Proposal: Ensemble/Hybrid Neural Operators in Modulus (e.g., FNO + DiffusionNet)

[YGMaerz]

Hello Modulus community and maintainers,

I propose introducing ensemble and hybrid neural operator examples within NVIDIA Modulus to demonstrate effective software engineering practices for integrating multiple model components. Using hybrid neural operators, such as combining Fourier Neural Operators with DiffusionNet, can showcase the framework’s flexibility and modularity, providing a powerful example for handling complex PDE problems.

1. Motivation — Software Perspective of Ensemble Models

Right now, NVIDIA Modulus showcases many PDE-constrained and physics-informed demos, but there’s no dedicated example illustrating how Modulus can be leveraged to build ensemble (or multi-operator) models. From a software engineering standpoint, offering such an example would demonstrate how to encapsulate multiple model components (or operators) under a single Modulus workflow. It would also show the design patterns for combining different blocks—essential for data scientists and engineers who deal with large-scale PDE problems and want the flexibility of ensemble or hybrid solutions.

A hybrid neural operator (e.g., merging Fourier Neural Operator and DiffusionNet-like layers) is an excellent test case because it highlights:

1. Global feature extraction (via FFT-based layers like FNO),
2. Local/mesh-based feature extraction (via diffusion or graph-based operators),
3. Integrating these components into a unified ensemble under Modulus’ existing PDE-solving framework.

Such a demonstration would have high impact—covering not only advanced modeling but also the software engineering patterns needed to build, train, and validate ensemble solutions.

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2. Why Modulus Should Feature Hybrid/Ensemble Neural Operators

1. Filling a Gap

   • Currently, the Modulus examples illustrate PINNs and PDE constraints but lack a direct example of multi-operator or ensemble approaches.
   • An ensemble tutorial (e.g., combining FNO for global context and DiffusionNet for local geometry) would expand Modulus’ coverage of cutting-edge PDE learning.

2. Supporting Multi-Scale PDE Solutions

   • Many users face PDEs that include global wave modes (e.g., large-scale flow) and local boundary phenomena (e.g., thin shear layers).
   • A hybrid operator inherently handles both scales, improving accuracy and generalization.

3. Showcasing Modulus’ Software Engineering Strength

   • Demonstrating an ensemble design reveals how to build modular pipelines within Modulus, highlighting clean code patterns and component-based architectures.
   • This helps the broader community adopt best practices in model composition (e.g., layering multiple neural operators, fusing outputs, orchestrating training loops).

4. Advancing Real-World Physics-AI

   • Many engineering or scientific workflows require robust surrogates or PDE solvers that can tackle irregular domains, variable boundary conditions, and multi-scale effects.
   • A curated example of an FNO+DiffusionNet ensemble in Modulus would accelerate adoption of these approaches in production-level simulations.

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3. Request for Community Input

We’d love to gather feedback or collaboration from Modulus users and maintainers:

1. Use Cases & Interest

   • Does anyone else have PDE problems or project ideas where local+global ensemble models would be useful?
   • Are there specific PDEs (e.g., Poisson, Helmholtz, Navier–Stokes) that you’d like to see in a tutorial?

2. Implementation Details

   • Should we interpolate unstructured data onto a grid for FFT, or attempt a graph Fourier transform directly on the mesh?
   • Are there recommended libraries or existing Modulus features that could ease building a multi-operator pipeline?

3. Example Scope

   • Start small (2D Poisson on a structured grid) or tackle a more advanced PDE domain?
   • Show how to incorporate PINN-style PDE constraints alongside the ensemble operators?

4. Performance & HPC

   • Any known HPC constraints or best practices for large 3D transforms in Modulus?
   • Strategies for multi-GPU or distributed training if we integrate multiple operator blocks?

Your insights would help shape a robust, community-driven approach to implementing and documenting ensemble/hybrid solutions in Modulus. Please share your ideas, experiences, or references that might guide us.

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4. Problem-Solving Perspective: Local + Global Operator Complexity

Below is a brief summary of key considerations when merging local and global operators—particularly around data formats (structured vs. unstructured) and fusion strategies:

High-Level Problem-Solving Insights

A. Two Axes of Complexity

1. Which operators to combine (Fourier-based, diffusion/graph-based, etc.).
2. Data format (uniform grid vs. unstructured mesh vs. partial/noisy data).

B. Structured vs. Unstructured Data

• Structured → straightforward for FFT layers, simpler local stencils (conv/diff).
• Unstructured → natural for graph/diffusion, but FFT may require domain embedding or graph spectral transforms.

C. Operator Fusion

• Interleaving layers (Diffusion → Fourier → repeat).
• Parallel paths with a fusion stage (sum, concat).
• Front-end vs. back-end ordering (global first, local second, etc.).

D. Handling Partial/Noisy Data

• Local diffusion can over-smooth.
• Global operators can fill in large-scale patterns but miss fine detail.
• Ensemble approach can mitigate both issues.

E. Balancing Complexity & Performance

• Start with simpler PDE/geometry.
• Keep code modular for easy swapping of operators or data representations.
• Watch out for HPC/memory constraints with large transforms or graph eigen-decompositions.

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Best,
Georg Maerz
YGMaerz