A Framework for Discovering and Amplifying Finite-Time Singularities in 3D Navier-Stokes Flows

The question of whether the three-dimensional, incompressible Navier-Stokes equations can develop finite-time singularities from smooth initial conditions remains one of the most profound open problems in mathematics. While a formal proof is elusive, computational investigations provide critical evidence and intuition. In this work, we present a novel computational framework that goes beyond direct simulation. We employ an AI-driven discovery engine, based on an evolutionary algorithm, to systematically find optimal spectral filters that, when applied to initial conditions, maximally amplify the growth of singularity proxies in low-resolution test simulations. This framework was applied to the classic case of anti-parallel vortices, using the variable vortex separation distance as a key parameter. Our results reveal a complex, non-linear relationship between the initial geometry and the estimated blow-up time, identifying a distinct "sweet spot" for singularity formation at a separation distance of approximately 1.26 units. We demonstrate that the AI-discovered filters are highly specific to the initial geometry, suggesting that the mechanism for singularity amplification is not universal but is instead intrinsically linked to the flow's initial structure. This work provides a powerful, data-driven methodology for mapping the landscape of potential Navier-Stokes singularities and offers a concrete, highly-unstable set of initial conditions as a target for future analytical study.