Zoom link: https://snu-ac-kr.zoom.us/my/youngjoonhong

Whether singularities can form in fluids remains a foundational unanswered question in mathematics. Historically, numerical approaches have primarily identified stable singularities. However, these are not expected to exist for key open problems, such as 3D Euler and Navier-Stokes 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 novel computational framework that leverages Physics-Informed Neural Networks (PINNs) to enable the first systematic discovery of new families of unstable singularities. Rather than treating neural networks as black boxes, we show how embedding key mathematical features of the problem as strong inductive biases into the network design enhances training efficiency and transforms the method into a powerful tool for resolving singularities across a wide range of fluid equations. In addition, we leverage the new second-order Gauss-Netwon optimizer with multi-stage training scheme that enables PINN training to resolve solutions with near–machine precision O(1e-14), a level constrained only by the GPU round-off errors. Such unprecedented precision not only uncovers the fine-scale structure of fluid blow-up but also meets the stringent requirements for rigorous computer-assisted mathematical proofs, offering a new pathway to resolving long-standing challenges in mathematical physics