Neural Operators (PINO / DeepONet / CViT)

Neural Operators (PINO / DeepONet / CViT)#

Unlike the point networks used elsewhere in underPINN (one collocation point in, one value out), neural operators map an entire function — sampled on a grid or at sensors — to another function. A single trained model generalizes across a distribution of initial conditions or PDE parameters, instead of re-solving from scratch for every new case.

Architectures#

Architecture	Description
FNO1D / FNO2D	Fourier Neural Operator (Li et al., 2020): truncated-spectrum convolution (`SpectralConv1d` / `SpectralConv2d`) plus a pointwise skip path per stage. `padding` zero-pads non-periodic (Dirichlet) domains before the FFT and trims after.
DeepONet1D	Branch/trunk operator (Lu et al., 2021): `s(u)(y) = branch(u) · trunk(y)`. Reuses the standard `MLP` / `GatedMLP` for both sub-networks.
CViT	Continuous Vision Transformer for Operator Learning (Wang et al., ICLR 2025): patch embed → learned sin/cos positions → latent time-aggregation → self-attention encoder → cross-attention decoder that queries continuous coordinates. `cvit_grid_predict` queries it on a regular grid (with a scaled-increment prediction trick) so it can pair with the same finite-difference residual used by FNO2D.

Select an operator architecture the same way as any other network — via YAML:

network:
  type: fno1d   # fno1d | fno2d | deeponet | cvit

All four are registered once in underPINN/nn/factory.py.

Training loops#

Solver	Loss	Used for
`OperatorSolver`	`OperatorLoss` — data MSE + weighted PDE grid-residual, optional warmup + RBA	FNO1D, FNO2D, CViT
`DeepONetSolver`	`DeepONetLoss` — IC/BC + autodiff residual, no full-field ground truth needed	DeepONet1D

OperatorSolver shares the same n_scan_steps GPU acceleration as FBPINNSolver (see Training System), plus a PDE-weight warmup ramp.

Operator-specific PDE residuals#

Residual class	Equation	Discretisation
`BurgersGrid1D`	`u_t + uu_x = νu_xx`	Finite-difference, periodic or Dirichlet
`BurgersGrid2D`	`u_t + u(u_x+u_y) = ν(u_xx+u_yy)`	Finite-difference, central or upwind stencil (selectable)
`DeepONetBurgersPDE`	`u_t + uu_x = νu_xx`	Autodiff (no grid required)
`CylinderNSGrid`	`∇·u = 0`, `u·∇u = -∇p + ν∇²u`	Finite-difference, with an obstacle mask

Worked 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`

Tip

The FNO2D cylinder-flow example ships its own datagen.py, which generates Chorin-projection reference solutions used as supervised targets for the operator’s data-fit loss term.