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

TL;DR

This work tackles the challenge of real-time, multifidelity PDE approximation across varying discretisations by introducing Graph Feedforward Networks (GFNs), which attach neural weights to mesh nodes and enable resolution-invariant learning. It then builds GFN-ROM, an autoencoder-based reduced order model whose encoder/decoder employ GFNs and whose latent mapper handles parameter-to-latent mappings, yielding a nonlinear, non-intrusive, multifidelity surrogate. The authors establish theoretical bounds for super-/sub-resolution errors and demonstrate strong empirical performance on Graetz, advection-dominated, and Stokes benchmarks, including adaptive training and significant efficiency gains over baselines. The framework yields a lightweight, interpretable approach for unstructured meshes with broad potential for time-dependent, multimodal, and experimental-data applications, while maintaining strong generalisation across fidelities and discretisations.

Abstract

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.

Figures (17)

• Figure 1: GFN-ROM is a nonlinear non-intrusive multifidelity ROM capable of dealing with unstructured data, interplaying between MOR and resolution-invariant techniques.
• Figure 2: GFN-ROM as a resolution-invariant ROM, capable of handling data from any discretisation, both in training and in testing modes.
• Figure 3: Illustration of the arrow notations used in this work.
• Figure 4: Single-layer graph feedforward autoencoder architecture. As shown via shading, the columns of the encoder weight matrix and the rows of the decoder weight matrix are associated to the individual nodes in the mesh $\mathcal{M}_o$.
• Figure 5: GFN approach to transform weights between meshes. Light blue shades denote the original mesh with its associated ${\hbox{\boldmath{$W$}}}^e$ and ${\hbox{\boldmath{$W$}}}^d$. The arrows depict how the new weight matrices (in darker blue shades) $\tilde{{\hbox{\boldmath{$W$}}}}^e$ and $\tilde{{\hbox{\boldmath{$W$}}}}^d$ can be created from the original weights and used for prediction on the new mesh.