Resources / Research

Lineages

Four families of field surrogates, from their founding idea to 2026.

Grid & voxel

The domain is a regular grid. Convolutions and spectral filters exploit the structure; geometry enters as a signed-distance or occupancy channel.

• U-Net (MICCAI 2015): Encoder–decoder with skip connections: global context and fine detail in one network.
• Deep flow U-Net (AIAA J. 2020): U-Nets reproduce RANS flow fields around airfoils from the geometry mask.
• FNO (ICLR 2021): Learns weights on the lowest Fourier modes: global mixing, resolution-independent evaluation.
• CNO (NeurIPS 2023): Convolutional neural operator that respects the continuous–discrete correspondence, avoiding aliasing.
• GINO (NeurIPS 2023): Graph operator maps irregular geometry onto a latent grid, where an FNO works.
• Poseidon (NeurIPS 2024): Multiscale operator transformer pretrained on many PDEs; fine-tunes with far fewer samples.

Graph & mesh

Nodes are mesh points and edges the mesh connectivity, so resolution follows the physics: fine near walls and shocks, coarse far away.

• GNS (ICML 2020): Message passing on particle neighbourhoods; noise injection keeps long rollouts stable.
• MeshGraphNets (ICLR 2021): Encode–process–decode on simulation meshes, with mesh-space and world-space edges.
• Multiscale MGN (ICML workshop 2022): Coarse graph levels carry information across the domain in fewer layers.
• BSMS-GNN (ICML 2023): Bi-stride pooling builds multiscale graphs automatically, without hand-made coarse meshes.
• GAOT (NeurIPS 2025): Multiscale attentional graph encoder and decoder around a transformer processor.
• GIST (arXiv 2026): Spectral embeddings of mesh connectivity for tightly packed geometries such as race cars.

Point cloud

Only coordinates (and normals), no connectivity. Works on any CAD tessellation or scan, and on surface and volume points alike.

• PointNet (CVPR 2017): Shared per-point MLP plus symmetric pooling: invariant to point order.
• PointNet++ (NeurIPS 2017): Hierarchical neighbourhoods add the local structure PointNet lacks.
• DGCNN (TOG 2019): Edge convolutions on dynamically recomputed k-NN graphs; a common car-drag baseline.
• Point Transformer V3 (CVPR 2024): Serialised neighbourhoods make attention on large point clouds fast.
• Transolver (ICML 2024): Physics-attention over learned slices of points in similar physical state.
• Transolver++ (ICML 2025): Adaptive slices and multi-GPU parallelism for million-point industrial meshes.

Latent transformers & operators

Any input (mesh, points, grid) is encoded into latent tokens; any output location is a query. The model learns the map between functions, not between grids.

• DeepONet (Nat. Mach. Intell. 2021): Branch net encodes the input function, trunk net the query location.
• GNOT (ICML 2023): Cross-attention over heterogeneous inputs with linear-complexity attention.
• UPT (NeurIPS 2024): Fixed-size latent space, Perceiver-style encoder, latent rollout in time.
• LNO (NeurIPS 2024): Physics-cross-attention into and out of a latent space; forward and inverse problems.
• AB-UPT (TMLR 2025): Anchored, branched decoders; surface and volume fields on up to 150M cells.
• Walrus (arXiv 2025): Cross-domain foundation model for continuum dynamics, fine-tuned per task.
• AeroJEPA (arXiv 2026): Predicts the latent of the flow from the geometry latent; semantic, probe-able latent space and mesh-size-invariant decoding.

Related

• Recent papers: Recent field-surrogate architectures from NeurIPS, ICML and ICLR, 2023 to 2026, in order.
• Model families: Neural fields, U-Nets, graph networks and transformers, and what each is good for.
• A short history: Seventy years from response surfaces and kriging to neural fields on any geometry.