Google KI könnte den Millenniums-Preis gewinnen. Ein chinesischer Doktor hat ein hundertjähriges mathematisches Rätsel gelöst und erstmals eine Singularität erfasst.

[Introduction] The century-old problem in fluid mechanics has finally been solved by AI! Google DeepMind, in collaboration with top institutions, has for the first time used AI to successfully discover a new family of mathematical "singularities" in three different equations, creating a brand - new research paradigm. Could the next Nobel Prize be pre - booked by AI?

The Millennium Prize Problem has finally seen a glimmer of hope!

Today, Google DeepMind, along with four top institutions including NYU and Stanford, has published a 20 - page high - impact paper

They used AI to discover a series of new families of unstable "singularities" in three different fluid equations.

These "singularities" are major mysteries in mathematical physics.

Generally, when mathematicians describe "fluid motion", they often use the Navier–Stokes equations.

In daily life, the airflow lifting an airplane wing or the formation of a whirlwind or hurricane all fall into this category.

However, in fluid mechanics, under certain extreme scenarios, these equations will "break down" and predict impossible infinite values.

A key challenge is how to find the "unstable singularities" in the equations?

Therefore, the Google DeepMind team used the "Physics - Informed Neural Network" (PINN) to directly encode the equations into the loss function of the neural network and minimize the difference between its output and the requirements of the equations.

Paper link: https://arxiv.org/pdf/2509.14185

As a result, they observed a clear and unexpected pattern: when the solution becomes more unstable, one of its key properties approaches a linear distribution infinitely.

This reveals a previously undiscovered new underlying mathematical structure in these equations.

Simply put, when the singularities become more and more "unstable", their behaviors converge into a linear distribution, showing amazing regularity.

This means that Google's AI has found a new solution to the century - old problem in fluid mechanics!

It will bring new breakthroughs to mathematics, physics, and engineering and is of great significance to weather forecasting, flood simulation, aerodynamics, and even cardiovascular research.

The Millennium Prize Problem, Unsolved for a Century

Everything follows certain laws.

For centuries, mathematicians have established various complex equations to describe the basic physical principles behind fluid dynamics.

They hope to carefully construct some scenarios where theory contradicts practice, thus predicting situations that are physically impossible to occur.

In these situations, physical quantities such as velocity and pressure tend to infinity, which are called "singularities" or "blow - ups".

Only by understanding the "singularities" can we see the fundamental limitations of fluid dynamics equations and accelerate human understanding of the way the physical world operates.

Among them, stability is a key characteristic in the formation process of singularities.

If a singularity remains stable under small perturbations, it is considered a "stable singularity".

Conversely, the formation of "unstable singularities" requires extremely harsh conditions.

Mathematicians believe that there are no stable singularities in the complex, boundary - less three - dimensional "Euler equations" and "Navier–Stokes equations".

In 1822, the French mathematician Henri Navier first proposed the basic equations describing fluid motion. Twenty - three years later, the Irish mathematician George Gabriel Stokes further improved them.

This is the real origin of the "Navier–Stokes equations".

All along, mathematicians have not solved it, and the core problem lies in -

Proving that the solutions of the equations are always "smooth" or that "singularities" occur under certain conditions.

Put simply, the reason why a tsunami suddenly occurs on a calm sea surface is closely related to the solution of this key problem.

The existence and smoothness of the solutions of the "Navier–Stokes equations" is one of the six "Millennium Prize Problems" set by the Clay Mathematics Institute.

Whoever solves this problem can win a $1 million prize.

Terence Tao and his co - author once studied the local and global behaviors of the solutions of the Navier–Stokes equations.

This time, Google DeepMind may be the first to pick the "holy grail" of this problem.

AI Has Found the Unstable Singularities

As early as three years ago, Google DeepMind, in collaboration with teams from NYU, Stanford, Brown University, etc., began to work on this secretly.

This team includes not only the world's top mathematicians but also famous geophysicists.

In the paper, the collaborators adopted a new AI method and for the first time systematically discovered a series of families of unstable "singularities" in three different fluid equations.

Research flow chart

The research process mainly consists of two main stages:

1. Solution - Finding Stage

First, "cast a net" in the space of self - similar blow - up solutions to find possible solutions. A key parameter is the scaling rate λ. Take the i - Burger's equation as an example.

Subsequently, an iterative method is used to optimize the machine - learning process (Figure ii) and improve the accuracy of the solutions.

The actually calculated candidate solutions (Figure iii) and their accuracy will help the collaborators adjust the mathematical model and the neural network structure.

For example, how to transform the input coordinates and how to design the output field all belong to "inductive bias".

The most important step comes. The researchers used the "Physics - Informed Neural Network" (PINN), combined with a Gauss - Newton optimizer and a multi - stage refinement training scheme, to generate high - precision candidate solutions while searching for the scaling rate λ.

2. Analysis Stage

After finding the candidate solutions, the team linearized them through partial differential equations to analyze their stability.

As a result, they found "unstable modes" - any small perturbation will cause the system to deviate from the blow - up solution trajectory.

Thus, by quantifying the degree of stability, high - precision stable/unstable singularities were finally found.

As shown below, the researchers unexpectedly found that as the "order of instability" of the solution (i.e., the number of unique ways the solution deviates from the blow - up) increases, the values of the parameter λ form a clear linear pattern.

This pattern is clearly visible in the Incompressible Porous Media (IPM) equation and the Boussinesq equation.

This implies that there may be more unstable solutions, and their corresponding λ values are expected to fall on the same straight line.

In addition, the research also shows more visual examples -

The following figure shows the vorticity (Ω) field calculated by one of the equations. Vorticity is a physical quantity that measures the intensity of rotation of a fluid at each point in space.

Another example is the one - dimensional slice diagram of the same vorticity field along an axis for all the instabilities discovered.

The figure shows the evolution process of the singularities as the instability increases.

Physics - Informed Neural Network: PINN Plays a Great Role

The reason why these singularities can be discovered is that Google DeepMind has integrated multiple ML technologies.

Specifically, the paper uses the "Physics - Informed Neural Network" (PINN) to capture unstable singularities.

Traditional neural networks need to learn from a large - scale dataset, while PINN is different.

It directly embeds physical laws and trains the grid to match the expectations of the equations. By minimizing the "residual", that is, the deviation between the network solution and the requirements of the equations, it "learns" to follow physical laws.

It is worth noting that the DeepMind team did not simply apply PINN. They embedded the intuition and insights of mathematicians into the training process of AI.

The team also integrated machine - learning technologies, such as second - order optimizers, and developed a high