# GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applications

arXiv: https://arxiv.org/abs/2406.03569 · HTML: https://mlsanity.com/paper/2406.03569/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 109 references that are in the map.
- First posted on arXiv 5 Jun 2024.

## Builds on (most cited first, 30 of 42)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Graph Neural Network for Traffic Forecasting: A Survey](https://mlsanity.com/paper/2101.11174/) (2021)
- [A Survey on Oversmoothing in Graph Neural Networks](https://mlsanity.com/paper/2303.10993/) (2023)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [A Review on Graph Neural Network Methods in Financial Applications](https://mlsanity.com/paper/2111.15367/) (2021)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [Tackling Over-Smoothing for General Graph Convolutional Networks](https://mlsanity.com/paper/2008.09864/) (2020)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [Graph Pooling for Graph Neural Networks: Progress, Challenges, and Opportunities](https://mlsanity.com/paper/2204.07321/) (2022)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [Inferring solutions of differential equations using noisy multi-fidelity data](https://mlsanity.com/paper/1607.04805/) (2016)
- [Deep learning of thermodynamics-aware reduced-order models from data](https://mlsanity.com/paper/2007.03758/) (2020)
- [Thermodynamics-informed graph neural networks](https://mlsanity.com/paper/2203.01874/) (2022)
- [Applications of physics informed neural operators](https://mlsanity.com/paper/2203.12634/) (2022)
- [Multi-fidelity regression using artificial neural networks: efficient approximation of parameter-dependent output quantities](https://mlsanity.com/paper/2102.13403/) (2021)
- [A Comparison of Neural Network Architectures for Data-Driven Reduced-Order Modeling](https://mlsanity.com/paper/2110.03442/) (2021)
- [Reduced operator inference for nonlinear partial differential equations](https://mlsanity.com/paper/2102.00083/) (2021)
- [TF-GNN: Graph Neural Networks in TensorFlow](https://mlsanity.com/paper/2207.03522/) (2022)

## Built on by (most cited first, 10 of 10)

- [Neural Chaos: A Spectral Stochastic Neural Operator](https://mlsanity.com/paper/2502.11835/) (2025)
- [Machine learning for modelling unstructured grid data in computational physics: a review](https://mlsanity.com/paper/2502.09346/) (2025)
- [Refine Drugs, Don't Complete Them: Uniform-Source Discrete Flows for Fragment-Based Drug Discovery](https://mlsanity.com/paper/2509.26405/) (2025)
- [Time Extrapolation with Graph Convolutional Autoencoder and Tensor Train Decomposition](https://mlsanity.com/paper/2511.23037/) (2025)
- [Surrogates for Physics-based and Data-driven Modelling of Parametric Systems: Review and New Perspectives](https://mlsanity.com/paper/2603.12870/) (2026)
- [A Resolution Independent Neural Operator](https://mlsanity.com/paper/2407.13010/) (2024)
- [Planning-Query-Guided Model Generation for Model-Based Deformable Object Manipulation](https://mlsanity.com/paper/2508.19199/) (2025)
- [Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEs](https://mlsanity.com/paper/2601.11259/) (2026)
- [Graph-Instructed Neural Networks for parametric problems with varying boundary conditions](https://mlsanity.com/paper/2603.08304/) (2026)
- [PGD-NO: A Neural Operator with Precomputed Geometry Decomposition for 3D Million-scale Physics Simulations](https://mlsanity.com/paper/2607.08025/) (2026)

## Abstract (arXiv)

This work presents a novel resolution-invariant model order reduction strategy for multifidelity applications. We base our architecture on a novel neural network layer developed in this work, the graph feedforward network, which extends the concept of feedforward networks to graph-structured data by creating a direct link between the weights of a neural network and the nodes of a mesh, enhancing the interpretability of the network. We exploit the method's capability of training and testing on different mesh sizes in an autoencoder-based reduction strategy for parametrised partial differential equations. We show that this extension comes with provable guarantees on the performance via error bounds. The capabilities of the proposed methodology are tested on three challenging benchmarks, including advection-dominated phenomena and problems with a high-dimensional parameter space. The method results in a more lightweight and highly flexible strategy when compared to state-of-the-art models, while showing excellent generalisation performance in both single fidelity and multifidelity scenarios.