Algebraic model prediction for ODE systems

在科学中使用数学模型几乎总是涉及从数据中估计未知参数值。“Sloppiness”已被提出来研究参数估计中的不确定性。迪弗雷纳、哈灵顿和拉曼为马虎引入了一个统一的数学框架，该框架植根于代数和几何。这项工作的核心是模型预测图[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。构建一个模型预测图需要（几乎）解决一个模问题。实际上，模型预测映射是由完全数据诱导的等价关系对参数空间的商的分离态射。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是中间步骤。对于具有时间序列数据的常微分方程系统，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的代数。本计画的目的是针对一类具有稳态资料的常微分方程系统，建构一个代数模型预测图。

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.