Architecture / Model families

Model families

Neural fields, U-Nets, graph networks and transformers, and what each is good for.

Neural field

The simplest full-field surrogate. An MLP looks at one point at a time: its position (and, on a surface, its normal) plus the design parameters go in, and the field value at that point comes out. Evaluate it at every point and you have the whole field.

• Fourier features turn coordinates into sines and cosines at several frequencies, so the network can draw sharp suction peaks.
• Three hidden layers of 48 SiLU units; 6,337 weights in total.
• Points are independent, so inference is embarrassingly parallel and maps naturally onto GPUs.

Good for: Smooth parametric families of one shape; arbitrary resolution; tiny and fast. Watch out: It never sees its neighbours, so it knows geometry only through coordinates and generalises poorly to new shapes.

U-Net / Fourier Neural Operator

A U-Net encodes the grid through coarser and coarser levels, then decodes back up with skip connections between matching resolutions, so it combines global context with local detail. An FNO replaces convolutions with learned filters on the lowest Fourier modes of the field.

• Input: the geometry as an occupancy or signed-distance grid, plus the flow condition.
• Encoder: convolutions and downsampling build a coarse, global picture.
• Decoder: upsampling with skip connections restores detail, as the coarse-to-fine animation shows.

Good for: Fields on regular domains; robust training; one of the first approaches applied to RANS airfoil flows. Watch out: Fixed resolution and N³ memory; surface quantities have to be read back from voxels.

MeshGraphNets

Encode node and edge features, run K rounds of message passing in which every node aggregates messages from its neighbours, then decode the field at each node. Watch the front in the animation: a node can only be predicted once information from the boundary has reached it, one hop per layer.

• Encoder: MLPs embed node features (position, normal, type) and edge features (relative position, length).
• Processor: K message-passing blocks with residual updates.
• Decoder: an MLP maps each node embedding to the output field.

Good for: Unstructured meshes and complex geometry; transient simulation by rollout. Watch out: The receptive field is K hops; long-range coupling needs deep or multiscale graphs, and cost grows with mesh size.

Transolver

Attention over millions of points is too expensive, so Transolver first assigns every point softly to a few learned slices: groups of points in a similar physical state, wherever they are. Attention runs between the slice tokens, then the result is broadcast back to every point.

• Slice: per-point weights over M slices; each token is a weighted average of point features.
• Attend: standard multi-head attention between the M tokens, which is cheap.
• Deslice: tokens are distributed back to points with the same weights. Cost is linear in N.

Good for: Large unstructured geometries with global coupling; strong results across PDE benchmarks. Watch out: Slices are learned, so they are not guaranteed to mean anything physical; it needs more data than a simple field.

Related

• Representations: Voxel grids, graphs or point clouds: how geometry and fields become numbers.
• Model atlas: Ten surrogate families at a glance, grouped by the data they consume.
• Lineages: Four families of field surrogates, from their founding idea to 2026.