Inferring bifurcation diagrams with transformers

The construction of bifurcation diagrams is an essential component of understanding nonlinear dynamical systems. The task can be challenging when one knows the equations of the dynamical system and becomes much more difficult if only the underlying data associated with the system are available. In this work, we present a transformer-based method to directly estimate the bifurcation diagram using only noisy data associated with an arbitrary dynamical system. By splitting a bifurcation diagram into segments at bifurcation points, the transformer is trained to simultaneously predict how many segments are present and to minimize the loss with respect to the predicted position, shape, and asymptotic stability of each predicted segment. The trained model is shown, both quantitatively and qualitatively, to reliably estimate the structure of the bifurcation diagram for arbitrarily generated one- and two-dimensional systems experiencing a codimension-one bifurcation with as few as 30 trajectories. We show that the method is robust to noise in both the state variable and the system parameter.