3 6 Ke·2025-09-19 08:48Core Insights - Google DeepMind collaborates with researchers from NYU, Stanford, and Brown University to discover new unstable singularities in fluid equations using machine learning and a high-precision Gauss-Newton optimizer [1][2] - The research aims to tackle the Navier-Stokes equations, one of the Millennium Prize Problems, with a potential reward of $1 million for a solution [1][2] - The project, named "Navier-Stokes Initiative," has been ongoing for three years and involves a 20-member team working under high confidentiality [1][2] Research Findings - The team has systematically identified new unstable singularities in three different fluid equations and established a simple empirical asymptotic formula linking blow-up rates to instability orders [1][2] - The method achieved significant precision improvements over existing works, with results approaching machine limits, constrained only by GPU rounding errors [2][5] Methodology - The research employs a two-phase structure for discovering and analyzing unstable singularities, focusing on high-precision solutions [5][6] - A candidate solution is used to search for self-similar blow-up solutions, followed by iterative optimization of the machine learning process [6][9] - The use of Physics-Informed Neural Networks (PINN) combined with the Gauss-Newton optimizer and multi-stage training strategies enhances the accuracy of the solutions [11][13] Training Enhancements - Key improvements in the training process include the introduction of the Gauss-Newton optimizer, which outperforms standard gradient optimizers in producing high-quality solutions [13][15] - Multi-stage training involves training one network to approximate solutions and a second network to correct errors, significantly enhancing precision [13][15] Results and Implications - The model's accuracy has reached new levels, with maximum errors comparable to predicting the Earth's diameter within a few centimeters [15] - The findings provide a new research paradigm for exploring the complexities of nonlinear partial differential equations (PDEs) and could pave the way for solving long-standing mathematical physics challenges [2][5]