Algebraic model prediction for ODE systems

数学模型在科学中的应用几乎总是涉及对数据中未知参数值的估计。“马虎度”一词被用来研究参数估计中的不确定性。杜弗雷纳、哈林顿和拉曼引入了一个统一的数学框架，以代数和几何为基础。这项工作的核心是模型预测映射[phi]的概念，它对应于完美数据的特定选择和将参数映射到相应数据预测的函数。构造模型预测图需要（几乎）解决一个模问题。实际上，模型预测映射是由完美数据诱导的等价关系对参数空间商的分离态射。模问题的解将给出集合论商的几何结构，分离态射的构造可以作为中间步骤。对于具有时间序列数据的ODE系统，已经有几种分离态射的构造；对于稳态数据，这是一个开放的问题。一个主要的挑战是，选择更接近实验现实的完美数据并不是自然代数的。本课题的目的是为一类具有稳态数据的ODE系统构建一个代数模型预测映射。

The use of mathematical models in the sciences almost always involves the estimation of unknown parameter values from data. "Sloppiness" has been proposed to study the uncertainty in parameter estimation. Dufresne, Harrington and Raman introduced a unified mathematical framework for sloppiness, rooted in algebra and geometry. Central to this work is the concept of model prediction map [phi], which corresponds to a specific choice of perfect data and a function mapping parameters to the corresponding data prediction. Constructing a model prediction map requires (nearly) solving a moduli problem. Indeed, a model prediction map is a separating morphism for the quotient of parameter space by the equivalence relation induced by perfect data. A solution of the moduli problem would give a geometric structure to the set-theoretic quotient, the construction of a separating morphism can be an intermediate step. For ODE systems with time series data, there are already several constructions of separating morphisms; for steady-state data it is an open problem. A main challenge is that the choice of perfect data that fits more closely to the experimental reality is not naturally algebraic. The objective of this project is to construct an algebraic model prediction map for a class of ODE systems with steady state data.