Physics Examples

22 worked examples across 8 physics domains. Each example folder is self-contained: one script + one YAML config living side by side. Run directly (python examples/burgers/burgers.py) or via the Command-Line Interface — both save predictions, plots, and a params.msgpack checkpoint automatically.

Core PINN examples

Problem	PDE	Network	Key Features	Config
Exponential Decay	du/dt + λu = 0	MLP [1,32,32,1]	ODESolver, TrainingConfig	examples/ode/config.yaml
Harmonic Oscillator	d²u/dt² + ω²u = 0	MLP [1,32,32,1]	ODESolver, IC derivative	examples/ode/config.yaml
FBPINN ODE	du/dx = cos(ωx)	FBPINN — 15 subnets [1,16,16,1]	Overlapping subdomains, partition-of-unity windows, hard IC constraint	examples/fbpinn_ode/config.yaml
1-D Burgers	u_t + uu_x = νu_xx	MLP [2,64,64,64,1]	FBPINN, RBA, cosine LR	examples/burgers/config.yaml
1-D Heat — Forward	u_t = αu_xx	MLP [2,64,64,64,1]	FBPINNSolver, exact Gaussian IC	examples/heat/heat_forward.yaml
1-D Heat — Inverse	u_t = αu_xx	MLP [2,64,64,64,1]	Recover α from 50 noisy observations	examples/heat/heat_inverse.yaml
1-D Wave	u_tt = c²u_xx	FourierMLP [2,128,128,1]	Dual IC (u, u_t), n_fourier=32	examples/wave/config.yaml
2-D Helmholtz	Δu + k²u = f	FourierMLP [2,128,128,1]	k=4, manufactured source term	examples/helmholtz/config.yaml
2-D Diffusion Inverse	u_t = α∇²u	MLP [3,64,64,64,1]	Log-param joint optimisation	examples/inverse/config.yaml

Fluid dynamics

Problem	PDE	Network	Key Features	Config
2-D Lid-Driven Cavity	Steady N-S, Re=100	FBPINN + SimpleGate	LDCSolver, attention	examples/LDC/config.yaml
2-D RANS k-ε	Turbulent channel, Re=10 000	FBPINN	RANSSolver, RBA	examples/K-Epsilon/config.yaml
NACA Airfoil	Steady N-S, Re=100	MLP / GatedMLP [2,128×6,3]	Cambered profiles, AoA via rotation, surface Cp	examples/airfoil/config.yaml
Cylinder Cross-flow	Steady N-S, Re=40	MLP [2,128×6,3]	Pure-PINN recipe, Cp(θ) vs inviscid, wake pool	examples/cylinder/config.yaml
3-D Pipe Flow	Steady 3-D N-S	MLP / GatedMLP [3,…,4]	Double-jacfwd Hessian, Hagen–Poiseuille exact	examples/pipe_flow/pipe_flow.yaml
3-D AAA Bulge	Steady 3-D N-S	GatedMLP [3,192×5,4]	Cosine² bulge R(x), flow-rate balance	examples/AAA/config.yaml
Carreau Pipe (blood)	Steady Carreau N-S	GatedMLP [3,128×4,4]	Shear-thinning μ(γ̇), 1-D Carreau exact	examples/pipe_flow_rheology/config.yaml
Carreau AAA (blood)	Steady Carreau N-S	GatedMLP [3,192×5,4]	Blood rheology in the bulge, apparent-viscosity maps	examples/AAA_rheology/config.yaml
3-D Pulsatile Pipe	Unsteady 3-D N-S	GatedMLP [4,…,4]	Time-marching transfer, per-window ckpts, window restart	examples/pipe_flow/pipe_flow_pulsatile_transfer.yaml
3-D Unsteady Pipe — Transfer	u_t = G + ν∇²u	MLP [3,64,64,64,64,1]	Bessel exact, Re & temporal transfer	examples/pipe_flow/pipe_flow_unsteady_transfer.yaml

Compressible flow (shock capturing)

Problem	PDE	Network	Key Features	Config
2-D Compressible Ramp	Steady Euler (conservative), M=3	MLP [2,80,80,80,80,80,4]	Oblique shock θ=10°, artificial viscosity (fixed/learnable), RAR	examples/ramp/config.yaml
2-D Compressible NS Ramp (SBLI)	Steady N-S (conservative), M=3	MLP [2,128×4,4]	Viscous shock–boundary-layer interaction, no-slip + isothermal walls, Re=10⁴, Pr=0.72	examples/ramp_ns/config.yaml
1-D Sod Shock Tube	Unsteady Euler (conservative)	MLP [2,80×5,3]	Learnable ε = softplus(log_av), exact Riemann reference, RAR	examples/sod_shock/config.yaml
1-D Toro Test 3 (blast wave)	Unsteady Euler (conservative)	MLP [2,128×4,3]	exp/log positivity, non-dimensionalisation, learnable ε, RAR	examples/toro3/config.yaml

Transfer-learning examples

Example	Network	Key Features	Config
Burgers Transfer	MLP [2,64,64,64,1]	Parameter transfer (ν) + temporal transfer	examples/transfer/burgers_transfer.yaml
Heat 2-D Transfer	MLP [3,64,64,64,1]	Cross-diffusivity transfer + temporal	examples/transfer/heat2d_transfer.yaml

Neural operator examples

Example	Architecture	Highlights	Config
FNO1D Periodic Burgers	FNO1D	Generalizes across ν, pde_weight warmup	examples/operators/fno1d_periodic/config.yaml
FNO1D Dirichlet Burgers	FNO1D	Zero-wall BCs, FNO domain-padding trick	examples/operators/fno1d_dirichlet/config.yaml
FNO2D Burgers	FNO2D	Central/upwind stencil selectable	examples/operators/fno2d_burgers/config.yaml
DeepONet Burgers	DeepONet1D	IC/BC + residual only, no full-field data	examples/operators/deeponet1d_burgers/config.yaml
CViT Burgers	CViT	Scaled-increment prediction, matched-upwind residual	examples/operators/cvit2d_burgers/config.yaml
FNO2D Cylinder Flow	FNO2D	Chorin-projection reference data, obstacle mask	examples/operators/fno2d_cylinder/config.yaml

See also

PDE Library Reference for the underlying residual classes and Geometry Reference for the domain samplers used across these examples.