In one lineTen surrogate families at a glance, grouped by the data they consume.
Scalar models predict a few numbers; full-field models predict the whole field.
A handful of design parameters → one or a few scalars
A probabilistic regression over the design space: every prediction comes with an uncertainty estimate.
Fields on a fixed mesh
Snapshots are compressed onto a few dominant modes (POD/PCA); a regression model predicts the mode coefficients.
Voxel or pixel grids: SDF or occupancy in, field out
Convolutional encoder–decoder with skip connections between matching resolutions.
Functions sampled on a regular grid
Layers mix information globally by learning weights on the lowest Fourier modes of the field.
Simulation mesh as a graph (nodes + edges)
Encode–process–decode: message passing along mesh edges, one hop per layer.
Particles with neighbour graphs rebuilt every step
Learns particle accelerations by message passing, then integrates in time.
Unordered points (x, y, z, normals…)
A shared MLP per point plus a symmetric global pooling, so point order does not matter.
Point cloud or mesh → latent regular grid → points
A graph neural operator maps points onto a latent grid, an FNO works there, and a second graph operator maps back.
Points from any mesh or point cloud
Physics-attention: points are softly grouped into learned slices, attention runs between slice tokens, results broadcast back to points.
Input function at sensor points + query coordinates
A branch net encodes the input function, a trunk net encodes where to evaluate; their dot product is the output.
In one lineThirty techniques that make surrogates work in practice, from basics to the frontier.
Classical and deep techniques, filterable by depth and by topic.
Latin hypercube, Sobol sequences or maximin designs spread simulations evenly over the design space.
+ why it mattersPolynomial regression and Gaussian-process interpolation over a few design parameters.
+ why it mattersScale coordinates, parameters and fields to unit range; predict non-dimensional quantities (e.g. a pressure coefficient rather than pascals).
+ why it mattersMeasure RMSE, relative L2 and R² on designs never used in training, by k-fold or a fixed hold-out set.
+ why it mattersGive each point its distance to the wall and the surface normal as input features.
+ why it mattersLift inputs to an embedding, process with a stack of blocks, read out the field.
+ why it mattersMap coordinates through sines and cosines at several frequencies before the first layer.
+ why it mattersEach node updates from aggregated messages of its neighbours, one hop per layer.
+ why it mattersLearned filters on the lowest Fourier modes of the field.
+ why it mattersCoarsen the grid, graph or point set, process, and refine with skip connections.
+ why it mattersTokens weight each other by query–key similarity; cross-attention reads one set from another.
+ why it mattersCorrupt inputs with noise during training of time-stepping models.
+ why it mattersCompress snapshots onto a few dominant modes and regress the mode coefficients.
+ why it mattersIn one lineFour families of field surrogates, from their founding idea to 2026.
Each family of full-field surrogates grew from a foundational idea. Hover the timeline to follow how each one evolved.
Physical view → Nodes are mesh points and edges the mesh connectivity, so resolution follows the physics: fine near walls and shocks, coarse far away.
Message passing on particle neighbourhoods; noise injection keeps long rollouts stable.
paper ↗Encode–process–decode on simulation meshes, with mesh-space and world-space edges.
paper ↗Coarse graph levels carry information across the domain in fewer layers.
paper ↗Bi-stride pooling builds multiscale graphs automatically, without hand-made coarse meshes.
paper ↗Multiscale attentional graph encoder and decoder around a transformer processor.
paper ↗Spectral embeddings of mesh connectivity for tightly packed geometries such as race cars.
paper ↗In one lineMore field-surrogate architectures from NeurIPS, ICML and ICLR, 2023 to 2026, in order; the explained highlights are on Papers.
The trend: encode any geometry into a compact latent space, process it with attention, and decode anywhere, at million-cell scale.
A longer reading list, with one line per paper. The papers that shaped the field, each explained with what it changed and the lesson that prepares you for it, are on Papers; to read any of them critically, see reading a surrogate paper.
Fixed-size latent space; perceiver encoder; latent rollout
paper ↗Physics-attention over learned slices
paper ↗Heterogeneous normalised cross-attention, linear complexity
paper ↗Graph operator to a latent grid, FNO there, graph operator back
paper ↗In one lineSeventy years from response surfaces and kriging to neural fields on any geometry.
