Active search for Bifurcations
Yorgos M. Psarellis, Themistoklis P. Sapsis, Ioannis G. Kevrekidis
TL;DR
This work tackles locating bifurcations in dynamical systems from scarce, possibly noisy data by formulating bifurcation detection as an active-learning problem solved with Bayesian Optimization and Gaussian Process surrogates. It derives analytical expressions for the uncertainty propagation from GP-predicted vector fields to steady-state derivatives, Jacobians, and eigenvalues, enabling uncertainty-aware acquisition functions that guide efficient sampling. The framework handles black-box timesteppers and extends to reduced-order PDE/ABM models, demonstrating robust performance across 1D folds, 2D Hopf bifurcations, 2D folds, 4D neuronal folds, and POD-reduced PDEs. The approach yields substantial computational savings over Monte Carlo methods while providing principled uncertainty quantification, with broad applicability to resource-constrained experimental and computational settings.
Abstract
Bifurcations mark qualitative changes of long-term behavior in dynamical systems and can often signal sudden ("hard") transitions or catastrophic events (divergences). Accurately locating them is critical not just for deeper understanding of observed dynamic behavior, but also for designing efficient interventions. When the dynamical system at hand is complex, possibly noisy, and expensive to sample, standard (e.g. continuation based) numerical methods may become impractical. We propose an active learning framework, where Bayesian Optimization is leveraged to discover saddle-node or Hopf bifurcations, from a judiciously chosen small number of vector field observations. Such an approach becomes especially attractive in systems whose state x parameter space exploration is resource-limited. It also naturally provides a framework for uncertainty quantification (aleatoric and epistemic), useful in systems with inherent stochasticity.