Fluids and learned simulators.

How a fluid moves is governed by the Navier–Stokes equations, and for all but the simplest cases they are solved numerically: the space is divided into a grid and the flow advanced in small time steps. The site’s solver uses the lattice Boltzmann method, which tracks how much fluid moves in each of a few directions at every grid point; it is simple, local and runs well on a graphics card.

Reynolds number

How much inertia matters against viscosity. Low (a biobot in water): smooth, sticky flow that stops when pushing stops. Higher: eddies, and behind an obstacle, vortices peeling off in turn.

Flow past a cylinder

A classic benchmark: fluid in a channel around a round obstacle, smooth at low Reynolds number and shedding vortices above about 50.

Prescribed flow

Where solving the fluid isn't the point, a fixed flow field stands in, as in the microfluidic chips of the Chip Archipelago.

A learned simulator, or emulator, is a neural network trained on a solver’s output to predict the next frame from the current one. Run on its own predictions, it can stand in for the solver, often much faster on suitable hardware. The catch is that it is reliable only on cases like those it was trained on: give it a faster flow, a new obstacle or a different channel and it can drift into flows that break physical laws, with nothing to say so. The questions are whether its failures can be seen coming, from its own output, and whether handing back to the solver in time is worth it.

Sources: Schäfer, Turek et al., the cylinder benchmark; Ohana et al., “The Well”; McGreivy & Hakim, on weak baselines in machine learning for fluids.

• Learned Simulators: Early Warnings: a lattice Boltzmann solver, a network trained to replace it, and the warning signals that come before it fails.
• Steering Active Matter: a self-driven fluid of filaments whose flow is solved each step (Stokes flow with friction), and light patterns designed through the simulation.
• Steering a Swarm: robots too small for inertia, carried by a simple channel flow and pushed by one shared field.
• Biobots: why low Reynolds number makes cilia-driven bodies stop the instant their cilia do.