Parameter Inference with Bifurcation Diagrams

TL;DR

This paper introduces a gradient-based method for inferring parameters of differential equation models using bifurcation diagrams, enabling parameter estimation from qualitative change points without differentiating through the solver.

Contribution

It presents a novel approach to parameter inference that leverages bifurcation diagrams and gradient optimization, avoiding the need for differentiating the solver's operations.

Findings

Successfully inferred parameters for saddle-node and pitchfork bifurcation models.

Applied method to the genetic toggle switch in synthetic biology.

Organized models based on topological and geometric equivalence.

Abstract

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.