  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.
