基于分叉图的参数推断
Parameter Inference with Bifurcation Diagrams
摘要：
微分方程模型中的参数估计可以通过将学习算法应用于定量时间序列数据来实现。然而，有时只能测量系统响应受控条件的定性变化。在动力系统理论中，这样的变化点被称为分叉，并且位于被称为分叉图的受控条件的函数上。在这项工作中，我们提出了一个基于梯度的方法来推断微分方程的参数，产生用户指定的分岔图。成本函数包含一个误差项，这是最小的模型分叉匹配指定的目标和分叉措施，它有梯度，推动优化器走向分叉参数制度。可以计算梯度，而不需要通过用于计算图的求解器的运算进行微分。我们展示了最小模型，探索鞍结和干草叉图的空间和合成生物学的遗传切换开关的参数推断。此外，成本景观允许我们根据拓扑和几何等价性组织模型。
英文摘要：
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.
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