Lotka-Sharpe Neural Operators for Control of Population PDEs
Authors: Miroslav Krstic, Iasson Karafyllis, Luke Bhan, Carina Veil
Organizations: University of California, San Diego, USA · National Technical University of Athens, Greece · KTH Royal Institute Technology, Stockholm, Sweden
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.
Figures & tables
Figure 1: Computational breakdown of the Lotka-Sharpe operator
Figure 2: Graph of the operator dependencies for the control construction in ( 28 ).
Figure 3: Explicit operator mappings in the predator-prey control law.
Figure 4: Example of various (k,μ) used in training and performance of learned operator GLS . (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 x1 , (b) population x2(t) , and (c) dilution control using (ζ^1,ζ2^)=GLS(k,μ) in the control law ( 28 ). k1,k2 and μ1,μ2 correspond with samples eight and nine in Figure 4 We consider the initial condition case where there are a large amount of species x1 compared to species x2 in the beginning, but become similarly concentrated due dilution choice u∗=0.83 ( x1∗(0)=8.45,x2∗(0)=7.42 ) is to obtain a larger species x1 .
Figure 6: Adaptive control simulation where k,μ,g are samples eight and nine from Figure 4 and k^i,μ^i are initialized with samples zero and one from Figure 4 . The blue lines in (a), (b), (d), (e) are the final states achieved and the red dashed lines are the target equilibrium and true k,μ functions for (a), (b) and (d), (e) respectively. All the initial conditions and the dilution point are the same as in Figure 5 and the controller uses the adaptive update laws in Section IX-A along with GLS(k^i(⋅,t),μ^i(⋅,t))=ζ^i(t) to compute the control law u(t) .