基于稀疏数据的捕食者-食饵系统参数识别
Parameter identification for predator-prey system with sparse data
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中文总结 AI 辅助
本文针对稀疏数据下捕食者-食饵系统参数识别的数值不稳定性问题，提出了结合自然梯度上升的计算框架，通过无量纲化与自适应求解器提升稳定性，可推广至多类动力学系统。
中文摘要 AI 辅助
从动力学系统的观测数据中进行参数识别是种群生物学中的一个基础问题。生态系统的机理模型依赖于优化方法，而这些方法需要准确的初始猜测才能保证收敛。在生态应用中，数据集包含观测噪声，且在稀疏时间点采集，这种稀疏性会产生不规则的似然函数，导致标准优化方法难以处理；同时，常微分方程求解器在参数空间的某些区域可能变得刚性或不稳定，这些不稳定性会导致运行时间过长或运行时错误。本文提出了一种用于参数识别的计算框架，该框架通过使用自然梯度上升（Natural Gradient Ascent）来解决这些数值不稳定性问题，并将其应用于经典的Lotka-Volterra捕食者-食饵模型。我们利用常微分方程的无量纲化处理，将缩放因子作为冗余参数处理，从而降低了优化问题的维度。为防止求解器步长过小，我们实现了一种自适应求解器，该求解器会在由模型两个组分推导得到的两个独立二阶方程之间切换。与标准梯度上升或BFGS方法相比，该方法使自然梯度上升能够在更少的迭代次数内以更高的稳定性收敛。当数据有限时，该框架为生态学中的参数估计提供了一种可靠方法；只要系统的不同组分不会同时出现数值问题，该方法就可推广到其他动力学系统。
英文摘要
Parameter identification from observations of dynamical systems is a fundamental problem in population biology. Mechanistic models of ecological systems rely on optimization methods that require accurate initial guesses to guarantee convergence. In ecological applications, datasets contain observation noise and are collected at sparse time points. This sparsity creates irregular likelihoods that cause standard optimization methods to struggle, while the ordinary differential equation solvers can become stiff or unstable in certain regions of the parameter space. These instabilities cause long running times or runtime errors. Here we present a computational framework for parameter identification that addresses these numerical instabilities by employing Natural Gradient Ascent, and we apply it to the classical Lotka-Volterra predator-prey model. We exploit the non-dimensionalization of the ordinary differential equations to treat scaling factors as nuisance parameters, reducing the dimensionality of the optimization problem. To prevent the solver step from becoming small, we implement an adaptive solver that switches between two independent second-order equations derived from the two components of the model. This approach allows Natural Gradient Ascent to converge in fewer iterations and with more stability than standard gradient ascent or BFGS methods. This framework provides a reliable method for parameter estimation in ecology when data is limited. The method can be generalized to other dynamical systems as long as the different components of the system do not become numerically problematic at the same time.
