Discovering new solutions to century-old problems in fluid dynamics

Our new method could help mathematicians leverage AI techniques to tackle long-standing challenges in mathematics, physics and engineering.

For centuries, mathematicians have developed complex equations to describe the fundamental physics involved in fluid dynamics. These laws govern everything from the swirling vortex of a hurricane to airflow lifting an airplane’s wing.

Experts can carefully craft scenarios that make theory go against practice, leading to situations which could never physically happen. These situations, such as when quantities like velocity or pressure become infinite, are called ‘singularities’ or ‘blow ups’. They help mathematicians identify fundamental limitations in the equations of fluid dynamics, and help improve our understanding of how the physical world functions.

In a new paper, we introduce an entirely new family of mathematical blow ups to some of the most complex equations that describe fluid motion. We’re publishing this work in collaboration with mathematicians and geophysicists from institutions including Brown University, New York University and Stanford University

Our approach presents a new way to leverage AI techniques to tackle longstanding challenges in mathematics, physics and engineering that demand unprecedented accuracy and interpretability.

The importance of unstable singularities

Stability is a crucial aspect of singularity formation. A singularity is considered stable if it is robust to small changes. Conversely, an unstable singularity requires extremely precise conditions.

It’s expected that unstable singularities play a major role in foundational questions in fluid dynamics because mathematicians believe no stable singularities exist for the complex boundary-free 3D Euler and Navier-Stokes equations. Finding any singularity in the Navier-Stokes equations is one of the six famous Millennium Prize Problems that are still unsolved.

With our novel AI methods, we presented the first systematic discovery of new families of unstable singularities across three different fluid equations. We also observed a pattern emerging as the solutions become increasingly unstable. The number characterizing the speed of the blow up, lambda (λ), can be plotted against the order of instability, which is the number of unique ways the solution can deviate from the blow up. The pattern was visible in two of the equations studied, the Incompressible Porous Media (IPM) and Boussinesq equations. This suggests the existence of more unstable solutions, whose hypothesized lambda values lie along the same line.