The idea is old: replace an expensive experiment or simulation with a cheap fitted model. What changed is what the model can predict, from a single number to the full field on any geometry.
Box & Wilson fit low-order polynomials to experiments: the first surrogate models, still used in design of experiments.
Gaussian-process interpolation from geostatistics: a surrogate that also says how uncertain it is.
Kriging, radial basis functions and POD reduced-order models enter aerospace and automotive design loops.
Convolutional U-Nets predict full flow fields on images and voxel grids; physics-informed networks appear.
Graph networks on simulation meshes, point-cloud networks and neural operators learn directly on engineering geometry.
Latent-token transformers scale to 100M-cell CFD; pretrained physics models fine-tune to new problems with little data.
In one lineWhere to start in code: open neural frameworks, plus classical surrogate and UQ toolkits.
Open implementations of the models, data pipelines and training recipes that most industrial surrogates build on.
A PyTorch framework for training and deploying physics-ML models on simulation data, from data loading to multi-GPU training. Formerly NVIDIA Modulus.
MeshGraphNet, Transolver, FNO, DoMINO, diffusion models, PINNs; reference examples for external aerodynamics
github.com/NVIDIA/physicsnemo ↗The reference library for neural operators: models that map between functions and work across discretisations.
FNO, tensorised FNO, U-shaped NO, GINO; data loaders and losses for PDE benchmarks
github.com/neuraloperator/neuraloperator ↗A library of neural PDE solvers from the authors of Transolver, with unified benchmarks for comparing architectures.
Transolver and many baselines on standard and industrial-geometry benchmarks
github.com/thuml/Neural-Solver-Library ↗Physics-informed neural networks and operator learning, with PyTorch, JAX and TensorFlow backends.
PINNs, DeepONet, inverse problems
github.com/lululxvi/deepxde ↗General-purpose graph neural network building blocks, often used to assemble mesh-based surrogates.
Message-passing layers, graph pooling, mesh and point-cloud utilities
github.com/pyg-team/pytorch_geometric ↗In one linePublic simulation datasets to benchmark on before you have your own.
Public simulation datasets are how new architectures are compared, and a starting point before you have your own. Check the licence: several are non-commercial.
Over 1,000 2D airfoils simulated with steady RANS in OpenFOAM: full volume and surface fields.
Airfoil aerodynamics, the 2D analogue of this page
github.com/Extrality/AirfRANS ↗500 parametric variants of the DrivAer car simulated with high-fidelity hybrid RANS-LES.
Road-car external aerodynamics; permits commercial use
caemldatasets.org/drivaerml ↗8,000 diverse car designs with CFD results, geometry and performance data.
Large-scale car aerodynamics; non-commercial (commercial licences offered separately)
github.com/Mohamedelrefaie/DrivAerNet ↗1,800 wall-modelled LES runs of 180 variants of the NASA CRM high-lift aircraft, at ten angles of attack from 4° to 22°, on grids of 300 to 500 million cells, compared with wind-tunnel data.
High-lift aircraft aerodynamics, attached to post-stall; used by AeroJEPA and CANTO
arxiv.org/abs/2605.19565 ↗500 variants of the Ahmed body, the classic bluff-body benchmark, with hybrid RANS-LES.
Low-speed bluff-body aerodynamics
caemldatasets.org/ahmedml ↗A benchmark suite of PDE datasets and baselines for scientific machine learning.
Advection, Burgers, Navier–Stokes, reaction–diffusion and more
github.com/pdebench/PDEBench ↗A very large collection of physics simulation datasets spanning fluids, plasmas, astrophysics and biology.
Multi-physics data for training general-purpose and foundation models
github.com/PolymathicAI/the_well ↗In one lineFrom open frameworks and datasets to a surrogate trained on your own simulations.
Four steps, from public data and reference code to a model fitted to your own product.
Start from an open dataset in the same regime (external aero, internal flow, thermal), or your own simulation archive.
Frameworks ship tested implementations and training recipes; reproduce a published baseline before changing anything.
Measure error on designs never seen in training, and report the training range alongside every prediction.
Adapt a pretrained model to your product line with a smaller set of your own simulations.
The surrogate on this site is a deliberately small neural field (6,337 weights) so it can be explained end to end. Industrial models are larger (graph, transformer or operator architectures trained on thousands of high-fidelity runs), but the workflow is the same.