Euler equations

The Euler equations are a set of mathematical formulas. They describe how air or water move. More precisely: they predict how velocity, density, and pressure at every point in a flow change from one moment to the next. They were formulated by the Swiss mathematician Leonhard Euler around 1757. Their crucial simplifying step is that they ignore the internal friction of the fluid — that is, the fact that neighboring layers of fluid stick to one another and slow each other down. This makes the formulas considerably easier to compute, but they only represent reality approximately.

The price of omitted friction

Without this simplification, one arrives at the Navier-Stokes equations. These are physically more complete because they account for friction as well. But they are also considerably more expensive to solve. For many problems, this extra effort is not worthwhile, because friction only plays a role in an extremely thin layer directly at the surface. Far away from walls, air behaves almost as if it had no friction at all.

That is precisely why the Euler equations are so widespread in engineering. Aircraft designers use them to estimate the lift of a wing profile. Astrophysicists use them to describe gas clouds in space, where friction plays practically no role. Weather models, too, are essentially built on related equations. One obtains a usable result in a fraction of the computation time.

There are limits, however, that one must be aware of. The aerodynamic drag of a car cannot be meaningfully calculated using the Euler equations, since it arises largely from friction. They also incorrectly predict flow separation on a wing. Anyone who uses them must therefore know exactly which part of the physics they are discarding.

Conservation laws as a computational principle

At their core, the Euler equations rest on three simple bookkeeping rules. First: mass does not disappear. Whatever flows out of an imagined small cube must have either flowed in beforehand or reduced the density within the cube. Second: motion changes only due to forces, here primarily due to pressure differences. Third: energy, too, is conserved overall.

A computer cannot apply these rules to infinitely many points simultaneously. Therefore, space is broken down into a grid of many small cells. For each cell, it is calculated how much mass, momentum, and energy migrate to the neighboring cells per tiny time step. Then the simulation advances by one time step. Millions of such steps ultimately yield the course of a flow.

One particular feature makes the whole thing difficult: the Euler equations produce shock waves. These are locations where pressure and density jump within an extremely short distance — for example, during the sonic boom of an aircraft. At such jumps, naive numerical methods fail and produce nonsensical zigzag values. Special numerical methods are needed that can represent jumps cleanly.

Why AI researchers compute with them

In recent years, the Euler equations have become a standard test case for AI. The idea: a neural network — a program that learns patterns from examples — is meant to predict flows without computing through the grid step by step. It is trained using the results of classical simulations. Afterward, it delivers an approximation in milliseconds instead of hours.

Research groups at Nvidia, DeepMind, and many universities are working on such models. Names like Fourier Neural Operator or Physics-Informed Neural Network come up regularly in this context. The Euler equations serve as a benchmark because their exact solutions are known for simple cases. One can thus measure how far off an AI model is.

Economically, this is quite relevant. Flow simulations consume a large share of the computation time in automotive, aircraft, and turbine development. If an engineer can test twenty variants per hour instead of one simulation per night, that changes the way work is done. That is why the term is now found not only in physics books, but also in reports about AI chips and simulation software.