Finding heteroclinic connections between simple invariant solutions using automatic differentiation

Our mechanistic understanding of fluid turbulence has substantially improved in recent decades due to the discovery of large numbers of unstable simple invariant solutions to the Navier-Stokes equations. Heteroclinic connections between these solutions have been hypothesised to play an important role in high-dissipation, intermittent `bursting' events. However, standard methods of detecting simple invariant solutions are not suited to finding these connecting orbits, and consequently only a few have been discovered to date. Here, we introduce automatic differentiation (AD) as a robust technique for finding connections via gradient-based minimisation of a suitable loss function. We first use a new, fully differentiable point vortex solver [jax-pv] as a playground to test the loss-based approach. We demonstrate that AD can successfully find the connection in the integrable 3 vortex system, and also find connections in non-integrable systems with higher numbers of vortices. We then extend our analysis to a two-dimensional Kolmogorov flow (monochromatically forced on the two-torus) using a fully differentiable Navier-Stokes solver (Kochkov et al, Proc. Nat. Acad. Sci. 118, 2021) to search for "bursting" connections between low dissipation relative equilibria.