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) .
Neural operators learn maps between function spaces, while hereditary network dynamics are described by Volterra resolvents with non-rational Laplace symbols. We introduce a fractional Laplace neural operator (fLNO) that embeds this structure in the learned map. For commuting excitation--Laplacian pairs, one block graph-spectral layer represents the full linear Volterra solution operator exactly. We establish an expressivity frontier for finite rational realizations: they approximate fractional memory geometrically on compact frequency windows, but cannot reproduce the non-integer critical asymptotics generated by a branch point, and on the half-line the best rational rate is root-exponential. The same theory yields trainable parametrizations that enforce a prescribed stability margin by construction, and a graphon-transfer theorem separates genuine operator consistency from parameter sharing. In a common-data benchmark, positive rational operators can match or exceed fLNO accuracy on finite horizons, whereas in controlled near-critical experiments fLNO recovers the branching coordinate more faithfully with far fewer parameters; unconstrained rational fits can cross the stability boundary, while certified parametrizations cannot. A four-parameter spectral law transfers without retraining from graphs of size 48 to 192 with 0.51--0.62% relative error. Applications to Chilean aftershock sequences and to renewal models for Chile and 21 Italian regions illustrate structured inference with explicit uncertainty. The contribution is an operator-learning architecture in which exact memory structure, physical coordinates and stability guarantees coexist with competitive accuracy.
Mauricio Herrera-Marín
Facultad de Ingenier´ıa, Universidad del Desarrollo, Santiago, Chile
Constructing Lyapunov functions for nonlinear dynamical systems is a central problem in stability analysis, yet remains challenging. Lyapunov functions are commonly characterized as solutions to first-order partial differential equations (PDEs), but these solutions are typically obtained for single systems, limiting their reuse across systems. In this paper, we study the Lyapunov solution operator that maps a vector field to the corresponding Lyapunov function defined by a dissipation-based Lyapunov PDE. We establish that, on compact subsets of the domain of attraction and under exponential stability assumptions, this operator is well-defined, unique, and continuous with respect to perturbations of both the vector field and the dissipation function. These results provide a theoretical foundation for approximating Lyapunov functions uniformly over families of nonlinear systems. Building on these theoretical foundations, we employ Fourier Neural Operators (FNOs) as a data-driven approximation of the Lyapunov solution operator. Numerical experiments demonstrate that a single trained operator can accurately approximate the numerical Lyapunov functions across parameterized families of dynamics. This illustrates the potential of neural operators for approximating Lyapunov functions.
Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fernández +1
Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Neural operators provide a framework for learning solution operators of partial differential equations (PDEs), enabling efficient surrogate modeling for complex systems. While universal approximation results are now well understood, quantitative approximation analysis specific to nonlinear reaction-diffusion systems remains limited. In this paper, we study Laplacian-based neural operators applied to the solution mapping from initial conditions to time-dependent solutions of nonlinear reaction-diffusion systems. By exploiting the Laplacian spectral representation of the diffusion semigroup, we obtain polynomial complexity bounds on the network depth, width, and spectral rank, demonstrating that Laplacian eigenfunction-based neural operators can exploit the structure of the underlying PDE while avoiding the curse of parametric complexity encountered in generic operator learning. Numerical experiments support the theoretical findings and illustrate the benefit of Laplacian eigenfunction representations compared with Fourier-based spectral representations.
Takashi Furuya, Ryo Ozawa, Jenn-Nan Wang
Doshisha University, RIKEN AIP · Tohoku University · National Taiwan University