[preprint] Lotka-Sharpe Neural Operator for Control of Population PDEs
Populations (in ecology, epidemics, biotechnology, economics, social processes) do not only interact over time but also age over time. It is therefore common to model them as age-structured partial differential equations, where age is the ‘space variable’. Since the models also involve integrals over age, both in the birth process and in the interaction among species, they are in fact integro-partial differential equations (IPDEs) with positive states and inputs, turning them into an extremely challenging control problem.
Even after developing suitable control strategies 1, 2, one challenge remains: the scalar ζ, defined implicitly by the Lotka-Sharpe nonlinear integral condition, as a mapping from age-dependent fertility and mortality rates to ζ. 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.
💡 Easy version: Gist
📃 Link: arXiv
ℹ️ More about this project: Population Dynamics
