Lotka-Sharpe Neural Operators for Control of Population PDEs

TL;DR

This paper develops neural operators to approximate the Lotka-Sharpe operator for controlling age-structured population PDEs, ensuring stability and enabling online application in ecological and biomedical models.

Contribution

It proves the Lipschitz continuity of the Lotka-Sharpe operator and demonstrates stable neural operator approximations for feedback control of population PDEs.

Findings

Neural operators can accurately approximate the Lotka-Sharpe operator.

The approximate feedback law maintains semi-global practical asymptotic stability.

The learned operator can be used online for control with estimated parameters.

Abstract

Age-structured predator-prey integro-partial differential equations provide models of interacting populations in ecology, epidemiology, and biotechnology. A key challenge in feedback design for these systems is the scalar $\zeta$, defined implicitly by the Lotka-Sharpe nonlinear integral condition, as a mapping from fertility and mortality rates to $\zeta$. To solve this challenge with operator learning, we first prove that the Lotka-Sharpe operator is Lipschitz continuous, guaranteeing the existence of arbitrarily accurate neural operator approximations over a compact set of fertility and mortality functions. We then show that the resulting approximate feedback law preserves semi-global practical asymptotic stability under propagation of the operator approximation error through various other nonlinear operators, all the way through to the control input. In the numerical results, not only do we learn ``once-and-for-all'' the canonical Lotka-Sharpe (LS) operator, and thus make it available for future uses in control of other age-structured population interconnections, but we demonstrate the online usage of the neural LS operator under estimation of the fertility and mortality functions.