11 — Operator Learning
Operator learning trains a network to map inputs (parameters, forcing functions, initial conditions) to outputs (PDE solutions) for an entire family of problems, instead of solving a single instance. jNO supports two complementary patterns:
- PDE-residual operator learning — the network sees one parametric instance per batch sample; the PDE residual is enforced at the collocation points. The network never sees ground-truth solutions, only the physics. Closest to "PINN with a parameter".
- Data-driven operator learning — the network is supervised on a dataset of
(input, solution)pairs. No PDE residual is computed during training; the solution operator is learnt purely from examples.
The three architecture tutorials showcase three foundax architectures on the same Poisson problem, so the only variable is the architecture itself. Two further tutorials cover FieldView — the physics API for operators that emit a full grid field in one shot, where PDE derivatives come from finite differences instead of autodiff:
| Tutorial | Architecture | Pattern | What it teaches |
|---|---|---|---|
| DeepONet — parametric Poisson | foundax.deeponet |
PDE-residual | Branch/trunk decomposition; the canonical operator-learning architecture |
| FNO2D — supervised Poisson | foundax.fno2d |
Data-driven | Spectral convolutions in Fourier space; resolution-independent in principle |
| U-Net 2D — supervised Poisson | foundax.unet2d |
Data-driven | Convolutional encoder-decoder with skip connections; multiscale by construction |
| FieldView — physics-informed FNO | foundax.fno2d |
Data + FD-physics | FD PDE residual on a live FNO's grid output, added to the loss as a gradient-carrying physics term |
| FieldView — wave-equation PINN audit | foundax.deeponet |
PINN + FD audit | Train a wave-equation PINN, then verify the trained network's grid prediction with second-order u.tt finite differences |
All three plug into the same jno.nn.wrap(...) interface, so the rest of your training pipeline (callbacks, schedules, checkpointing, evaluation) is identical. Foundation models (PROSE, Poseidon, PDEformer-2) plug in the same way but are out of scope here — see the Foundation Models page for the fine-tuning workflow.