Our new methods 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 hurricane vortex to the airflow lifting the wings of planes.
Experts can carefully develop scenarios in which theory opposes practice, leading to situations that could not occur physically. These situations are called “singularity” or “blow-up,” such as when the amounts of velocity or pressure are infinite. They help mathematicians identify the fundamental limitations of the equations of fluid dynamics and develop an understanding of how the physical world works.
In a new paper, we present a whole new family of mathematical explosions to some of the most complex equations describing fluid motion. He has published the work in collaboration with mathematicians and geophysicists from institutions such as Brown University, New York University and Stanford University.
Our approach presents new ways to leverage AI techniques to tackle long-standing challenges in mathematics, physics and engineering that require unprecedented accuracy and interpretability.
The importance of unstable singularities
Stability is an important aspect of specificity formation. Singularities are considered stable when they are robust to small changes. Conversely, unstable singularities require very accurate conditions.
Because mathematicians believe that there is no stable singularity in 3D Euler and Naviestokes equations without complex boundaries, it is expected that unstable singularities will play a major role in the fundamental problems of fluid dynamics. Finding singularities in the Navier-Stokes equation is one of six well-known Millennium Award problems that have yet to be resolved.
Using our new AI method, we presented the first systematic discovery of a new family of unstable singularities across three different fluid equations. We also observed patterns of patterns appearing as the solution became increasingly unstable. The number lambda (λ), which characterizes the velocity of the explosion, can be plotted against the order of instability. This pattern was presented in two studied equations: the incompressible porous media (IPM) and the Boussinesq equation. This suggests the existence of a more unstable solution, with the hypothesizing lambda values along the same line.
These specificities were discovered by incorporating machine learning techniques such as quadratic optimizers for training neural networks. These methods have allowed us to improve accuracy to an unprecedented level. For reference, the largest error addressed corresponds to predicting the Earth’s diameter within a few centimeters.
Here we present an example of the vorticity (ω) field found in one of the equations studied. This is a measure of how much the liquid is rotating at each point.
Also, for all the instability we discovered, we present one-dimensional slices passing through the same field along the axis, indicating the evolution of increasingly unstable singularities.
New methods navigate vast landscapes of singularity
Our approach is based on the use of physics-based neural networks (PINNs). Unlike traditional neural networks that train from vast data sets, the models were trained to fit equations that model physics laws. The output of the network is always checked against what the physical equation expects and learns by minimizing its “residuals.”
The use of PINNs goes beyond its typical role as a general purpose tool used to solve partial differential equations (PDEs). Embedding mathematical insights directly into training allowed us to capture elusive solutions (such as volatile singularities) that traditional methods have long been challenging.
At the same time, we have developed a high-precision framework that pushes PINN to almost machine accuracy, enabling the level of accuracy required for strict computer aided proofs.
A new era of computer-aided mathematics
This breakthrough combines deep mathematical insights with cutting-edge AI to represent a new way to conduct mathematical research. This work looks forward to helping pioneer a new era where long-standing challenges tackle evidence of AI and computer aid.
