Estimation of parameters in differential equation models can be achieved by applying learning algorithms to quantitative time-series data. However, sometimes it is only possible to measure qualitative changes of a system in response to a controlled condition. In dynamical systems theory, such change points are known as bifurcations and lie on a function of the controlled condition called the bifurcation diagram. In this work, we propose a gradient-based approach for inferring the parameters of differential equations that produce a user-specified bifurcation diagram. The cost function contains an error term that is minimal when the model bifurcations match the specified targets and a bifurcation measure which has gradients that push optimisers towards bifurcating parameter regimes. The gradients can be computed without the need to differentiate through the operations of the solver that was used to compute the diagram. We demonstrate parameter inference with minimal models which explore the space of saddle-node and pitchfork diagrams and the genetic toggle switch from synthetic biology. Furthermore, the cost landscape allows us to organise models in terms of topological and geometric equivalence.
翻译:通过对定量时间序列数据应用学习算法,可以对差异方程模型中的参数进行估计。但是,有时只能根据受控条件衡量系统的质量变化。在动态系统理论中,这种变化点被称为双形,并取决于控制条件的函数,即双形图。在这项工作中,我们提出了一个基于梯度的方法,用以推断产生用户指定的双形图的差别方程参数。成本函数包含一个最小的错误术语,当模型双形匹配指定目标时,以及一个具有梯度,将选取者推向双形参数体系的双形测量尺度时,该错误术语是最小的。在计算梯度时,不必通过用于对图进行比较的求解器的操作加以区分。我们用最小模型来显示参数的推推力,这些模型探索马鞍-圆形和Groidfork图的空间,以及合成生物学的基因切换。此外,成本景观使我们能够用表和几何等等法来组织模型。