Algebraic identifiability of partial differential equation models

Differential equation models are crucial to scientific processes. The values of model parameters are important for analyzing the behaviour of solutions. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. To determine if a parameter estimation problem is well-posed for a given model, one must check if the model parameters are globally identifiable. This problem has been intensively studied for ordinary differential equation models, with theory and several efficient algorithms and software packages developed. A comprehensive theory of algebraic identifiability for PDEs has hitherto not been developed due to the complexity of initial and boundary conditions. Here, we provide theory and algorithms, based on differential algebra, for testing identifiability of polynomial PDE models. We showcase this approach on PDE models arising in the sciences.

微分方程模型对科学过程至关重要。模型参数的值对于分析解的行为很重要。如果一个参数的值可以由输入和输出函数唯一确定，那么它就被称为全局可识别的。要确定对于给定模型参数估计问题是否适定，就必须检查模型参数是否全局可识别。对于常微分方程模型，这个问题已经得到了深入研究，并且已经发展了相关理论以及若干高效算法和软件包。由于初始条件和边界条件的复杂性，偏微分方程的代数可识别性综合理论迄今尚未建立。在这里，我们基于微分代数提供了用于测试多项式偏微分方程模型可识别性的理论和算法。我们在科学中出现的偏微分方程模型上展示了这种方法。