用神经算子学习种群控制中的关键参数，实现高效反馈设计。
Lotka-Sharpe Neural Operators for Control of Population PDEs
- 通过证明洛特卡-夏普算子的利普希茨连续性，确保神经算子可高精度逼近。
- 学成的算子在在线估计下仍能保持系统的半全局实用渐近稳定。
- 一次训练即可复用于多种种群系统控制，适合生态与生物技术应用。
年龄结构化的捕食者-猎物积分-偏微分方程为生态学、流行病学和生物技术中的相互作用种群提供了模型。反馈设计的关键挑战在于标量 ζ，它由洛特卡-夏普非线性积分条件隐式定义，是从生育率和死亡率到 ζ 的映射。为解决此问题，我们首先证明洛特卡-夏普算子在生育率与死亡率函数的紧集上具有利普希茨连续性，从而保证了任意精确的神经算子近似存在。随后，我们证明了由此产生的近似反馈律在算子近似误差通过多个非线性算子传播至控制输入的过程中，仍能保持半全局实用渐近稳定性。数值结果表明，不仅可‘一次性’学习标准洛特卡-夏普（LS）算子，供未来其他年龄结构种群互联系统的控制使用，还展示了在生育率与死亡率函数估计下的在线使用效果。
原文摘要 · Abstract (English)
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.
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