Steven Brunton
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Lex Fridman Podcast · #503 – AI Crisis in Math: Navier-Stokes, Riemann Hypothesis &…
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What Did AI Prove About Navier-Stokes?
Steven Brunton says the proven result is the forced version on the Clay website, one of four versions, not every Navier-Stokes problem. Smooth external forcing is arranged so terms cancel and a vortex forms, and the construction only works in three dimensions.
The AI proof builds a specialized vortex that stretches and thins until the radius is infinitely small, like a figure skater pulling their arms in during a spin. Energy in the fluid stays finite, while the flow at that filament blows up and the vorticity becomes infinite. Physical intuition says viscosity should damp that growth. Mathematicians only started to think a counterexample was possible about 15 years ago.
Day-to-day enginee
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Why Learn Linear Algebra If a Computer Does It Better?
Steven Brunton says you should still learn linear algebra even if a computer can do linear algebra better than you, because the point is to build intuition, and the same is true for calculus and physics. Curiosity, reading, and the fundamentals have always been good advice. Understanding history also matters, so the present shift can sit next to other great changes across human existence.
The mind's curiosity is under constant pressure from the engagement economy. The same engine that makes a person curious also makes that person doom scroll, so the work is to avoid distractions and let curiosity stay with one subject long enough to soak it in and actually learn it. High school and college
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Why Is Turbulence So Hard?
Steven Brunton says turbulence is a complex fluid roiling, mixing, and moving, and it is hard because it is multi-scale.
Da Vinci drew fluids at many sizes at once. The big vortices, the medium vortices, and the little vortices all matter for predicting lift and drag. Gas in a room is the opposite of turbulence. The molecules move in a frenetic way, and there are 10 to the 23 gas particles, yet the individual configurations do not matter. Thermodynamics, a great success of classical physics, shows that a small handful of numbers determines how the gas behaves: temperature, pressure, and density. No simple description like that exists for turbulent fluids. Flow over a wing, flow inside a hea
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What Is the Basic Building Block of a Fluid?
Steven Brunton says the basic building block of a fluid is a velocity field, together with the pressures that force it to move. A fluid moves, and it moves differently in different places and at different times. At every point in space, a velocity vector tells how the fluid is moving. Where the flow is fast in one place and slow in another, a vortex can form. The picture gets richer when velocity changes a lot over space. A tornado is a fluid velocity field that changes very rapidly in both space and time, and the core of a tornado is low pressure, so pressure pushes the velocity field along.
The equations then tell how that velocity changes in time and in space according to the fluid's own
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Why Is a Fluid So Hard to Solve?
Steven Brunton says force equals mass times acceleration is simple to write down, yet a fluid becomes very complicated to solve in any practical sense. A single pendulum is easy to write equations for. A second pendulum on the end gives a more complicated equation and chaos. Chop the air in a room into a billion little boxes, each with a differential equation for momentum, and the system is simple to write down and simple to understand conceptually, while how the fluid actually moves stays very complicated to solve.
He read that airlines could save something like tens of billions of dollars by reducing drag by 1 percent, and drag is maybe easier than turbulence. A relatively small improveme
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How Are Research Papers Changing Because of AI?
Steven Brunton says an academic contribution in engineering and applied science is already shifting, and with AI a small reusable piece is worth more than one result or one paper.
Writing code inside a larger reusable infrastructure already counts. LIGO, billion dollar astronomy projects, and cancer research projects need hundreds of people and huge resources. The work is not necessarily one 15 page PDF. It is code, data, and reusable bits that fit a larger framework.
A workable widget, an API, a code base, or a data set that the community or AI can leverage is increasingly valuable. How the same change looks in pure math is still open.
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Are Machines Already Finding Physical Laws?
Steven Brunton says that search is already happening for physical laws. Lex Fridman had asked whether a machine could loop through and find every simple, general law, on the scale of E equals mc squared.
Research programs systematically and automatically look for minimal governing equations. They observe a natural system and test whether a simple physical law models how it operates, then distill the observations down to basics until known theory comes back out. Balls dropping yield gravity and F equals ma. Fluids yield the Navier-Stokes equations. The same pass over data is aimed at systems where simple equations have not been written down, including weather, turbulence, and solar flares.
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Why Does Steven Brunton Stay Hopeful About AI?
Steven Brunton says most of his feelings are hopeful. He is an engineer who likes solving problems that make technology more realistic for humans, including artificial hearts and airplanes. The ability to solve real applied problems is going to get much better, and scope and reach are going to increase.
Uncertainty is still large. Fear of losing meaning, losing control, and being cut out of the process people love is a little scary. Over the next five or ten years, two paths would surprise nobody: humans are dwarfed by other intelligence, or progress levels off and AI becomes an incredible tool for curiosity and human problem solving. The two worlds are very different. Uncertainty is diffic