Optimization of randomized neural networks for transfer operator approximation

2026-05-22

AI Summary

This work addresses the limited expressivity of basis functions in randomized neural networks, which stems from fixed random weights and biases and hinders accurate approximation of complex dynamical system transfer operators. To overcome this limitation, the authors propose a novel optimization approach that treats the activation function as a learnable component within the randomized neural network framework—termed RaNNDy—while keeping hidden-layer weights and biases fixed. By constructing a more adaptive function dictionary through trainable activations and combining it with closed-form output-layer training, the method significantly enhances the data-driven approximation accuracy of transfer operators. Its effectiveness and superiority are demonstrated on benchmark tasks including stochastic differential equations and random walks on graphs.

Abstract

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.

Problem

• transfer operator approximation
• randomized neural networks
• activation function optimization
• dynamical systems
• basis functions

Innovation

• randomized neural networks
• transfer operator approximation
• activation function optimization
• data-driven dynamics
• operator learning