PDE Library Reference

PDE Library Reference#

Every PDE in underPINN implements a single residual() method conforming to BasePDE (see API Reference). This page lists every registered residual class, its governing equation, and the examples that use it.

PDE	Equation	Key method	Used in
Burgers (1-D)	`u_t + uu_x = νu_xx`	`BurgersPDE.residual`	`examples/burgers/`, `examples/transfer/`
Diffusion / Heat (1-D)	`u_t = αu_xx`	`DiffusionPDE.residual`	`examples/heat/`
Heat (2-D unsteady)	`u_t = α(u_xx + u_yy)`	`Heat2DPDE.residual`	`examples/inverse/`, `examples/transfer/`
Wave (1-D)	`u_tt = c²u_xx`	`WavePDE.residual`	`examples/wave/`
Helmholtz (2-D)	`Δu + k²u = f`	`HelmholtzPDE.residual`	`examples/helmholtz/`
Navier-Stokes (2-D steady)	`∇·u=0`, `u·∇u = -∇p + ν∇²u`	`NavierStokesPDE.residual`	`examples/LDC/`, `examples/airfoil/`, `examples/cylinder/`
Navier-Stokes (3-D steady)	Same + z-momentum	`SteadyNS3DPDE.residual`	`examples/pipe_flow/`, `examples/AAA/`
Navier-Stokes (3-D unsteady)	`u_t + (u·∇)u = −∇p + ν∇²u`	`UnsteadyNS3DPDE.residual`	`examples/pipe_flow/` (pulsatile)
Carreau N-S (3-D steady)	`∇·[μ*(γ̇)(∇u+∇uᵀ)]` stress	`CarreauNS3DPDE.residual`	`examples/pipe_flow_rheology/`, `examples/AAA_rheology/`
Pipe unsteady	`u_t = G + ν(u_yy + u_zz)`	`PipeUnsteadyPDE.residual`	`examples/pipe_flow/`
RANS k-ε	N-S + `k` + `ε` transport	`KEpsilonPDE.residual`	`examples/K-Epsilon/`
Compressible Euler (2-D steady)	`∂F/∂x + ∂G/∂y = ε∇²U` (conservative)	`CompressibleEulerPDE.residual`	`examples/ramp/`
Compressible Navier–Stokes (2-D steady)	`∂x(F−Fv/Re) + ∂y(G−Gv/Re) = 0`	`CompressibleNS2DPDE.residual`	`examples/ramp_ns/`
Compressible Euler (1-D unsteady)	`∂U/∂t + ∂F/∂x = ε∂²U/∂x²`	`Euler1DUnsteadyPDE.residual`	`examples/sod_shock/`
Exponential Decay	`du/dt + λu = 0`	`ExpDecayODE.residual`	`examples/ode/`
Harmonic Oscillator	`d²u/dt² + ω²u = 0`	`HarmonicODE.residual`	`examples/ode/`
Burgers grid (1-D, FNO)	`u_t + uu_x = νu_xx` (FD residual)	`BurgersGrid1D.residual`	`examples/operators/fno1d_periodic/`, `fno1d_dirichlet/`
Burgers grid (2-D, FNO/CViT)	`u_t + u(u_x+u_y) = ν(u_xx+u_yy)` (FD residual)	`BurgersGrid2D.residual`	`examples/operators/fno2d_burgers/`, `cvit2d_burgers/`
Burgers (DeepONet)	`u_t + uu_x = νu_xx` (autodiff residual)	`DeepONetBurgersPDE.residual`	`examples/operators/deeponet1d_burgers/`
Navier-Stokes grid (2-D, FNO)	`∇·u=0`, `u·∇u = -∇p + ν∇²u` (FD residual)	`CylinderNSGrid.residual`	`examples/operators/fno2d_cylinder/`

Note

Module paths live under underPINN/pde/ — e.g. underPINN.pde.burgers.BurgersPDE, underPINN.pde.navier_stokes_3d.SteadyNS3DPDE. See API Reference for the package layout.