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Lotka-Sharpe Neural Operators for Control of Population PDEs

Miroslav Krstic, Iasson Karafyllis, Luke Bhan, Carina Veil

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 $ζ$, defined implicitly by the Lotka-Sharpe nonlinear integral condition, as a mapping from 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. 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.

Lotka-Sharpe Neural Operators for Control of Population PDEs

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 , defined implicitly by the Lotka-Sharpe nonlinear integral condition, as a mapping from fertility and mortality rates to . 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.

Paper Structure

This paper contains 25 sections, 7 theorems, 145 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Age-structured population model
  3. Four Operators
  4. Lotka-Sharpe operator (output sits within an integral --- it has to be solved for).
  5. Three easier operators (outputs = direct evaluations of integrals).
  6. Control Laws: Nominal and Neuro-approximated
  7. Nominal controller ensures stabilization
  8. Approximate controller introduces a perturbation
  9. Neural Approximability of Lotka-Sharpe Operator
  10. Stabilization with Neural Operator
  11. Stability theorem under errors in Lotka-Sharpe parameters $\zeta_1,\zeta_2$
  12. Main result---stabilization under Lotka-Sharpe NO
  13. Proofs of the Theorems
  14. Proof of Lipschitzness of Lotka-Sharpe operator
  15. Proof of stability under Lotka-Sharpe parameter errors
  16. ...and 10 more sections

Key Result

Proposition 1

The equilibrium state $(x_1^*(a),x_2^*(a))$ of the population system (eq:pred-prey-model), along with the equilibrium dilution input $u^*$, is given by with unique parameters $\zeta_i(k_i,\mu_i)$ resulting from the Lotka-Sharpe condition sharpe1911problem, and the positive concentrations of the newborns Moreover, for $u^*$ to be positive, the prey birth concentration must be commanded to be lar

Figures (6)

  • Figure 1: Computational breakdown of the Lotka-Sharpe operator
  • Figure 2: Graph of the operator dependencies for the control construction in \ref{['eq-eta-cont']}.
  • Figure 3: Explicit operator mappings in the predator-prey control law.
  • Figure 4: Example of various $(k, \mu)$ used in training and performance of learned operator $\widehat{\mathcal{G}}_{\rm LS}$. (c) highlights the residuals of all $100$ test examples during training in blue and the samples corresponding to (a) and (b) in orange.
  • Figure 5: Simulation profiles of the (a) population $x_1$, (b) population $x_2(t)$, and (c) dilution control using $(\hat{\zeta}_1, \hat{\zeta_2}) = \widehat{\mathcal{G}}_{\rm{LS}} (k, \mu)$ in the control law \ref{['eq-eta-cont']}. $k_1, k_2$ and $\mu_1, \mu_2$ correspond with samples eight and nine in Figure \ref{['fig:fno']} We consider the initial condition case where there are a large amount of species $x_1$ compared to species $x_2$ in the beginning, but become similarly concentrated due dilution choice $u^\ast = 0.83$ ($x_1^\ast(0) = 8.45, x_2^\ast(0) = 7.42$) is to obtain a larger species $x_1$.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Proposition 1: Equilibrium veil2025stabilization
  • Theorem 1
  • Corollary 1
  • proof
  • Theorem 2
  • Corollary 2
  • proof
  • proof : Proof of Theorem \ref{['thm-LS-Lip']}
  • proof : Proof of Theorem \ref{['thm-spas']}
  • Lemma 1: Lipschitz continuity of $\mathcal{G}_\gamma$, $\mathcal{G}_\kappa$, $\mathcal{G}_\pi$ in $\zeta$
  • ...and 3 more