Models

PDEForge ships 41 models. Each one has its own page below, with the equation it solves, the operator-learning task it defines, its parameters and their ranges, a runnable snippet, and a figure generated by the package itself.

Most models run on the base installation. Eight need FEniCSx, which is marked in the tables and covered in FEniCSx setup; airfoil_euler_2d is finite-volume and pure NumPy.

Published benchmark setups ship as presets rather than as separate models, so the coefficients, the domain and the input measure travel together. See the preset list.

Diffusion and transport

Linear and near-linear baselines. Everything here is cheap, and the answers are predictable enough to make these the models you debug a pipeline against.

Model	Task	Backend
`heat_1d`	\(u(x,0) \mapsto u(x,T)\)	spectral
`heat_2d`	\(u(x,y,0) \mapsto u(x,y,T)\)	spectral
`heat_3d`	\(u(\mathbf{x},0) \mapsto u(\mathbf{x},T)\) on the cube	spectral
`advection_1d`	exact translation; the sanity anchor	spectral
`darcy_2d`	\(\kappa \mapsto u\), periodic	spectral

[`heat_1d`](heat_1d.md): four inputs and their images at T

[`advection_1d`](advection_1d.md): translation, shape intact

[`darcy_2d`](darcy_2d.md): pressure over a heterogeneous medium

Waves and dispersion

Non-dissipative dynamics, where nothing is smoothed away and a low-pass bias in an operator shows up immediately.

Model	Task	Backend
`wave_1d`	\(u(x,0) \mapsto u(x,T)\)	spectral
`wave_2d`	\(u(x,y,0) \mapsto u(x,y,T)\)	spectral
`heterogeneous_wave_2d`	medium \(c \mapsto\) wavefield	spectral
`helmholtz_2d`	source \(f \mapsto\) field, frequency domain	spectral
`kdv_1d`	solitons and undular bores; four presets	spectral
`schrodinger_1d`	complex field, norm conserved exactly	spectral

[`wave_1d`](wave_1d.md): one pulse splits into two characteristics

[`kdv_1d`](kdv_1d.md): solitons overtake and survive

[`heterogeneous_wave_2d`](heterogeneous_wave_2d.md): refraction through a random medium

[`helmholtz_2d`](helmholtz_2d.md): a smooth, sub-resonant source and its field

[`schrodinger_1d`](schrodinger_1d.md): a focusing condensate

[`wave_2d`](wave_2d.md): amplitude conserved, pattern rearranged

Nonlinear advection and turbulence

Where the difficulty concentrates: shocks, filaments and chaos.

Model	Task	Backend
`burgers_1d`	shock formation; six presets	spectral
`burgers_2d`	vector self-advection, no pressure	spectral
`ks_1d`	spatiotemporal chaos	spectral
`ns_vorticity_2d`	the canonical FNO benchmark	spectral / JAX
`kolmogorov_flow_2d`	forced, statistically steady turbulence	spectral / JAX
`shallow_water_2d`	three coupled fields, mass conserved exactly	spectral

[`burgers_1d`](burgers_1d.md): steepening until viscosity holds the front

[`ks_1d`](ks_1d.md): chaos in space and time

[`kolmogorov_flow_2d`](kolmogorov_flow_2d.md): forced 2D turbulence

[`burgers_2d`](burgers_2d.md): fronts in two dimensions

[`ns_vorticity_2d`](ns_vorticity_2d.md): patches merge and shear into sheets

[`shallow_water_2d`](shallow_water_2d.md): gravity waves spreading

Pattern formation and phase separation

Models whose output structure is generated by the dynamics rather than supplied in the input.

