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 that describe the fundamental physics involved in fluid mechanics. These laws apply to everything from the swirls of a hurricane to the air currents that lift the wings of an airplane.
Experts can carefully create scenarios where theory contradicts practice, giving rise to situations that could never occur physically. Such a situation, where a quantity such as velocity or pressure becomes infinite, is called a “singularity” or “explosion.” These help mathematicians identify the fundamental limits of fluid mechanics equations and improve our understanding of how the physical world works.
A new paper introduces an entirely new family of mathematical extensions of some of the most complex equations that describe fluid motion. We collaborated with mathematicians and geophysicists from Brown University, New York University, Stanford University, and others to publish this research.
Our approach presents a new way to leverage AI techniques to tackle long-standing challenges in mathematics, physics, and engineering that require unprecedented precision and interpretability.
The importance of unstable singularities
Stability is an important aspect of singularity formation. A singularity is considered stable if it is robust to small changes. Conversely, unstable singularities require very precise conditions.
Because mathematicians believe that stable singularities do not exist in complex, unbounded, 3D Euler and Navier-Stokes equations, unstable singularities are expected to play an important role in fundamental problems in fluid mechanics. Finding the singularity of the Navier-Stokes equations is one of the six famous Millennium Prize problems that remain unsolved.
Using our new AI method, we present the first systematic discovery of a new family of unstable singularities across three different fluid equations. We also observed a pattern that emerged as the solution became increasingly unstable. The numerical value lambda (λ) characterizing the velocity of the explosion can be plotted against the order of instability. This is the number of unique ways a solution can deviate from explosion. This pattern was confirmed for two of the equations investigated, the incompressible porous media (IPM) and Boussinesq equations. This suggests the existence of a more unstable solution where the hypothetical lambda values are on the same line.
