Our latest method could help mathematicians leverage AI techniques to tackle long-standing challenges in mathematics, physics and engineering.
For hundreds of years, mathematicians have developed complex equations to explain the basic physics involved in fluid dynamics. These laws govern every thing from the swirling vortex of a hurricane to airflow lifting an airplane’s wing.
Experts can rigorously craft scenarios that make theory go against practice, resulting in situations which could never physically occur. These situations, resembling when quantities like velocity or pressure develop into infinite, are called ‘singularities’ or ‘blow ups’. They assist mathematicians discover fundamental limitations within the equations of fluid dynamics, and help improve our understanding of how the physical world functions.
In a latest paper, we introduce a wholly latest family of mathematical blow ups to a few 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, Latest York University and Stanford University
Our approach presents a brand new technique 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 vital aspect of singularity formation. A singularity is taken into account stable if it is strong to small changes. Conversely, an unstable singularity requires extremely precise conditions.
It’s expected that unstable singularities play a significant role in foundational questions in fluid dynamics because mathematicians consider no stable singularities exist for the complex boundary-free 3D Euler and Navier-Stokes equations. Finding any singularity within the Navier-Stokes equations is certainly one of the six famous Millennium Prize Problems which can be still unsolved.
With our novel AI methods, we presented the primary systematic discovery of latest families of unstable singularities across three different fluid equations. We also observed a pattern emerging because the solutions develop into increasingly unstable. The number characterizing the speed of the blow up, lambda (λ), will be plotted against the order of instability, which is the variety of unique ways the answer 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 means the existence of more unstable solutions, whose hypothesized lambda values lie along the identical line.
