In this paper we develop a new method for numerically approximating sensitivities in parameter-dependent ordinary differential equations (ODEs). Our approach, intended for situations where the standard forward and adjoint sensitivity analyses become too computationally costly for practical purposes, is based on the Peano-Baker series from control theory. Using this series, we construct a representation of the sensitivity matrix $\mathbf{S}$ and, from this representation, a numerical method for approximating $\mathbf{S}$. We prove that, under standard regularity assumptions, the error of our method scales as $\mathcal{O}(\Delta t^2_{\text{max}})$, where $\Delta t_{\text{max}}$ is the largest time step used when numerically solving the ODE. We illustrate the performance of the method in several numerical experiments, taken from both the systems biology setting and more classical dynamical systems. The experiments show the sought-after improvement in running time of our method compared to the forward sensitivity approach. In experiments involving a random linear system, the forward approach requires roughly $\sqrt{n}$ longer computational time, where $n$ is the dimension of the parameter space, than our proposed method.

翻译：本文针对参数依赖常微分方程（ODEs）的灵敏度数值近似问题，提出了一种新方法。当标准前向与伴随灵敏度分析的计算成本过高而无法实际应用时，本方法基于控制理论中的Peano-Baker级数构建解决方案。利用该级数，我们构造了灵敏度矩阵$\mathbf{S}$的表示形式，并据此提出一种近似计算$\mathbf{S}$的数值方法。我们证明，在标准正则性假设下，该方法的误差阶为$\mathcal{O}(\Delta t^2_{\text{max}})$，其中$\Delta t_{\text{max}}$为数值求解ODE时采用的最大时间步长。通过从系统生物学和经典动力系统中选取的多个数值实验，我们展示了该方法的性能。实验结果表明，相较于前向灵敏度方法，本方法在运行时间上取得了预期的改进。在涉及随机线性系统的实验中，前向方法的计算时间约为本方法的$\sqrt{n}$倍，其中$n$为参数空间的维度。