Implement Sequential DeepONet (S-DeepONet)

Summary

Implement Sequential DeepONet, which uses recurrent networks (GRU) in the branch to capture temporal dependencies in time-dependent input functions.

Reference

• "Predictions of transient vector solution fields with sequential deep operator network," Acta Mechanica, 2024. DOI: 10.1007/s00707-024-03991-2

Description

Sequential DeepONet replaces the standard feedforward branch network with a Gated Recurrent Unit (GRU) to process time-series input functions. This enables the model to learn operators where the input function has a temporal structure (e.g., time-varying boundary conditions or forcing terms).

Since the existing DeepONet already accepts arbitrary Lux layers for the branch, this may primarily require documentation/examples showing how to use Lux.Recurrence(Lux.GRUCell(...)) as the branch network, plus any necessary adjustments to handle sequence inputs.

Start by reading the existing DeepONet branch-layer interface and how it handles sequence inputs, then consult the referenced paper and the Lux.Recurrence(Lux.GRUCell(...)) usage. Document or implement the GRU branch path, including any required input-shape adjustments, and demonstrate that a time-dependent example runs as intended.