Gross Elizabeth, Davis Brent, Ho Kenneth L, Bates Daniel J, Harrington Heather A
Department of Mathematics, San José State University, San José, CA 95112, USA.
Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA.
J R Soc Interface. 2016 Oct;13(123). doi: 10.1098/rsif.2016.0256.
Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation and model selection. These are all optimization problems, well known to be challenging due to nonlinearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data are available. Here, we consider polynomial models (e.g. mass-action chemical reaction networks at steady state) and describe a framework for their analysis based on optimization using numerical algebraic geometry. Specifically, we use probability-one polynomial homotopy continuation methods to compute all critical points of the objective function, then filter to recover the global optima. Our approach exploits the geometrical structures relating models and data, and we demonstrate its utility on examples from cell signalling, synthetic biology and epidemiology.
使用数学模型的研究人员经常面临参数估计、模型验证和模型选择等相关问题。这些都是优化问题，众所周知，由于非线性、非凸性和多个局部最优解，这些问题具有挑战性。此外，当只有部分数据可用时，挑战会更加复杂。在这里，我们考虑多项式模型（例如稳态下的质量作用化学反应网络），并描述一个基于数值代数几何优化的分析框架。具体来说，我们使用概率一多项式同伦延拓方法来计算目标函数的所有临界点，然后进行筛选以恢复全局最优解。我们的方法利用了模型和数据之间的几何结构，并在细胞信号传导、合成生物学和流行病学的实例中证明了其效用。
