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

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 Lᵖ-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.