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

TL;DR

This work tackles Neumann-series problems for non-expansive Perron-Frobenius operators on $L^p$ spaces with damping $0<\alpha<1$, focusing on approximating the solution to $u - α\mathcal{P}u = f_0$ via a Neumann series. It develops two neural-network frameworks: PINNs, which minimize the strong-form residual, and RVPINNs, which minimize a variational, discrete dual-norm residual; a reformulation using the Koopman operator avoids requiring $S^{-1}$. The authors establish a well-posed variational formulation in $L^2$ and extend to $L^p$ settings, derive a priori error estimates for quasi-minimizers, and prove a local Fortin-type condition for RVPINNs. Numerical experiments on 1D tents maps, 2D circular and standard maps, and a two-cavity interior-density problem demonstrate that PINNs and RVPINNs achieve accurate solutions with modest network sizes and outperform fixed-grid methods, indicating strong potential for high-dimensional transfer-operator problems and PDE contexts.

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.

Figures (12)

• Figure 1: PINNs approximation for $u^{[1]}$ with $\alpha = 0.5$.
• Figure 2: PINNs approximation for $u^{[2]}$ with $\alpha = 0.5$.
• Figure 3: RVPINNs approximation for the tent map example.
• Figure 4: The discrepancy between the loss function computed using different numbers of quadrature points and the true value of the loss function.
• Figure 5: Diagram of a ray trajectory in a circular domain. The boundary map $S_1$ sends $(\varphi,\psi)$ to the new ray coordinates $(\varphi',\psi')$.

Theorems & Definitions (6)

• Theorem 1: Well-posedness of variational problem in $L^2$
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5: Well-posedness of variational problem in $L^p-L^q$ spaces
• Lemma 6