Model	Task	Backend
`allen_cahn_1d`	interfaces annihilating in pairs	spectral
`allen_cahn_2d`	curvature-driven coarsening	spectral
`allen_cahn_3d`	the same on the cube	spectral
`cahn_hilliard`	spinodal decomposition, 2D or 3D, mean conserved	spectral
`eggshell_droplets_3d`	coalescence vs Ostwald ripening, two droplets in a shell	spectral
`gray_scott_2d`	the Pearson parameter plane	spectral / JAX
`fitzhugh_nagumo_1d`	excitable pulses past a threshold	spectral
`fitzhugh_nagumo_2d`	broken fronts, spirals, and when they die	spectral
`lotka_volterra_2d`	predator-prey with diffusion	spectral

[`allen_cahn_1d`](allen_cahn_1d.md): interfaces drift together and annihilate

[`cahn_hilliard`](cahn_hilliard.md): the spinodal interface in 3D

[`gray_scott_2d`](gray_scott_2d.md): every (feed, kill) pair grows a different pattern

[`fitzhugh_nagumo_1d`](fitzhugh_nagumo_1d.md): one stimulus, two pulses

[`allen_cahn_2d`](allen_cahn_2d.md): domains separated by thin interfaces

[`lotka_volterra_2d`](lotka_volterra_2d.md): predator trails prey

Porous media and solids

Coefficient-to-solution maps, which is the shape most inverse problems take.

Model	Task	Backend
`darcy_fno_2d`	the canonical benchmark, bit-exact	direct sparse
`darcy_fno_3d`	the same measure on the cube; no frozen dataset exists	direct / CG
`elasticity_2d`	modulus \(\mapsto\) displacement and stress	FEniCSx
`porous_darcy_fem`	flow through a grown microstructure	FEniCSx

[`darcy_fno_2d`](darcy_fno_2d.md): coefficient and the pressure it induces

[`darcy_fno_3d`](darcy_fno_3d.md): pressure isosurfaces on the cube

[`elasticity_2d`](elasticity_2d.md): stress concentrating at stiff inclusions

Viscous and compressible flow

Walls, obstacles and geometry. Everything here except stokes_2d and airfoil_euler_2d needs FEniCSx.

Model	Task	Backend
`stokes_2d`	force \(\mapsto\) velocity and pressure, creeping flow	spectral
`cylinder_flow_2d`	steady wake behind a cylinder	FEniCSx
`cylinder_flow_2d_unsteady`	the von Kármán vortex street, as a trajectory	FEniCSx
`cylinder_flow_2d_parameterized`	cylinder position as an input	FEniCSx
`cylinder_flow_2d_turbulent`	Re 2000 under a Smagorinsky closure	FEniCSx
`naca_flow_2d`	airfoil geometry \(\mapsto\) laminar flow, with \(C_l\) and \(C_d\)	FEniCSx
`rayleigh_benard_2d`	convection in a closed cavity, Nusselt-validated	FEniCSx
`airfoil_euler_2d`	transonic Euler with a shock, on a C-grid	finite volume

[`naca_flow_2d`](naca_flow_2d.md): laminar flow past a drawn airfoil

[`cylinder_flow_2d_unsteady`](cylinder_flow_2d_unsteady.md): the vortex street

[`stokes_2d`](stokes_2d.md): forcing and the flow it drives

[`rayleigh_benard_2d`](rayleigh_benard_2d.md): convection rolls in the cavity

[`porous_darcy_fem`](porous_darcy_fem.md): flow through a grown labyrinth

Stochastic PDEs

Each sample carries several realisations of the same solve, so the target is a distribution. This is the part of the catalogue the calibration protocol exists for.

Model	Task	Backend
`stochastic_heat_1d`	Gaussian, analytically known covariance	spectral
`stochastic_heat_2d`	the same on the square	spectral
`stochastic_burgers_1d`	spread concentrated at the fronts	spectral
`stochastic_allen_cahn_2d`	multimodal: realisations branch between wells	spectral

[`stochastic_heat_1d`](stochastic_heat_1d.md): one input, an ensemble out

[`stochastic_heat_2d`](stochastic_heat_2d.md): ensemble mean and spread

[`stochastic_allen_cahn_2d`](stochastic_allen_cahn_2d.md): two realisations, one input

Finding a model from code

from pdeforge import list_models, describe_model

list_models()                    # every registered name
print(describe_model("kdv_1d"))  # parameters, ranges, field names, backend

Every figure on this page is package output, seeded and regenerable: scripts/make_model_figures.py builds the per-model set and scripts/make_gallery.py builds the gallery.