Neural network methods for Neumann series problems of Perron-Frobenius operators

TL;DR

This paper introduces neural network techniques, specifically PINNs and RVPINNs, to approximate solutions of Perron-Frobenius operator problems, providing error estimates and demonstrating effectiveness through numerical examples.

Contribution

The paper develops neural network methods for Perron-Frobenius operator problems, including error analysis and practical demonstrations in multiple dimensions.

Findings

Effective approximation of Perron-Frobenius operator solutions in 1D and 2D

Error estimates for neural network solutions

Successful application to a two-cavity system density approximation

Abstract

Problems related to Perron-Frobenius operators (or transfer operators) have been extensively studied and applied across various fields. In this work, we propose neural network methods for approximating solutions to problems involving these operators. Specifically, we focus on computing the power series of non-expansive Perron-Frobenius operators under a given $L^p$-norm with a constant damping parameter in $(0,1)$. We use PINNs and RVPINNs to approximate solutions in their strong and variational forms, respectively. We provide a priori error estimates for quasi-minimizers of the associated loss functions. We present some numerical results for 1D and 2D examples to show the performance of our methods. We also demonstrate the applicability of our methods by approximating interior densities in a two-cavity system.