Physics as data / From geometry to numbers

Representations

Voxel grids, graphs or point clouds: how geometry and fields become numbers.

The geometry is rasterised onto a regular 3D grid, usually as an occupancy or signed-distance field, and the physical field (pressure, velocity, temperature) is stored per voxel. Grids are what convolutional networks and FFT-based operators expect. Strength: Simple, GPU-friendly, mature architectures. Limit: Memory grows with the cube of resolution; thin features like a trailing edge disappear below the voxel size. Models: U-Net, FNO.

Nodes sit on mesh points and edges connect neighbours, either from the simulation mesh or by k-nearest neighbours. Field values live on the nodes. The graph follows the geometry exactly at any resolution. Strength: Mesh-native; handles unstructured CFD meshes and complex geometry. Limit: Information moves one hop per layer, so long-range effects need many layers or multiscale graphs. Models: MeshGraphNets, GNS.

Only points with coordinates (and often normals), with no connectivity. It is the lightest representation and works for any geometry source, from CAD tessellations to scans. Strength: No meshing, no grid; any number of points, in any order. Limit: The network has to learn spatial relations itself, with attention, a learned latent grid or by conditioning on coordinates. Models: Neural fields, PointNet, Transolver, GINO.

Related

• Model families: Neural fields, U-Nets, graph networks and transformers, and what each is good for.
• Tokens: Each mesh point becomes a vector of numbers describing where it is and what it carries.
• Physical fields: Decide which quantities the model must reproduce before choosing the model.