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 two architecture tutorials showcase two foundax architectures on the same Poisson problem, so the only variable is the architecture itself:

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

Both plug into the same jno.nn(...) 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 Models page for the fine-tuning workflow.