Discovery of unstable singularity with machine precision.

Whether singularities can form in fluids remains a foundational unanswered question in mathematics. This phenomenon occurs when solutions to governing equations, such as the 3D Euler equations, develop infinite gradients from smooth initial conditions. Historically, numerical approaches have primarily identified stable singularities. However, these are not expected to exist for key open problems, such as the boundary-free Euler and Navier-Stokes cases, namely the Millennium Prize problem. For these problems, the true challenge lies in finding unstable singularities, which are exceptionally elusive, as any tiny perturbation can divert the system from its blow-up trajectory.

In this talk, I will present a new computational framework which has led to the first systematic discovery of new families of unstable singularities in various fluid equations. Our approach merges curated machine learning architectures with a multi-stage training scheme and a high-precision Gauss-Newton optimizer, creating a powerful tool for navigating the complex landscape of nonlinear PDEs. Beyond discovering these singularities, the precision of this method is another key breakthrough, achieving unprecedented accuracies on the order of $O(10^{-13})$—a level constrained only by the round-off errors of the GPU hardware. This level of precision meets the stringent requirements for rigorous mathematical validation of the discovered solution via computer-assisted proofs, offering a new pathway to resolving long-standing challenges in mathematical physics.