Reduced Order Modelling with Neural Operators in Gridap.jl

THE PROBLEM: Solving partial differential equations (PDEs) for parametrical studies is a fundamental part of scientific research, but traditional many-query Finite Element simulations are highly computationally expensive and suffer from performance bottlenecks. THE SOLUTION: This project aims to drastically reduce simulation times by integrating modern machine learning into the Gridap.jl ecosystem. The plan is to augment GridapROMs.jl with nonlinear neural operators. The work plan is split into three phases: 1. Benchmarking: Generating high-fidelity training data from 1D test cases to evaluate and compare external neural operators (via NeuralOperators.jl) against standard linear Reduced Order Models (ROMs). 2. Implementation: Integrating the best-performing neural architectures to solve more complex, real-world physics problems, such as heat equations with nonlinear coefficients and turbulent fluid dynamics. 3. API Extension: Extending the GridapROMs.jl API to unify traditional linear models and the new nonlinear neural operators under a single interface. DELIVERABLES: - A public repository containing data generation scripts, training loops, and a comparative performance report of the tested models. - Validation scripts showing the core implementation works on complex physics problems. - A final Pull Request to GridapROMs.jl containing the extended API, unit tests, documentation, and user tutorials demonstrating the end-to-end workflow.