Source: https://arxiv.org/abs/2509.14185

The discovery of new families of unstable singularities in fluid dynamics, a long-standing challenge in mathematics. Researchers utilized a novel framework combining curated machine learning architectures with a high-precision Gauss-Newton optimizer to achieve unprecedented accuracy in solving relevant partial differential equations (PDEs).

Specifically, the study identified unstable self-similar solutions for the incompressible porous media equation and the 3D Euler equation with a boundary, providing an empirical asymptotic formula for blow-up rates. This approach significantly surpasses previous numerical work, reaching near double-float machine precision for certain solutions, which is crucial for rigorous mathematical validation through computer-assisted proofs.

The findings offer a new methodology for exploring complex nonlinear PDEs and tackling fundamental problems in mathematical physics.