avoiding common problems associated with deep learning such as sensitivity to hyperparameters and slow convergence. Additionally, the proposed framework allows us to compute a closed-form solution for the output layer which directly represents the eigenfunctions of the operator. Moreover, it is possible to estimate uncertainties associated with the computed spectral properties via ensemble learning. We present results for different dynamical operators, including Koopman and Perron-Frobenius operators, which have important applications in analyzing the behavior of complex dynamical systems, and the Schrödinger operator. The numerical examples, which highlight the strengths but also weaknesses of the proposed framework, include several stochastic dynamical systems, protein folding processes, and the quantum harmonic oscillator.