Available Models

PDEForge ships 41 models under one API. Every one of them is called the same way, and each has its own page under Models covering the equation, the operator task, the parameters and their ranges, a runnable snippet, and a figure.

from pdeforge import generate_dataset

dataset = generate_dataset(
    model="burgers_1d",
    n_samples=1000,
    resolution={"x": 256},
    params={"viscosity": 0.01, "time_horizon": 1.0},
    seed=42,
)

This page is the catalogue. Start from the Models index if you would rather browse by physics with the figures alongside.

What you need installed

Group	Count	Requirement
Spectral	30	base installation
Finite-difference elliptic	2	base installation
Finite volume	1	base installation (pure NumPy)
Finite element	8	FEniCSx setup or the Docker image

Two of the spectral models, ns_vorticity_2d and kolmogorov_flow_2d, run considerably faster with backend="jax", and gray_scott_2d benefits at long horizons.

The catalogue

Diffusion and transport

Model	Operator task
`heat_1d`	\(u(x,0) \mapsto u(x,T)\)
`heat_2d`	\(u(x,y,0) \mapsto u(x,y,T)\)
`heat_3d`	the same on the periodic cube
`advection_1d`	exact translation; the sanity anchor
`darcy_2d`	\(\kappa(x,y) \mapsto u(x,y)\), periodic

Waves and dispersion

Model	Operator task
`wave_1d`	\(u(x,0) \mapsto u(x,T)\), energy conserved
`wave_2d`	the same on the square
`heterogeneous_wave_2d`	\(c(x,y) \mapsto\) wavefield; the medium is the input
`helmholtz_2d`	\(f \mapsto \operatorname{Re} u\), frequency domain
`kdv_1d`	solitons, undular bores, benchmark regimes
`schrodinger_1d`	complex field as two real channels

Nonlinear advection and turbulence

Model	Operator task
`burgers_1d`	shock formation; the regularity ladder
`burgers_2d`	vector self-advection, no pressure
`ks_1d`	spatiotemporal chaos
`ns_vorticity_2d`	\(w(\cdot,0) \mapsto w(\cdot,T)\); the canonical benchmark
`kolmogorov_flow_2d`	forced, statistically steady turbulence
`shallow_water_2d`	\((h, hu, hv)\); mass conserved exactly

Pattern formation and phase separation

Model	Operator task
`allen_cahn_1d`	interfaces annihilating in pairs
`allen_cahn_2d`	curvature-driven coarsening
`allen_cahn_3d`	the same on the cube
`cahn_hilliard`	spinodal decomposition, 2D or 3D
`eggshell_droplets_3d`	which coarsening mechanism wins, in a catalyst shell
`gray_scott_2d`	the Pearson parameter plane
`fitzhugh_nagumo_1d`	excitable pulses past a threshold
`fitzhugh_nagumo_2d`	broken fronts, spirals
`lotka_volterra_2d`	predator-prey with diffusion

Porous media and solids

Model	Operator task	Needs
`darcy_fno_2d`	\(a \mapsto u\); bit-exact against the published data	base
`darcy_fno_3d`	the same measure on the cube	base
`elasticity_2d`	\(E \mapsto (u, v, \sigma_{vM})\)	FEniCSx
`porous_darcy_fem`	\(k \mapsto (p, u_x, u_y)\) through a grown microstructure	FEniCSx

Viscous and compressible flow

Model	Operator task	Needs
`stokes_2d`	\((f_x, f_y) \mapsto (u, v, p)\), creeping flow	base
`cylinder_flow_2d`	inlet scale \(\mapsto (u, v, p)\)	FEniCSx
`cylinder_flow_2d_unsteady`	the vortex street, as a trajectory	FEniCSx
`cylinder_flow_2d_parameterized`	cylinder position as input	FEniCSx
`cylinder_flow_2d_turbulent`	Re 2000, Smagorinsky LES	FEniCSx
`naca_flow_2d`	airfoil geometry \(\mapsto\) flow, with \(C_l\) and \(C_d\)	FEniCSx
`rayleigh_benard_2d`	convection in a cavity, Nusselt-validated	FEniCSx
`airfoil_euler_2d`	transonic Euler with a shock, on a C-grid	base

Stochastic PDEs

Each sample carries several realisations of the same solve, so outputs have an extra realisation axis and the target is a distribution. See the calibration protocol and the stochastic systems guide.

Model	Operator task
`stochastic_heat_1d`	\(u_0 \mapsto \{u_T^{(i)}\}\), Gaussian
`stochastic_heat_2d`	the same on the square
`stochastic_burgers_1d`	spread concentrated at the fronts
`stochastic_allen_cahn_2d`	multimodal: realisations branch

Presets

Published benchmark setups ship as presets rather than as separate models, because they differ from the base model only in coefficients, domain and input measure. A preset pins all three together, so the measure travels with the physics.

dataset = generate_dataset(preset="fno_darcy_2d", n_samples=1000, seed=0)

Preset	Model	What it pins
`fno_darcy_2d`	`darcy_fno_2d`	Canonical Darcy421, log-normal; bit-exact against the distributed data
`fno_darcy_clean_2d`	`darcy_fno_2d`	The Darcy421 measure on the node grid, with no resampling
`fno_darcy_piececonst_2d`	`darcy_fno_2d`	The two-phase pushforward, \(\{12, 3\}\)
`fno_burgers_1d`	`burgers_1d`	Sine prior at \(\nu = 0.01/\pi\)
`fno_burgers_grf_1d`	`burgers_1d`	The official GRF measure \(N(0, 625(-\Delta + 25)^{-2})\)
`fno_ns_vorticity_2d`	`ns_vorticity_2d`	Forced NS, \(\nu = 10^{-3}\), \(T = 50\)
`burgers_smooth_1d`	`burgers_1d`	Regularity ladder, smooth end
`burgers_canonical_1d`	`burgers_1d`	Regularity ladder, paper-baseline fronts
`burgers_rough_1d`	`burgers_1d`	Regularity ladder, front-dominated
`pdebench_burgers_1d`	`burgers_1d`	PDEBench-style low viscosity, shock-rich
`kdv_dsw_1d`	`kdv_1d`	Undular bore, vigorous and un-resolvable at \(n_x = 512\)
`kdv_dsw_epistemic_1d`	`kdv_1d`	Undular bore, near-resolvable
`mp_pde_kdv_1d`	`kdv_1d`	The Brandstetter et al. MP-PDE regime
`mp_pde_kdv_easy_1d`	`kdv_1d`	The same at \(T = 50\)

from pdeforge import list_presets
from pdeforge.presets import get_preset

list_presets()
get_preset("kdv_dsw_1d")     # the full pinned configuration

Model information

describe_model reports what a model accepts without you reading its source:

from pdeforge import describe_model
print(describe_model("burgers_1d"))

It shows the physical parameters you can modify, their defaults and valid ranges, the input and output field names, and the backend.