Parameter Inference with Bifurcation Diagrams

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 semi-supervised approach for inferring the parameters of differential equations that produce a user-specified bifurcation diagram. The cost function contains a supervised error term that is minimal when the model bifurcations match the specified targets and an unsupervised 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.Preprint. Under review.

Multi-parametric optimization for controlling bifurcation structures

“…This issue is here circumvented by using a combination of geometric and arithmetic means as introduced in [20]. It allows one to consider all possible combinations of targets and detected bifurcations, with the added benefit of mitigating the risk of two bifurcations being matched to the same target as the term goes to zero only when all targets are matched with at least one bifurcation.…”

“…In this paper, we propose a general computational methodology to control, through multiparametric optimization, the local bifurcations of engineered systems, moving them at desired locations in parameter space or even removing them if necessary. The method is inspired by the work of Szep et al [20] where an optimization problem was formulated to estimate normal form model parameters of a gene model based on bifurcation diagrams. Here, we repurpose this work to the context of bifurcation control and generalize it to handle multiple types of bifurcations simultaneously.…”

Model-Free Dominant Pole Placement for Restabilizing High-Dimensional Network Systems via Small-Sample-Size Data

“…. , l. The optimizer of ( 14) satisfies that i∈I B1 n j=1 (B(i, s i )K (s i , j) − αv 1 (i)r(j)) 2 = 0 since we can chose K * gen (s i , j) = αv 1 (i)r(j) for any possible r. Thus, we can ignore the second term of ( 17) and focus on minimizing i∈I B0 v 1 (i) 2 (r(1) 2 + . .…”

“…By choosing I B1 = I v,l , the index i with a large value of v 1 i 2 is not included in I B0 . Thus, we can minimize the coefficient v 1 (i) 2 for every i ∈ I B0 and achieve the goal of minimizing (18). □…”

“…The research on predicting and preventing the deterioration of a high-dimensional network system has always been a hot topic. An interesting phenomenon in the progress of system deterioration is that there is a critical transition just before the system falls into a drastic deterioration [1], [2]. One key…”