标题： 基于插值的微分方程参数估计的双层优化 显示英文标题

标题： Bi-Level optimization for parameter estimation of differential equations using interpolation

摘要： 常微分方程的反问题或参数估计是一种利用状态的实验测量值获得最佳参数的过程。 单次（多次）打靶法是一种顺序优化方法，旨在最小化测量值与数值积分状态之间的误差。 然而，这种方法需要计算灵敏度，即通过数值积分器计算状态相对于参数的导数，这可能会导致计算成本高昂。 为了解决这一挑战，许多基于插值的方法被提出，目的是减少灵敏度计算的计算成本或消除其需求。 本文采用了一种双层优化框架，该框架利用插值并利用微分方程的结构来解决内部凸优化问题。 我们将此方法应用于两种不同的问题形式。 首先，对于已知模型结构但参数未知的微分方程和时滞微分方程进行参数估计。 其次，对于模型结构和参数均未知的模型发现问题。 显示英文摘要

摘要： Inverse problem or parameter estimation of ordinary differential equations is a process of obtaining the best parameters using experimental measurements of the states. Single (Multiple)-shooting is a type of sequential optimization method that minimizes the error in the measured and numerically integrated states. However, this requires computing sensitivities i.e. the derivatives of states with respect to the parameters over the numerical integrator, which can get computationally expensive. To address this challenge, many interpolation-based approaches have been proposed to either reduce the computational cost of sensitivity calculations or eliminate their need. In this paper, we use a bi-level optimization framework that leverages interpolation and exploits the structure of the differential equation to solve an inner convex optimization problem. We apply this method to two different problem formulations. First, parameter estimation for differential equations, and delayed differential equations, where the model structure is known but the parameters are unknown. Second, model discovery problems, where both the model structure and parameters are unknown.