# Parameter Inference with Bifurcation Diagrams

arXiv: https://arxiv.org/abs/2106.04243 · HTML: https://mlsanity.com/paper/2106.04243/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 36 references that are in the map.
- First posted on arXiv 8 Jun 2021.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Fashionable Modelling with Flux](https://mlsanity.com/paper/1811.01457/) (2018)
- [DiffEqFlux.jl - A Julia Library for Neural Differential Equations](https://mlsanity.com/paper/1902.02376/) (2019)
- [Differentiable Implicit Layers](https://mlsanity.com/paper/2010.07078/) (2020)
- [Bayesian Smooth-and-Match strategy for ordinary differential equations models that are linear in the parameters](https://mlsanity.com/paper/1604.02318/) (2016)

## Built on by (most cited first, 2 of 2)

- [Symbolic Neural Ordinary Differential Equations](https://mlsanity.com/paper/2503.08059/) (2025)
- [Adaptive Sampling for Hydrodynamic Stability](https://mlsanity.com/paper/2512.13532/) (2025)

## Abstract (arXiv)

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.