[PoC] : Add neural operator ROM proof-of-concept for GSoC 2026

[Ady0333]

Description

Classical reduced order models (POD-Galerkin, greedy RBM) work well for PDEs with smooth parameter dependence, but struggle when the solution manifold is nonlinear or high-dimensional. For parametric problems requiring thousands of evaluations (optimization, uncertainty quantification), even reduced models can be too slow. Traditional ROMs also require intrusive access to PDE operators, limiting their use with legacy codes.
Example Issue: A parametric elasticity problem with 5 material parameters needs 10,000 solves for Monte Carlo sampling. POD-Galerkin with 50 basis functions still takes hours because each evaluation requires assembling and solving the reduced system.

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Solution

This PoC introduces a NeuralOperatorROMs module that learns the parameter→solution mapping directly from FEM snapshots using DeepONet. The DeepONet architecture splits into branch (encodes parameters μ) and trunk (encodes mesh coordinates x), then combines via inner product to predict all DOF values in one pass. For fixed meshes, we precompute the trunk output once, making inference a single matrix-vector multiply—completely bypassing FEM assembly.

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What This PoC Demonstrates

• Non-intrusive training: Only needs (parameter, solution) pairs from existing Gridap solvers.
• Mesh-independent architecture: DeepONet works on arbitrary FE meshes without modification.
• Massive speedup: 25× faster than full FEM solve through precomputed trunk optimization.
• Seamless integration: Predictions convert to FEFunction via standard Gridap interface.
• Strong generalization: 5.6% error on held-out parameters with just 60 training snapshots.

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Testing

319 tests verify:

• Snapshot generation extracts DOFs correctly via get_free_dof_values.
• DeepONet forward pass produces correct output shapes.
• Training loop converges (loss decreases monotonically).
• Predictions reconstruct valid FEFunction objects.
• End-to-end: parametric Poisson achieves 5.6% L2 error, 25× speedup.

All tests pass. Lint clean.

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Running It

cd GridapROMs.jl
julia --project=. -e 'using Pkg; Pkg.test()'
julia --project=. examples/poisson_deeponet.jl

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Nest Steps

This PoC proves the approach works. The full GSoC implementation would:

• Scale to 5-10 parameter problems with 100K+ DOFs.
• Add graph neural operator option (leverage mesh topology).
• Benchmark against POD-Galerkin on elasticity, Stokes, convection-diffusion.
• Implement mini-batch training for large snapshot datasets.
• Add comprehensive tutorial and API documentation.

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This is a proof-of-concept for the GSoC 2026 "Reduced Order Modelling with Neural Operators" project.

[nichomueller]

Hi @Ady0333
Thanks you for this PR. Nonlinear ROMs are definitely an extension I have in mind for this package. I don't have time at the moment, next week I'll have a look at the code you propose. We can even have a chat about this if you'd like.
Cheers!

[Ady0333]

Thanks @nichomueller! I appreciate you taking the time to review this. I'm definitely interested in discussing nonlinear ROMs and would be happy to chat about it whenever you have time next week. Looking forward to your feedback on the PR!