Nonlinear Scale-Local Geometric Deformations of Vortex Rings in Smooth Euler Flows via Bayesian Optimization and Adjoint Methods

We consider the incompressible three-dimensional Euler equations for a vortex ring with Kelvin waves undergoing radially expanding Lagrangian transport. To clarify the fundamental mechanisms underlying nonlinear scale-local deformations of the vortex structure, we develop a geometric Lagrangian framework that avoids singular integral representations of the pressure and yields a novel wave equation governing the axis of swirling particles. Within this framework, we identify intrinsic nonlinear mechanisms that drive scale-local deformations of the vortex structure, supported by a machine-learning-based analysis. Specifically, we propose a hybrid optimization framework that combines Bayesian global exploration with adjoint-based local refinement. The resulting optimization problem exhibits a highly non-convex loss landscape, in which the adjoint method alone fails to escape local minima.