Optimization of randomized neural networks for transfer operator approximation
For researchers studying complex dynamical systems, this work provides a method to enhance randomized neural networks for transfer operator approximation without increasing training costs.
RaNNDy is a randomized neural network for approximating transfer operators, but its performance depends on the fixed random weights and biases. This work optimizes the activation function while keeping weights fixed, improving approximation accuracy across benchmarks including stochastic differential equations and random walks on graphons.
RaNNDy is a randomized neural network architecture for the data-driven approximation of transfer operators associated with complex dynamical systems. The weights and biases of the hidden layers of the network are randomly initialized and kept fixed, only the output layer is trained. This has several advantages over fully optimized neural networks, notably a closed-form solution for the output layer and significantly lower training costs. Despite these advantages, RaNNDy is restricted to the initial selection of weights and biases that parametrize the basis functions required for the operator approximation. Since the basis functions are determined by the activation function, choosing an appropriate activation function for the hidden layers is crucial. In this work, we propose an algorithm that optimizes the activation function itself, while keeping the weights and biases in the randomized neural network fixed, providing a more suitable dictionary. We illustrate the efficacy of the approach using various benchmark problems, including stochastic differential equations and random walks on graphons.