Implement MIONet (Multiple-Input Operator Network)

Summary

Implement MIONet, the first neural operator architecture for learning multiple-input operators on products of Banach spaces.

Reference

• Jin et al., "MIONet: Learning multiple-input operators via tensor product," SIAM Journal on Scientific Computing, 2022. arXiv:2202.06137

Description

MIONet extends DeepONet to handle operators with multiple input functions (e.g., both initial condition and forcing term). It uses separate branch networks for each input function and a trunk network for query locations, merging outputs via Hadamard product (element-wise multiplication) rather than simple dot product. The paper includes a universal approximation theorem for this architecture.

This is architecturally close to DeepONet but requires:

• Support for an arbitrary number of branch networks
• Hadamard product merging of branch outputs with trunk output
• The existing DeepONet in src/models/deeponet.jl could be extended or a new MIONet struct could be added

Start by reading the existing DeepONet implementation in src/models/deeponet.jl and the referenced MIONet paper at arXiv:2202.06137. Determine whether to extend DeepONet or add an MIONet struct, then define how arbitrary branch networks combine with the trunk through Hadamard products. Done means MIONet supports multiple input functions and query locations as described in the issue.