Our new method could help mathematicians use artificial intelligence techniques to solve long-standing challenges in mathematics, physics and engineering.
For centuries, mathematicians have developed complex equations that describe the fundamental physics involved in fluid dynamics. These laws govern everything from the swirling vortex of a hurricane to the airflow that lifts an airplane's wing.
Experts can carefully develop scenarios in which theory conflicts with practice, leading to situations that could never physically occur. Such situations, for example when quantities such as velocity or pressure become infinite, are called “singularities” or “blowouts”. They help mathematicians identify fundamental constraints in fluid dynamics equations and help better understand how the physical world works.
In new paperwe introduce a completely new family of mathematical magnifications of some of the most complex equations describing fluid motion. We are publishing this work in collaboration with mathematicians and geophysicists from institutions such as Brown University, New York University and Stanford University
Our approach presents a new way to use artificial intelligence techniques to address long-standing challenges in mathematics, physics and engineering that require unprecedented accuracy and interpretability.
The importance of unstable singularities
Stability is a key aspect of singularity formation. A singularity is considered stable if it is resistant to small changes. Conversely, an unstable singularity requires extremely precise conditions.
Unstable singularities are expected to play a major role in fundamental fluid dynamics problems because mathematicians believe that there are no stable singularities in complex, boundaryless 3D Euler AND Navier-Stokes equations. Finding any singularity in the Navier-Stokes equations is one of the six famous ones Problems with the Millennium Prize which are still unresolved.
Using our novel artificial intelligence methods, we presented the first systematic discovery of new families of unstable singularities in three different fluid equations. We also observed an emerging pattern as solutions become increasingly unstable. The number characterizing the blowing speed, lambda (λ), can be plotted against the order of instability, which is the number of unique ways in which the solution can deviate from the blowing. This pattern was visible in two of the equations studied, the incompressible porous medium (IPM) equations and the Boussinesq equations. This suggests the existence of more unstable solutions whose hypothetical lambda values ​​lie on the same line.
