GRAPHIC APPAREL SHOPResearchers at Google DeepMind are using AI to identify new solutions to challenging fluid dynamics equations. Their achievement offers fresh hope for solving some of the most persistent challenges in mathematics and physics.
Singularities: The Enigmas of Fluid Equations
Fluid dynamics governs how liquids and gases move, relying on intricate equations like Euler and Navier-Stokes. These equations describe everything from weather patterns to aircraft design. Yet, they also hide "singularities," cases where variables such as velocity or pressure become infinite. Although these extremes don't occur in nature, they highlight the limits and potential flaws in our models.
Unstable singularities are particularly intriguing. Mathematicians believe they reveal deep insights about fluid behavior, but finding them is exceptionally difficult. Proving even one singularity in the Navier-Stokes equations remains a Millennium Prize Problem, one of mathematics' greatest unsolved mysteries.
AI Unveils New Singularities
In an innovative collaboration with academic partners, DeepMind introduced an AI-driven system to systematically identify new families of unstable singularities. They discovered an unexpected trend: as singularities became more unstable, the rate at which they formed (represented by lambda) followed a distinct pattern. This finding, observed in equations like Incompressible Porous Media (IPM) and Boussinesq, points to new avenues for future discoveries.
Key to this progress was advanced machine learning, particularly second-order optimizers for neural network training. These tools enabled the team to reach astonishing levels of accuracy, equivalent to measuring the diameter of the Earth within a few centimeters. Such precision exposed subtle mathematical errors previously undetectable.
Physics-Informed Neural Networks: A Precision Engine
The heart of DeepMind's approach is the use of Physics-Informed Neural Networks (PINNs). Unlike traditional machine learning models that learn from huge datasets, PINNs are designed to obey the laws of physics set by partial differential equations. They continuously adjust their predictions to minimize mathematical errors, or "residuals," aligning their learning process with the equations themselves.
By embedding these equations directly into their models and striving for near-machine precision, the researchers turned PINNs into powerful tools for discovering rare, unstable singularities. This approach not only advances the search for new solutions but also opens the door to rigorous, AI-assisted mathematical proofs.
Seeing the Results: Visualizing Singularities
To illustrate their findings, the team shared visualizations mapping fluid properties like vorticity, how much a fluid spins at each point. These visuals track how singularities emerge and evolve, offering compelling evidence for the effectiveness of their AI methods and revealing new structural patterns within the equations.
Line graph showing a one-dimensional slice through the same two-dimensional vorticity (Ω) field shown above, showing the evolution of increasingly unstable singularities. Credit: Google Deepmind
Transforming Scientific Discovery
DeepMind's breakthrough signifies a turning point in mathematical research. It demonstrates how deep mathematical understanding, combined with sophisticated AI, can unlock solutions previously out of reach. The implications extend to fields like theoretical physics, engineering, and even the development of computer-assisted proofs, where human ingenuity and machine precision join forces.
AI as a Partner in Science
This latest advance proves that AI's role in science goes far beyond routine applications. With tools like PINNs, artificial intelligence is emerging as a collaborator in unraveling the universe's fundamental mysteries. As these methods continue to evolve, we can anticipate further breakthroughs that shed light on the deep structures underpinning our scientific knowledge.
How AI Is Revolutionizing Fluid Dynamics and Mathematical Discovery